O. If if = 0, then we get the result by replacing if by E> 0 for every E> 0; thus h and hence f is zero. •
4.1.. Lemma. Let X be as in Lemma 4.1.6. Let F, (x o ) denote the solution (-integral curve) of X'(I)=X(x(t»,x(O)=x o' Then there is a neighborhood V of Xo and a number Il > 0 such Ihal for every y E V there is a unique integral curve x(t) = F,(y) salisfying x'(t) = X(x(t) for IE [- E, II and
190
VECTOR FIELDS AND DYNAMICAL SYSTEMS
x(O) -= y. Moreover. 1IF,(x)- F,( Y)II" eK1'Ilix -
ylI·
Proof. The first part is clear from Lemma 4.1.6. For the second. let /(1) -1IF,(x)- F,(Y~I. aearly
/(1) =- Ilfo'X(~(X»)- X(~(y» dr + x -
so the result foUows from Lemma 4.1.7.
yll" IIx - yll+ Kfo'/{s}
dr.
•
This result shows that F,(x) depends in a continuous. indeed Lipschitz. manner on the initial condition x and is jointly continuous in (t. x). Again. the same result holds if X depends explicitly on 1 and on a parameter p; the evolution operator F!.,o(x) is the unique integral curve x( t) satisfying x'(t) - X(X(I). I, p) and X(lo) = x. Then F,~,,,(x) is jointly continuous in (to. I. P. x). and is Lipschitz in x. uniformly in (to. I. p). The next result shows that F, is C· i~ X is. and completes the proof of 4.1.5.
4.1.8 Lemma. LeI X in Lemma 4.1.6 be 0/ class C k • I ~ k ~ 00. and leI F,(x) be defined as be/ore. Then locally in (t. x). F,(x) is 0/ class C* in x and is Ck+ I in the I-variable. Proof. We deCine "'(I. x) E L(E. E). the continuous linear maps of E to E. to be the solution of the "linearized" or "first variation" equations:
d
dl"'(I.x)-DX(F,(x»)o",(t.x).
'" (0. x) = identity where DX( y): E .... E is the derivative of X taken at the point y. Since the vector field"' ..... DX(F,(x»o '" on L(E, E) (depending explicitly on I and on the parameter x) is Lipschitz in "'. uniformly in (I. x) in a neighborhood of every ('0' x o ). by 4.1.8 it foUows that "'(I. x) is continuous in (I. x) [using. the norm topology on L(E. E»).
VECTOR FIELDS AND FLOWS
We claim that DF,(x) == 1/1(/, x). To show this, lIet 8(t. h) = F,(x F,(x) and write
191
+ h)-
'(I, II )-1/1(/, x )·11 - fo'{X( F,(x + II» - x(f~( x»} ds - fo'[DX(F,(x» 0 1/I(s, x»)·hds· - fo'DX( F,(x»· ['(s, 1I)-1/I(.f. x )·11] tis
+ !'<X(f,(x + h»- X( F,(x»- DX(F,(x»'[F,(x + h)- F,(x)]} tis. Since X is of class C l , given e> 0, there is a I) > 0 such that IIhll < I) implies the second term is dominated in norm by fJellF,(x + h)- F,(x~1 dv, which is, in tum, 'smaller than Aellhll for a positive constant A. By GroDwall's inequality we obtain 118(1,11)-1/1(1, x)'hll" (constant)ellhll. It follows that DF,(x)·1I = 1/I(t,x)·h. Thus both partial derivatives of F,(x) exist and are continuous; therefore, F,(x) is of ~Iass CI. We prove F,(x) is c" by induction on k. Now
d dt F,(x) =
X( F,(x»
so d d . dl dlF,(x) - DX(F,(x»'X(F,(x»
and d dl DF,(x)'" DX( F,(x »·DF,(x).
Since the right-hand sides are Cit -~, so are the solutions by induction.· Thus F itself is Cit. • Again there is an analogous result for the evolution opera~(X) for a time-dependent vector field X( x, I, p), wbK:h may also .. extra parameters p in a Banach space P. II X is ~", then:~tX) is C~ -. •. . variables ~ is C" + I in , and '0' The variable p can be • dealt with .., _pending X to a new vector rleld obtained by appendiDa." ..... . differential equation p. 0; this dermes a vector rleld on BX r ... 44$" may be applied to iL The now on'E X Pisjuat F,(x.p)-(I1{X~;lJ;;'
192
VECTOR FIELDS AND DYNAMICAL SYSTEMS
For another more "modem" proof of 4.1.5 see Box 4.1C. This proof has a technical advantage: it works easily for other types of differentiahility on X on F,. such as Holder or Sobolev differentiability; see Ehin and Marsden (1970] for details. The mapping F gives a locally unique integral curve e. for each u E ~). and for each 1 E I. F, = FI(uo X (t}) maps ~) to some other set. It is convenient to think of each point u being allowed to .. flow for time t" along the integral curve c.. (see Fig. 4.1.2 and our opening motivation). This is a picture of a Uo "nowmg." and the system (Uo• a. F) is a local flow of X. or flow box. The analogous situation on a manifold is given by the following.
Figure 4.1.2
4.1.10 DeflnlUon. Lei M be a manifold and X a r
~
(i)
(ii) (iii)
I. A
cr tleelor field on
M
f10w box of X al me M is a triple (Uo• a. F). where
Uo eM is open. me Uo. and a E R. where a> 0 or a = + 00; F: Uo X I" -+ M is of class cr. where la = (- a. a); for each u E Uo. eu : I" curve of X al u;
-+
M defined by eu(t) = F(u./) is an integral
(iv) if F,: Uo -+ M is defined by F,(u) = F(u.n. Ihen for open. and F, is a C' diffeomorphism onto its image.
1E
la. F,(Uo) is
Before proving the existence of a flow box. it is convenient first to establish the following. which concerns uniqueness.
4.1.11
Prop08lUon. Suppose c. and c 2 are two integral curves of X at
me M. Then c. = c 2 on the intersection of their domains.
VECTOR FIELDS AND FLOWS
193
Proof. This does not follow at once from 4.1.5 for c, and C2 may lie in different charts. (Indeed. if the manifold is not Hausdorff. Exercise 4. t L shows that this proposition is false.) Suppose c,: I, - AI and c;: 12 . - M. Let I = I, n 12, and let K = (III E I and c,(t) = C,2(1»; K is closed since M is Hausdorff. We will now show that K is open. From 4.1.5. K contains some neighborhood of O. For IE K consider cl and c~. where c'(s) = c( I + of). Then c: and c~ are integral curves at c.(t) = c2(1). Again by 4.1.5 they agree on some neighborhood of O. Thus some neighborhood of I lies in K. and so K is open. Since I is connected, K"" I. • The next proposition gives elementary properties of now boxes.
4.1.12 Prop08It1on. Suppose (Uo• a, F) is a Iriple .,alis/i'ing (i), (ii), lind (iii) 0/2.1.3. Then lor I, s, and I + s E la' we have and
Fo is Ihe identUv map.
Moreover, i/ U, - F,(Uo) and U, n Uo *0, Ihen F,I U., n Uo: U _/ n Uo Uo n U, is a diffeomorphism and ils inverse is F _,I UO n u,.
Proof.
F,+.(u)" COl(1 + s), where cu is the integral curve defined by Fat u. But d(t) = F,(F,(u» = F,(cu(s» is the integral curve through cues) and /(/) -'C,,(I + s) is also an integral curve at cu(s). Hence by 4.1.11 we have F,( F,(u» - C.,( 1+ s) .. F,+.( u). For F, •• = F. 0 F,. merely note that F, • • = F,+,-F,oF,. Since C.,(/) is a curve at u, c,,(O)=u, so f;, is the identity. Finally, the Jast statement is a consequence of F, 0 F / = F .. , 0 F, = identity. Note, however, that F,(Uo)n Uo .. 0 can occur. •
Now we are ready to state and prove the existence and uniqueness of flow boxes. (E%i8~ _ U.~ of Flo .., Bo.'(es). C' veclor field on a ""''Ii/old M. For each m E M there is a flow m. Suppose (Uo• a. F), (Uo• a', F) are two flow boxes at m E M. F are equal on (UO n Uo)x(/.. n I ... ).
4.1.13 Theorem
Lei X be a box 0/ X at Then F and .
Proof (Uniqueness ). Again we emphasize that this does not follow at once
from 4.1.5, for Uo;Uo need not be chart domains. However, for each u E uon Uo we have FI{u}X I .. F'l{u}X I, where I -I.. n I .... This follows from 4.1.11 and 4.1.10 (iii). Hence F= F' on (UonUo)x I. (Exislence). Let (U, cp) be a chart in M with m E U. It is enough to establish the result in cp( U) by means of the local representation. That is, let (Uci. a, F') be a flow box of i. the local representative of X, at cp(m) as.
194
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Biven by 4. J.5. with
and let
F: UoX!" ..... M; Since F is continuous, there is abE (0, a) c R and ~, c Uo open. with mE Vo. such that F(Vo x I h ) c Uo. We contend that h. F) is a now box at m (where F is understood as the restriction of F to x/,.). Parts (i) and (ii) of 4.1.10 follow by construction and (iii) follows from the n:marks following 4.1.4 on the local representation. To prove (iv). note Iha. for t E Ih' F, has a C' inverse, namely. F _I as V, () Uo = v,. It follows thai /';1 is open. And. since F, and F I are both of class C. F, is a C' diffeomorphism. •
(v.,. v.,
v.,)
As usual. there is an analogous result for time- (or parameter-) dependent vector fields. The following result. called the "straightening out theorem." shows that near a point m that is not a critical point. that is. X( m) "" O. the now can be modified by a change of variables so that the integral curves oecome straight lines.
••1.14 Theorem.
Let X he a vector field on a manifold M and suppose at
me M, X( m) ... O. Then there is a local chart (U, cp) with m E U such that
(i) cp(U) = V x IcE X R, VeE. open, and 1 = (- a, a) c R. a > 0; (ii) cp-ll{v}X I: I ..... M is an integral curve of X at cp-I(V,O),for aI/v E V; (iii) the local representative K has the form K(y, t) = (y,t; 0,1).
Proof. Since the result is local. by taking any initial coordinate chart. it suffices to prove the result in E. We can arrange things so that we are working near 0 E E and X(O) = (0.1) E E = E1eR where EI is a complement to the span of X(O). Letting (~,. h, F) be a now box for X at 0 where Uo = VO X( - e. e) and Vo is open in E •. define
But Dfo{O.O) = Identity since
aF,(O,o)
at
I,-0 -
X(O)
=
(0,1)
and
Fo = Identity.
VECTOR FIELDS AND FLOWS
195
By the inverse mapping th~rem there are open neighhorhoods V X 10 c Vo X 1b and U- f(V X la) of (0,0) such thatfX I,,: V X 10 -+ U is a difrcomorphism. Then, I: U -+ V X la can serve as chart for (i). Notice that c ... /I{y}X I: 1-+ U is the integral curve of X through (y.O) for all y E V. thus p'roving (ii). Finally. the expression of the vector field X in this is D,I(y./)·X(f(y.I)I=D/ l(c(t)I·c'(I)= local chart given by (f- 1 0 c)'( I) = (0. I). since (f- 1 0 c)( I) = (y. I). thus proving (iii). •
",IV
,I
Now we turn our attention from local nows to global considerations. These ideas center on considering the now of a vector rield as a whole. extended as far as possible in the I-variable.
4.1.15 Definition. Given a manifold M and a tle('tor field X on M. let "i)x c M X R be the .ret of (m. I) E M X R such Ihat there is an integral ('urtJe
c: 1-+ M of X at m with (E I. The vector field X is complete if6fJx = M xR. Also. a poinl mE M is called (J complfle. where (J = +, -. or ±. if ('ilx n({m}XR) contains all (m. t) for I > 0,1 < 0,' or IE R. respeclively. Lei T': (resp. T;) denole Ihe sup (resp. in/) of Ihe limes of exiSlence of Ihe inlegral curves through m: T': (resp. T';;) is called Ihe positiw (".,me) life,u.ofm.
Thus. X is complete iff each integral curve can be extended so that its domain becomes ( - 00.(0): i.e.• T': - 00 and T; - - 00 for all m E M.
4.1.11 Exempt... A. For M - R 2. let X be the constant vector field. whose principal part is (0,1). Then X is complete since the integral curve of X through (x, y) is' .... (x.y+t). B. On M = R 2 \{O}. the same vector field is not complete since the integral curve of X through (0. - I) cannot be extended beyond' = I; in fact as I -+ I thiS. integral curve tends to (0.0). Thus ~; _I) = I. while ~o: -1)-00.
e.
On R consider the vector field X(x) = I + x 2 • This is not complete since the integral curve c with c(O) - 0 is c( (I) - tan (I and thus it cannot be continuously extended beyond - .,,/2 and .,,/2; i.e.• To t = ± .,,/2. •
4.1.17
P~ltIon.
Let M be a manifold and X
E
'X'( M), r ~ l. Then
(i) 6j)x::> M x {O}; (ii) 6fJx is open in M X R; (iii) Ihere is a unique C' mapping Fx: oDx -+ M such ,hal Ihe mapping I .... Fx(m,/) is an inlegral curve al m. for all mE M. (iv) for(m.t)E 6j)x.lhe pair (Fx(m. I).S)E oTJx'iff(m. (+ s)E "i)x: in this CtUe Fx (";, t + s) - Fx(Fx(m, I). s).
196
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Parts (i) and (ti) follow at once from the flow box existence theorem. and (iii) by the uniqueness of integral curves: "Dx is the union of all now box domains Uo x I" and Fx is the unique extension of the now box maps to UDx . Thus (iv) is a reformulation of the first part of 4.1.12. • Proof.
Thus. if X is complete, (M. 00. Fx) is a flow box. 4.1.18 DeflnlUon. Let M be a manifold and X E <X( M), r ~ 1. Then the mapping Fx is called the ;"tegl'Gl of X. and the curoe t ..... Fx (m, t) is called the mtIXimtII ;"tegrwl CfIrW of X at m. In case X is complete. Fx is called the /10M' ofX.
Thus. if X is complete with now F. then the set {F,I t E R} is a group of diffeomorphisms on M. sometimes called a one - parameter group of diffeomorphisms.t For incomplete flows. (iv) says that F, 0 F. = F,+ .• wherever it is defined. Note that F,(m) is defined for t E ]T'; • T,;! (. The reader should write out similar definitions for the time-dependent case and note that the lifetimes depend on the starting time to' Next we describe some basic criteria that ensures completeness of a vector field. 4.1.19 ProposlUon. Let X be C. where r ~ 1. Let c(t) be a maximal integral curoe of X such that for every finite open interoal ]a. b( in the domain 14(0)' T.·;O)( of c. c(]a. b() lies in a compact subset of M. Then C' is defined for alitER. Proof. It suffices to show that a E I. bEl. where I is the interval of definition of c. Let ttl E ]a, b(. ttl .... b. By compactness we can assume some subsequence c(l,,) converges. say. to a point x in M. Since the domain of the flow is open. it contains a neighborhood of (x,O). So there are e> 0 and T > 0 such that integral curves starting at points (such as c(l",) for large k I closer than f! to x persist for a time longer than T. This serves to extend c to a time greater than b, so bel since c is maximal. Similarly, a E I. •
The support of a vector field X on a manifold M is the closure of the set (1ft E MIX(m). O}.
4.1.20 Corollary. A C vector field with compact support on a manifold M is complete. In particular. a C' vector field on a compact manifold is complete. Since F" - (F,)" (the ntb power), the notation F' is sometimes convenient and is used where we use F,.
t
VECTORFIELOSANOFLOWS
197
Completeness corresponds to well-defined dynamics persisting eternally. In some circumstances (shock waves in fluids and solids. singularities in general relativity. etc.) one has to live with incompleteness or overcome it in some other way. Because of its importance we give two additional criteria. In the first result we use the notation X[ fl == df· X for the derivative of f in the direction X. Here 1: E - Rand df stands for the derivative map. In standard coordinates on R".
af af ) df(x) ... ( - , •...• ax" ax
and
X(f] ==
n / af LX -,' ax
/ _,
4.1.21 Proposition. Suppose X is a c k vector field on E. k ~ I. and f: E - R is a C' proper map (that is. the inverse images of compact sefs are compact). Suppose there are constants K. L IX(f](m)1 ~ Klf(m)1
~
0 such that
+ L forall mE E.
Then the flow of X is complele. Proof. that
From the chain rule we have (a/at)f(F,(m»== X[fJ(F,(m», so
f{F,(m»)- f(m) ;'l'X(f](~(m» dT. o Applying the hypothesis and Gronwall's inequality we see that 1f(F,(m»1 is bounded on any finite I-interval. so as f is proper, F,(m) lies in a compact set. Hence 4.1.19 applies. • Note that the same result holds if we replace "properness" by "inverse images of compact sets are bounded" and assume X has a uniform existence time on each bounded set. This version is useful in many infinite dimensional examples. The next theorem gives a completeness criterion in terms of the norm' on a Banach space.
4.1.22 ProposlUon. Let E be a Banach space and X a C' vector field on E. r ~ 1. Let" be any integral curve of X. Assume IIX( ,,(t)~1 is bounded on , finite t-intervals. Then the flow of X is complete. Proof. have
Suppose IIX(,,(t)~1 < A for tEla, b[ and let
"''(In)-''(I,,,)I~j'''lfJ'(t)ldl= I"
In
~ h. For
tn
< I", we
j'm Il X(,,(t»IIc/I< AI/",-I.I. 'n
198
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Hence G( t n) is a Cauchy sequence and therefore. converges. Now argue as in 4.1.19. • The following is a typical application. 4.1.23 IXllmple.
Let E be a Banach space. Suppose X(x) "" .... ·x + B(x).
where .... is a bounded linear operator of E to E and B is sublinear; i.e.• B: E .... E is C' with r;;. I and satisfies IIB(x~11Iit Kllxll+ L for constants K and. L. We shall show that X is complete by using 4.1.22 (In RIO. 4.1.21 can also be used withf(x) -lIxIl 2 ). Let x(t) be an integral curve of X. Then x(t) == x(O)+
fo'( .... ·x(s)+ B(x(s») dx.
Hence
IIx(t)lIlIitllx(O)II+ 1'(11 .... 11+ K)lIx(s)lItb + Lt. D
By Gronwall's inequality.
IIx(t)II'" (LT+llx(O)U)eUIAII+KI'. Hence x(t) and so X(x(t» remain bounded on bounded t-intervals.
•
BOX "-1A PRODUCT FORMULAS
A result of some importance in both theoretical and numerical work concerns writing a flow in terms of iterates of a known mapping. Let X E ~(M) with flow F, (maximally extended). Let K,(x) he a given map defined in some open set of [0. co[ X M containing {O} X M and taking values in M. and assume that (i) 'Ko(x)-x
(ii) K.(x) is C' in the "algorithm."
I
with derivative continuous in (e, .t). We call K '
Let X be a C' vector field. r> I. Assume that the algorithm K,(x) is ,....,., with X in the sense that X(x)-
a
a,K,(x)J.-o
VECTOR FIELDS AND FLOWS
199
Then, if (I, x) is in Ihe domain of F,(x), K,i,,(x) is defined for n sufficienlly large and converges 10 F, (x) as n --> 00 . Conversely.. if K,i,,(x) is defined and converges for 0 Et lEtT, then (T. x) is in Ihe domain of F and the limit is F,(x).
In the following proof the standard notation O( x ). x E R is used for any continuous function in a neighborhood of the origin such thaI O( x)/ x is bounded. Recall from Section 2.1 that o( x) denotes a. continuous function in a neighborhood of the origin satisfying limx_oo(x)/x - O. Proof. First, we prove that convergence holds locally. We begin by showing that for any x o, the iterates K,i,,(x o ) are defined if I is sufficiently small. Indeed, on a neighborhood of x o' K,(x) == x + O(e), so if K!/j(x) is defined for x in a neighborhood of x(). for j = 1,00" n -I, then
+ ... +(K,/,,(x)-x) ==O(t/n}+ .. · +O(I/n) =
OCt}
This is small, independent of n for I sufficiently small; so. inductively, K,i,,(x) is defined and remains in a neighborhood of -"0 for x near xo' Let P be a local Lipschitz constant for X so that 1IF,(x)- F,(yXI ~ e~ltlllx - yll. Now write
200
VECTOR FIELDS AND DYNAMICAL SYSTEMS
wherey.. " K,~,,(x). Thus 1IF,(x)-K,i,,(x)lI.;;
E" elll,l(n-k)lnllF,ln(Yn~kl)-K,/nCVn
~ 1)11
.. -I
..;nelll'lo(l/n)-O
as n ..... oo
since F.(y)- K,(y) - o(e) by the consistency hypothesis. Now suppose F,(x) is defined for 0.;; I .;; T. We shall show K,i,,(x) converges to F,(x). By the foregoing proof and compactness. if N is large enough, F,/N -lim n _ oa K,inN uniformly on a neighborhood of the curve t>-+ F,(x). Thus, for 0.;; I';; T,
By uniformity in t. Fr(x)= limKL(x). I
~oo
Conversely. let K,in{x) converge to a curve c(t).O~/~T. Let S = (t IF,( x) is defined and c( I) = F,( x )}. From the local result. S is a nonempty open set. Let E S.l k - t. Thus F,,(x) converges to c(t). so by local existence theory. F,(x) is defined, and by continuity, F,(x)= C(/). Hence S= [0. T) and the proof is complete. •
'4
4.1.25 Corollary. LeI X. Y E 'X,(M) wilhflows F, and G,. Let S, be + Y. Then for x E M.
the flow of X
The left-hand side is defined iff the right-hand side is. This follows from 4.1.24 by setting K,(x) - (F. 0 G,)(x). for example. for n X n matrices A and B. 4.1.25 yields the classical formula e(A+8)= lim (e A1 "e 8/,,)". n-oo
To see this. define for any n X n matrix C a vector field Xc E 'X,(R") by XC< x) - Cx. Since Xc is linear in C and has now F,(.r) - e'c.r. the formula follows from 4.1.25 by letting I = I. The topic of this box will continue in Box 4.20. The foregoing proofs were inspired by Nelson (1969) and Chorin et al. (1978).
VECTOR FIELDS AND FLOWS
201
BOX 4.18 INVARIANT SETS If X is a smooth vector field on a manifold M and N c M is a submanifold. the now of X will leave N invariant (as a set) ifr X is tangent to N. If N is not a submanifold (e.g.• N is an open subset together with a non-smooth boundary) the situation is not so simple; however. ·for this there is a nice criterion going hack to Nagumo [1942]. Our proof follows Brezis (1970).
4.1.21 Theorem. Let X be a locally Upschitz vector field on an open set U c E. where E is a Bana-:h space. Let G cUbe relatively closed and set d(x. G) = infOlx - Yilly E G}. The following are equivalent: Iimlr~o(d(x+hX(x).G)/h)=O locally uniformly in xEG (or just pointwise if E == R n); and . (ii) if x(t) is an integral curve of X starting in G. then xU) E G for all t'~ 0 in the domain of x(·).
(i)
Note that x(t) need not lie in G for t.s; 0; so G is only + invariant. (We remark that if X is only continuous the the~rem fails.) We give the proof assuming E = R n for simplicity. Proof. Assume (ii) holds. Setting x(t) = F,(x), where F, is the now of X. we get
d(x+hX(x).G).s;lIx(h)-x-hX(x)II=lhl II x(h) h - x -X(x) II
from which (i) foIlows. Now assume (i). It suffices to show x(t)E G for small t. Near x'" 'x(O). say on a ball of radius r. we have
and
We can ~ssume 1IF,(x)-xll
202
VECTOR FIELDS AND DYNAMICAL SYSTEMS
needed here.) Thus,IIY, - xII < r. For small h,IIF"y, - xII < r, so
cp(1 + h)" 1IF,+,,(x)- F"y,II+IIF,,(y,)- y, - hX(y,)1I + d(y, + hX(y,).G) "eK"IIY, - F,xll+IIF,,(y,)- y, - hX(y,)II+ d(y, + hX(y,). G) or
cp(1 + hl- cp(l) " ( e K: -I )CP(I)+ II
FIr<'l~)- Y,
- X( y,)11
+ d(y, + hX(y,).G)/h. Hence lim sup cp(l+h)-cp(l) .r;;Kcp(l) "lO h As in Gronwall's inequality, we may conclude that
so cp(t) = O. • 4.1.27 Example. Let X he a smooth vector field on R n. let g: R n --0 R he smooth. and let ~ E R he a regular vullie for g. so g I( ~) is a submanifold. Let G = f{ I( 1- 00. ")) and suppose that on g 1(,,). (X. grad g) ,,0. Then G is + invariant under F, as may be seen by using 4.1.26. This result has been generalized to the case where fJG might not be smooth by Bony (1969). See also Redheffer (1972) and Martin (1973). Related references are Yorke (1967). Hartman (1972). and Crandall (1972).
BOX 4.1C A SECOND PROOF OF THE EXISTENCE AND UNIQUENESS OF FLOW BOXES We shall now give an alternative "modern" proof of Theorems 4.1.5 and 4.1.13, namely. if X E 'X'( M). k ~ I. then for each mE M there exists a unique c' now box at m. The basic idea is due to Robbin (1961) altbouah similar alternative proofs were simultaneously
VECTOR FIELDS AND FLOWS
, 203
discovered by Abraham and Pugh [unpublished) and Marsden [1968b, p. 36S). The present exposition follows Robbin (1968) and Ebin arid Marsden [1'970).
SUp 1. Existence and Uniqueness of Integral Curves for Vector fields.
c·
Proof. Working in a local chart, we may assume that X: Dr(O) ...... E, where Dr(O) the open disk at the origin of radius r in the Banach space E. Let U = Dr/ 2 (O), 1 = [ - I, I) and define
41: R x CJ(I, U) ...... C°(l, E) by
4>(s,y)(t)
=
dy di(t)-sX(y(t».
where Ci(l, E) is the Banach space of Ci-maps of 1 into E, endowed with the II'II;-norm (see Box 2.4.B), qU, E) = (f E CiU, E) 1/(0) = O} is a closed subspace of CiU. E) and Cj(l, U) .... (f E CjU, E)I/U) C U} is open in CjU, E). We first show that 41 is a C·-map. The map dldt: CJu, E) -+ C°(l, E) is clearly linear and is continuous since IIdldtll ~ I. Moreover, if dyldt = on I, then y is constant and since yeO) = 0, it follows y =- 0; i.e., d I dt is injective. Also, given 8 E C°(l, E), yet) = Iri8(s) ds defines an element of CJ(l. E) with dYldt = 8, i.e.• dldt is a Banach space isomorphism from CJu, E) to C°(l. E). From these remarks and the O-Iemma (2.4.I'S), it follows that 4> is a C· map. Moreover, Dy4>(O,O) = dldt is an isomorphism of CJU. E) to Co( 1. E). Since 4)(0,0) = o. by the implicit function theorem there is an E> such that 4>(E. y) = 0 has a unique solution y.(t) in CJ(l,U). The unique integral curve sought is y(t)=y,(lIE), -f~
°
°
t
~f• •
The same argument also works in the time-dependent case. It also shows that y varies continuously with X.
Step 1. Proof.
setting
The local flow of a C" vector field X is C k . First, suppose k = I. Modify the definition of
+: R x U x CJ(I, U) -+ COU, E),
c)l
in step I by
+(s, x. y )(t) = y(t)-sX(x + y(t ».
°+
As in step I.
is a C· map and Dy+(O,O.O) is an isomorphism, so can be locally ~lved for y giving a map H,: U-+ cJ(I.U).e> 0. The local flow is F(x.t)=x+ H,(x)(tlr), as in step I. By' 2.4.17 (differentiability of the evaluation map), F is C·. +( E. x, y) ..
204
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Next we prove the result for k ~ 2. Considet the Banach space F-Ck-'(cl(U),E) and the map "'x:F .... F; 1J ..... Xo1J. This map is C' by the O-lemma (remarks following 2.4.18). Regarding "'x as a vector field on F, it has a unique C' integral curve 1J, with 1Jo ... identity. by Step I. This integral curve is the local flow of X and is C k - . since it lies in F. Since k ~ 2, 11, is at least C' and so one sees that D1J, =", satisfies .,Idt ... DX( 11;)''',. so by Step I again, II, lies in C k - '. Hence 11, is Cit. • The following is a useful alternative argument for proving the result for k - I from that for k ~ 2. For k -I. let X" .... X in C', where X" are C 2 • By the above, the flows of X" are C 2 and by Step I. converge uniformly i.e. in Co, to the now of X. From the equations for DlI,", we likewise see that DTI," converges uniformly to the solution of dM,ldt,. DX(TI,)·",. "0" identity. It foUows by elementary analysis (see Exercise 2.41 or Marsden [1974a, p. 109]) that TI, is C· and DTI, = II,. • This proof works with minor modifications on manifolds with vector fields and flows of Sobolev class H S or Holder class C k ~"; see Ebin and Marsden (1970) and Bourguignon and Brezis [1974]. In fact the foregoing proof works in any function spaces for which the O-Iemma can be proved. Abstract axioms guaranteeing this are given in Palais [1968].
Exercl... Find an explicit formula for the flow F,: R 2 .... R2 of the harmonic oscillator equation .f+ ",2X "" O. '" ERa constant. 4.lB Show that if (UO. a. F) is a flow box for X. then (Uo• a. F _) is a flow box for - X. where F _Cu. t) == F(u. - t) and (- X)(m) = -(X(m». 4.1 C Show that the integral curves of a C' vector field X on an n-manifold can be defined locally in the neighborhood of a point where X is nonzero by n equations ,,;(m, t) = cj == constant. i .. l •... ,n. Such a system of equations is called a local complete system 0/ integrals. (Hint: Use the straightening-out theorem.) 4.1 D Prove the foUowing generalization of Gronwall's inequality. Suppose vet) ~ 0 satisfies v(t)" c + fJ Ip(s)lv(s) tis, c ~ O. Then 4.1A
v(t) < cexP({IP(s)1 tis ). 1 I",.
this to generalize 4.1.23
,- -iilX.
In
allow A to be a time-dependent
VECTOR FIELDS AND FLOWS
4.1H
205 '
now
(VariatiON of COIUltmlllomtIIIa). Let F, - e'x be the of a linear vector field X on E. Show that the solution of the equation
.t- X(x)+ I(x) with initial condition Xo satisfies tbe integral equation
4.1 F
Let F( m. I) be a efta mal>ping of R x M to M such that F, •• - F, 0 F, and Fo - identity (wbere F,(m) - F(m, Show that there is a unique efta vector field X wbose now is F.
I».
4.10 Let a(l) be an integral curve of a vector field X and let g: M .... R. Let T(I) satisfy T'(I)= g(a(T(I))). Then show t .... CJ(T(t» is an inte&ral curve of gX. Show by example that even if X is complete. gX need not be. 4.1H
(i) (Gradienl Flows) letf: R" -+R be e l and let X= (iJf/ax l ••••• ilf/ ax") be the gradient of f. Let F be the now of X. Show that f(F,(x» ~ f(~(x» if I ~ s. (ii) Use (i) to find a vector field X on R" such that X(O) = o. X'(O) = 0, yet 0 is globally attracting; i.e.. every integral curve converges to 0 as I -+ 00.
4.11
let X= y 2 a/ax and Y=x 2 il/ay. Show that X and Yare complete on R 2 but X + Y is not. (ii) Prove the following theorem: Let H be a Hilbert space and let X and Y be locally Lipschitz vector fields tbat satisfy the following: (a) X and Yare bounded and Lipschitz on bounded sets; (b) there is a constant fJ ~ 0 sucb tbat (i)
(Y(x). x) < fJllxlf' (c)
for all
x EX
there is a locally Lipschitz monotone increasing function C(I) > O. I ~ 0 such that JOOdx/c(x)'" + 00 arid if X(I) is an integral curve of X.
!lIx(t)1I < c(lIx(t)lI) Then X. Y and X + Yare positively complete. NOle: One may assume IIX(xo~I'" c(lIxolI) in (c) instead of (d/dl~I%(t~1 < cOI%(I~I>. (Hinl. Find a differential inequality
206
VECTOR FIELDS AND DYNAMICAL SYSTEMS
for t
4.1J
+ fo'C(f(S»ds.
IE
[0.11
Then /(/)" r(/) for IE [0, T[.) Prove the following result on the convergence 0/ flows: Let X .. be locally Lipschitz vector fields on M for 0 in some' topological space. Suppose the Lipschitz constants of X" are locally bounded as 0"'" 00 and X .. ..... X .... locally uniformly. Let (.( I) be an integral curve of X .... 0 " 1 < T and e > O. Then the integral curves C.(/) of X. with c..(O) ... c(O) are defined for IE [0. T - e) for 0 sufficiently close to 00 and C..(/) ..... c(/) uniformly in 1 E [0. T - e) as 0"'" Go. If the flows are complete, P,'''''' F, locally uniformly. (The vector fields may be time dependent if the estimates are locally I-uniform.) (Hin/: Show that
IIc.. (/)-c(/)II" Kl'lIc.( .,)-c( ")lId., +l'IIX.(c( .,)- XII (c( "»lId., o 0 • and conclude from Gronwall's inequality that c.( I) ..... c""( t) for since the second term ..... O. This estimate shows that C.. (/) exists as long as c(/) does on any compact subinterval of [0. T[.) 4.1K Prove that the C flow of a C+' vector field is a C' function of the vector field by utilizing Box 4.1C. (Caulion. It is known that the C" flow of a e" vector field cannot be a C· function of the vector field; see Ebin and Marsden (1970] for the explanation and further references). 4.1 L (Nonunique inlf!gral curves on non-Hausdorff mani/olds). Let M be the line with two origins (see Exercise 3.5H) and consider the vector field X: M ..... TM defined by X[x. i] = x. i = 1.2: here Ix. i] denotes a point of the quotient manifold M. Show that through every point other than (0.0] and [0.1], there are exactly two integral curves of X. Show that X is complete. (Hin/: The two distinct integral curves pass respectively through (0,0) and [0,1).) 0 ..... 00
4.1 M Give another proof of Theorem 4.1.5 using Exercise 2.5J.·
· VECTOR FIELDS AS DIFFERENTIAL OPERA TORS
207
4.2 VECTOR FIELDS AS DIFFERENTIAL OPERATORS
In the previous section vector fields were studied from the point of view of dynamics-the flows they generate. Before continuing the development of dynamics, we shall treat some of the algebraic aspects of vector fields. The specific goal of the section is the development of the Lie derivative of functions and vector fields and its relationship with flows. One important feature is the behavior of the constructions under mappings. The operations should be as natural or covariant as possible when subjected to a mapping. Let. us begin then with a discussion of the action of mappings on functions and vector fields. First. recall some notation. Let' C(M. F) denote the space of C maps f: M - F, where F.is a Banach space. and let ' X '( M) denote the space of C vector fields on M. Both are vector spaces with the obvious operations of addition and scalar multiplication. For brevity we write '!i'(M)=C"'(M.R), '!i"(M)=C(M.R) and ~X(M)= ~OO(M). Note that '!i"( M) has an algebrastructwe; that is. forf. g E IS'(M) . the product fg defined by (fg)(m) = f(m)g(m) obeys the usual algebraic properties of a product such as fg = gf and f( g + h) = fg + fh. 4.2.1 Definition.. (i) Let cp: M - N be a C' mapping of manifnld.~ and f E IS'(N). Define the puIl-lNIek off by cp by cp.f =
f
0
cp E IS' ( M).
If cp is a C' diffeomorphism and X E CX'( M). the puS"- forwtlrd of cp is defined by
(ii)
X
~y
CP.X= TIP, Xo cp-I E CX'(N). 0
The reason for the names is shown in Figure 4.2.1. Consider local charts (V,X). x: V-V'cE on M and (V.I{I). I{I: V -+ V'c Fon N, and let (TX 0 X 0 X-I)(u) = (u, X(u». where X: V' -+ E is the local representative of X. Then from the chain rule and the definition of push-forward. the local representative of CP. X is
where v = (I{I cp X-I)(u). The different point of evaluation on each side of the equation corresponds to the necessity of having cp- I in the definition. If M and N are finite dimensional, Xi are local coordinates on M and Xi local coordinates on N. the preceding formula gives the components of CP.X by 0
0
(cp.X)i(X) where X- cp(x).
=
iJcpi . (x)X'(x) iJx
-I
208
VECTOR FIELDS AND DYNAMICAL SYSTEMS
or- =
pull back
Figure 4.2.1
We can interchange "pull-back" and "push-forward" by changing qJ to I, i.e., defining qJ* (resp. qJ*) by qJ* ... (qJ-I)*(resp. qJ* = (qJ-I ).). Thus the push-forward of a function f on M is qJ. f = f 0 qJ - 1 and the pull-back of a vector field Yon N is qJ·Y-(TqJ)-loY0qJ. Notice that qJ must be a diffeomorphism in order that the pull-back and push-forward operations make sense, the only exception being pull-back of functions. Thus vector fields can only be pulled back and pushed forward by diffeomorphisms. ' However, even when qJ is not a diffeomorphism we can talk about qJ-related vector fields as follows.
qJ-
4.2.2 DeflnlUon. Let qJ: M -+ N be a C' mapping of manifolds. The vector fi~1ds XE'X,-I(M) and YE~X'-I(N) are called cp-n/l"ed, denoted
"
X-
Y-
Y,
if TqJ ° X = yo qJ.
Note that if qJ is a diffeomorphism and X and Yare qJ-related, then In general however, X can be qJ-related to more than one vector
qJ. X.
VECTOR FIELDS AS DIFFERENTIAL OPERA TORS
209
field on N. cp-relatedness means that the following diagram commutes: TfP TM---'TN X t fP I Y M N
4.2.3 PrOposition. (i) Pull-back and push-for....ard are linear maps. and . 'I' fP*(fg) = (fP*f)( fP*g), fP.(fg) = (fP*f)( fP*g)· Moreover. if X, - Y,. i = 1.2,
'"
and a, b e R. then aX, + bX2 - aY, + bY2 • (ii) For fP: M -+ Nand 1/1: N -+ P, we have ( 1/1 0 fP )* = fP* 0 1/1* and ( 1/1 0 fP ). = 1/1* 0 fP *.
'"
'"
IfXe~Y..(M), Ye'X(N),Ze'X(P),X- YandY-Z.thenX ",. -
'" Z.
In, this proposition it is understood that all maps are diffeomorphisms with the exception of the pull-back of functions and the relatedness of vector fields. Proof.
'" Y"
X, -
(i)
This consists of straightforward verifications. Por example. if
i =-1,2, then
TfP o(aX,
+ bX2 ) ... aTfP 0 X. + bTfP 0 X2 - aY. 0 fP +
bY2 0 fP. i.e., aX. + bX2 ! aY. + bY2 • (ii) These relations on functions are simple consequences of the definition, and the ones on 'X.(P) and 'X.(M) are proved in the following way using the chain rule: T( 1/1 0 fP) 0
x ... TI/I 0 TfP
0
X = TI/I 0 yo fP
=
Z 01/1 0 fP.
•
In this development we can replace
4.2.4 ProposlUon. Let fP: M -+ N be a C'-mapping of manifolds and X e <X'(M) and Y e 'X.'(N). Let F,x and F,Y denote the flows of X and Y. respectively. Then X - Y iff fP 0 F,x = F,Y 0 fP. In particular. if fP is a diffeomorphism, then Y == fP * X iff the flow of Y is fP 0 F,x 0 fP - I.
'"
Proof. Taking the time derivative of the relation (fP F,x)(m) = (F,Y 0 fP)(m), for m eM, using the chain rule and definition of the flow, we 0
210
VECTOR FIELDS AND DYNAMICAL SYSTEMS
get Ttp ( (Ttp
0
aF.X(m») . aF. Y I
a,
"" -t,- (tp( m»,
i.e.
X 0 F,X)( m ) - (y 0 F, Yo tp )( m) .. (y 0 tp a F,X)( m ),
which is equivalent to Ttp a X = yo tp. Conversely, if this relation is satisfied, let C(I) - F,x(m) denote the integral curve of X through me M. Then d( tp. C)(I) _ Ttp( «(I) ) ... Ttp( X(C(I»)
dl
dl
== Y« tp a C )(1»
says that tp. C is the integral curve of Y through tp( c(O» = tp( m). By uniqueness' of integral curves, we get (tp • F,X)( m) = (tp a c)( I) = F, Y( tp( m»•
•
We call tp. F,. tp-I the push-forward of F, by tp since it is the natural way to construct a diffeomorphism on N out of one on M. See Fig. 4.2.2. Thus, 4.2.4 says that the flow of the push-forward of a veclor field is Ihe push forward of ils flow. Next we define how vector fields operate on functions. This is done by means of the directional derivative. Let f: M - R, so Tf: TM - TR == R xR. Recall that a tangent vector to R at a base point). E R is a pair ()., ,,), the number" being the principal part. Thus we can write Tf acting on a vector., E T", M in the form
Tf·"= (J(m),df(m)·.,). F,
"o/'r . q'
I
conjugation ---......
c-inlqnl curve of X
'" • c - integral curve
or",.x
F..... 4.2.2
VECTOR FIELDS AS DIFFERENTIAL OPERA TORS
211
This defines, for each m EM the element df(m) E T.:M. Thus dfis a Section of -r M. -a covector field, or one-form. 4.2.5 Definition. The C01.'ector field df: M..,. T* M defined called the differential of f.
thi.~
way is
For F-valued functions. f: M -+ F, where F is a Banach space. a similar definition gives df(m) E L(TmM, F) and we speak of df as an F-valued ' one-form.
Clearly if f is C'. then df is C'- I. Let us now work out df in local charts for f E ~if( M). If q1: U c M -+ VeE is a local chart for M. then the local representative of f is the map j: V -+ R defined hy i = f q1 I. The local representative of Tf is the tangent lJlap for local manifold~: 0
Tj(x, v) = (j(x), Dj(x)·v).
Thus'the local representative of df is the derivative of the local representative of f. In particular. if M is finite dimensional and local coordinates are denoted (Xl, . .. ,x"), then the l~al components of df are
( df),
=
aax~ .
The introduction of df leads to the following. 4.2.6 DefinlUon. Lei fE'!t'(M) and Xe'X' '(M).r;21:l. Define the directiolull or Lie derivative, 0/ f along X by
Lxf(m) == X(f](m) =df(m)·X(m). for any me M. Denote by X[f]=df(X) the map m EM· .. X[f)(m)E R. If fis F-valued. the same definition is used, but now X[f] is F-vahied. The local representative of X[f] in a chart.is given by the real valued function x ...... Dj(x)'X(x), where j ,and X are the local representatives of I and X. In particular, if M is finite dimensional then we have
X(f] == LxI =
t
i-I
af X'.
ax'
Evidently if I is C' and X is C,-I then Xlf) is
C'~ '.
212
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Let us observe that from the chain rule, d( f qJ) = df TqJ where qJ: r ~ l. For real-valued functions, Leibniz' rule gives 0
0
N ..... M is a C' map of manifolds,
dUg) - fdg
+ gdf·
(If f is F-valued, g is G-valued and B: F X G ..... H is a continuous bilinear map of Banach spaces, this generalizes to d(B(f, g» = B(df, g)+ B(f, dg).)
4.2.7 Propolltlon. (i) Suppose qJ: M ..... N is a diffeomorphism. Then Lx is natural with respect to push- forward by qJ. That is, for each f L".x( qJ.f) = qJ.Lxf; or the following diagram commutes: qJ •
\'t'( M)
E
'!f(M),
• \1"( N)
1L".x
Lxl ~(M)
qJ.
~(N)
(ii) Lx is IfIItIIrtII witll rnpect to rntmiOlU. That is, for V open in M andfE~(M), Lx1u(fl V )"" (Lxf)IV; or, if IV: ~(M)--+~(V) denotes
restriction to V, the following diagram commutes: IV ~(M)-~(V)
Lx!
!Lxlu
CJ(V)
CJ(M) IV
Proof.
For (i), if n E N then
L".x( qJ.f)(n)'" d(t 0 cp-I)( qJ.X)(n) = d(t 0 cp-I )(n)· (TqJ 0 X 0 qJ-l)(n) = df{ qJ-l(n»)-( X
0
qJ-l)(n) = qJ.(Lxf)(n)
Then (ii) follows from the fact that d(fl V) = (df)1 V, which is clear from the definition of d. • This proposition is readily generalized to F-valued C' functions. Since qJ. = (cp - I)., the Lie derivative is also natural with respect to pull-back by cpo This has a generalization to qJ-rl'lated vector fields as follows.
VECTOR FIELDS AS DIFFERENTIAL OPERA TORS
213
4.2.1 Proposition. Let cp: M ..... N be a C' map. X E 'X,-I( M) and
"
Y e ~'-I(N). II X - Y. then Lx(cp*f) - cp*LI'I for all / E C(N. F); i.e .• the lollowing diagram commute,,:
".
C'(N.f)---· C'(M.F)
Ly
I ".
C-I(N.F)
Proof.
For m
E
l
Lx
• C-I(M, F)
M,
Lx( cp·f)(m) = dU 0 cp)(m )·X(m) = df(cp(m»'(T",cp( X( m») = df( cp(m»· Y( cp(m» = df( Y)( cp(m»' = (cp·Lrf)(m) . • Next we show that Lx satisfies the Leibniz rule.
4.2.9 Propoaltlon. (i) The mapping Lx: C( M, F) ..... C- I ( M, F) is a tkriwltion. That is Lx is R-linear and for / E C'(M, F), g E C(M.G) and B: F X G ..... H a bilinear map. Lx(BU,
g» = B(Lx/. g)+ BU. Lx1d.
For real-valued functions, LxUg) = gLxf + fLxB. '(ii) If c is a constant function, Lxc'" O. Proof. (i) This follows from the product rule and the definition of Lx/. , Part (ii) results from the definiiion. •
,The connection between Lx f and the flow of X is as follows.
4.2.10 Propoaltlon. Let/ E C'(M, F), X E
ex' 'I(M) and suppose X has
a flow F,. Then Proof. By the chain rule, the definition of the differential of a function and the flow of a vector field.
d d dF.(m) dt(Fif)(m) = dtU°F,)(m)=df(F,(m». 'dt = df( F,(m »·X( F,(m» = df( X)( F,( m» == (Lxf)( F,( m»
= (FiLxf)( m).
•
214
VECTOR FIELDS AND DYNAMICAL SYSTEMS
As an application of the Lie derivative, we consider a partial differential equation on R" + 1 of the form
(P) {
al
.
n'
i
af
a/(X./)=i~IX(X)aXi(X,/)
f(x,O) = g(x)
for given smooth functions Xi(x), i = 1, . .. ,n, g(x), and a scalar unknown f(x, I).
4.2.11
Proposition. Suppose X = ( X l , ••• , X n) has a complete flow F,.
Then f(x,t)=g(F,(x)) is a solution of the foregoing problem (P). (See Exercise 4.2C for uniqueness.) Proof.
Thus one can solve this scalar equation by computing the orbits of X and pushing (or "dragging along") the graph of g by the flow of X; see Fig. level curves of g • F,
Flgurw 4.2.3
VECTOR FIELDS AS DIFFERENTIAL OPERATORS
215
4.2.3. These trajectories of X are called characteristics of (P). (As we shall see below. the vector field X in (P) can be time dependent.) 4.2.12 Example. Solve the partial differential equation af =(x+ y )( af _ af), at ax ay f(x. y.o) = x 2 + y2.
The vector field X(x. y) = (x + y, - x - .1') has a complete flow F,(x. y) = «x + y)t + x. -(x + y)t + y). so that the solution of the previous partial differential equation is given by Solution.
Now we tum to the question of using the Lie derivative to characterize vector fields. We will prove that any derivation on functions uniquely defines a vector field. Because of this, derivations can he (and often are) used to define vector fields. (See the introduction to Section 3.4.) In the proof we shall need to localize things in a smooth way, hence the following lemma of general utility is proved first. 4.2.13 Lemma. Let E be a C', Banach space. i.e" one whose norm is C' on E\{O}, r;;, I. Let VI be an open ball of radius r l ahout Xo and V2 an open ball of radius r2 • r l < r2 • Then there is a C' function II: E -+ R such that II is one on VI and zero outside V2 •
We call h a bump function. Later we will prove more generally that on a manifold M. if VI and V2 are two' open sets with cJ(VI ) c V2 • there is an he 'J'(M) such that h is one on VI and is zero outside V2 • Proof. By a scaling and translation. we can assume that VI and V2 are balls of radii 1 and 3 and centered at the origin. Let 8: R -+ R be given by 8(x) = {exp( -1/(I-l x I2
o.
»).
(See the remarks folJowing 2.4.15.) Now set
8 1(s ) =
t1-
8(t) dt
0000
-- 00
8( t) dt
Ixl
216
VECTOR FIELDS AND DYNAMICAL SYSTEMS
so ',(s) is a Coo function, 0 if s < -1, and 1 if s > 1. Let
so
'2 is a Coo function that is 1 if s < 1 and 0 if s > 3. Finally, let
The nonn on a real Hilbert space is Coo away from the origin. The order of differentiability of the norms of some concrete Banach spaces is also known; see Bonic and Frampton (I966J and Yamamuro (1974).
4.2.14 II",
Carol'' ' '
Let M be a manilold modeled on a C' Banach space. II
e r:,M, then there is an I e ~'( M) such that d/(m) =
11m'
.
Proof. If M .. E, SO TmE g; E, let I(x) = II",(X), a linear function on E. Then dl is constant and equals II",. The general case can be reduced to E using a local chart and a bump function as foUows. Let cp: U ..... U' c E be a local chart at m with cp( m) = 0 and such that U' contains the ball of radius 3. Let a", be the local representative of II", and let h be a bump function 1 on the ball of radius 1 and zero outside the ball of radius 2. Let i( x) = am (x) and let
1_ {(hJ)oCP, 0,
on U on M\U
It is easily verified that I is C' and d/( m) -
II",.
•
4.2.15 Propoeltlon. (i) Let M be a manifold modeled on a C' Banach space. The collection 01 operators Lx lor X e CX'(M), defined on C'+ '(M, F) and laking values in C'(M, F) lorms a real vector space and IJ(M) module with (/Lx )(g) -/(LxS), and is isomorphic to CX(M) as a real vector space and as an ~(M) module. In particular, Lx =0 iff X=O; and LfX = /Lx. (ii) Let M be any manilold.11 LxI = olor aliI e C'(U, F),lorall open subsets U 01 M, then X = O.
I'rrIof. (i) Consider the map a: X .... Lx. It is obviously R and
IJ(M)
linear; i.e.
Lx,+/x,'" Lx, + /Lx, To show that it is one-to-one, we must show that Lx = 0 implies X ... O. But if Lx/(m)'" 0, then 4j(m)X(m) == 0 for all f. Hence, 1I",(X(m»'" 0 for all II", e r:,( M) by 4.2.14. Thus X( m) .. 0 by the Hahn- Banach theorem.
VECTOR FIELDS AS DIFFERENTIAL OPERA TORS
217
(il) This has an identical proof with the only exception that one works in a local chart, so it is not necessary to extend a linear functional to the entire manifold M as in 4.2.14. Thus the condition on the differentiability of the norm of the model space of M can be dropped. •
Now we are ready to give the main result on derivations.
4.2.18 Theorem. (i) If M is finite dimensional, the collection of all derivations on iJ(M) is a real vector space isomorphic to ~x'(M). In particular, for each derivation 8 there is a unique X E ~ ( M) such that 8 = Lx. (ii) Let M be a manifold modeled on a Coo Banach space E i.e. E has a Coo norm away from the origin. The collection of all (R-linear) derivations defined on Coo(M, F) (for all Banach spaces F), furms a real vector .fpace isomorphic to ~(M).
We prove (ii) first. Let 8 be a derivation. We wish to construct X such that 8, = Lx. First of all, we note that 8 is a loca/ operator; that is, if he Coo(M, F) vanishes on a neighborhood V of m, then 8(h)(m)= o. Indeed, let g be a bump function equal to one on a neighborhood of m and zero outside V. Thus h = (1- g)h and so Proof.
8(h)(m)
=
8(1- g)(m)·h(m)+8(h)(m)(l- gem»~ = O.
(I)
If U is an open set in M, and f E Coo(U, F) define (8/uXf)(m) = 8( gf)( m), where g is a bump function equal to one on a neighborhood of m and zero outside U. By the prevlOi.1~ remark, (81 UXf)(m) is independent of g, so 81 U is well defined. For convenience we write 8 = 81 U. Let (U, 'P) be a chart on M, m E U, and f E C oo ( M, F) where 'P: U -+ U' c E; we can write, for x E U' and a '"" 'P( m), ('P.f)(x)= ('P.f)(a)+1 1 aa ('P.f)[a+t(x-a)]dt 0 t
,
= ('P.f)(a)+ foID('P.f)[a+t(x-a)].(x-a)dt.
This formula holds in some neighborhood 'P(V) of a. Hence for u E V we have (2) feu) = f(m)+ g(uH'P(u)-a), whereg E Coo(V, L(E, Applying 8 to (2) gives
F» is given by g(u) = fJD('P.f){a + t('P(u)~ a)]dt.
, 8f(m) = g(m)· (8!p )(m) = D( 'P.f)(a)· (8'P)(m)
(3)
218
VECTOR FIELDS AND DYNAMICAL SYSTEMS
since 8 was given globally, (3) is independent of the chart. Now define X on U by its local representative
X.(x) - (x,e(4p)(u». where x - cp( u) E U'. It follows that XI U is independent of the chart cp and hence X E 'X(M). Then, for IE C«J(M, F), the local representative of LxI is
Hence Lx = 8. Finally, uniqueness follows from 4.2.14. The vector derivative property was used only in establishing (I) and (3). Thus, if M is finite dimensional and e is a derivation on ~ ( M), we have as before
I(u) - I(m)+ g(u)'(4p(u)-a) -/(m)+
"
L (c,I(u)-a')8,(u). ,-,
where g, E Cj(V) and
0= (0', ... ,0"). Hence (3) becomes
8/(m) =
L" gi(m)8(4pi)(m)
i-'
and this is again independent of the chart. Now define X on U by its local representative
(x, 8( 4p')( u), ... ,e( 4p")( u» and proceed as before. • There is a difficulty with this proof for derivations mapping C .. , to C. Indeed in (2), 8 is only C if I is C+', so e need not be defined on g.
VECTOR FIELDS AS DIFFERENTIAL OPERATORS
219
The result is. however. still true in this case. hut rcquires a different argument (see Box 4.2C). For finite-dimensional manifolds. the preceding theorem provides a local basis for vector fields. If (U. "'). "': U - VcR" is a chart on M defining the· coordinate functions Xi: U --+ R. define n derivations a/ax' on ~(U) by
These derivations are linearly independent with coefficients in
(Lli~)(XJ)=f'=O
Eli-;=O. then ax
ax
i_I
~'f(U).
for if
forall j=I ..... n.
since (a/axi)x i = 8/. By 4.2.16, (a/ax i ) can be identified with vector fields on U. Moreover, if X E 'X( M) has components Xl •...• X" in the chart ",. then
Lxl=X(fJ=
E
Xi
i-I
ali ax
=
(E, _ X'~)/. ax I
i.c.
X=
EX'~. ax'
i_I
Thus the vector fields ( a/ ax I). i - I •...• n form· a local basis for the vector fields on M. It should be mentioned however that a global basis of 'X( M). i.e.• n vector fields. XI ..... X n E 'X(M) that are linearly independent over ~(M) and span 'X(M). does not exist in general. Manifolds that do admit such a global basis for ~ (M) are called parallelizable. It is straightforward to sho~ that a finite-dimensional manifold is parallelizable iff its tangent bundle is trivial. For example. it is shown in differential topology that S3 is parallelizable but S2 is not (see Hirsch (1976». This completes the discussion of the Lie derivative of functions. Turning to the Lie derivative of vector fields. let us begin with the following.
4.2.17 Propoaltlon.
II X
and Yare C' vector fields on M. then [~x. Ly) H I( M. F) to C,-I(M. F).
... Lx Ly-:- Ly Lx is a derivation mapping C 0
0
Proof. More generally, let 81 and 6z be two derivations mapping C'· I to C and C to C- I • Clearly [8 1, 6z1 = 81 06z - 6z 08 1 is linear and maps C+ I to C- I . Also. if IE C'+ I (M. F), g E C+ I (M,G). and BE L(F,G; H),
220
VECTOR FIELDS AND DYNAMICAL SYSTEMS
then
(1 •• 8z](B(f. g» ... (I. o8z)(B(f. g»-(8z o8.)(B(f. g» - 1.{B(8z(f), g)+ BU. 8z(g))}- 8z{B(8.(f). g)+ BU.I. (g))} ==
B(I.(8z(f». g)+ B(8z(f), 8.(g»+ B(I.(f). 8z( g» + BU. 1.(8z( g»)- B( 8z(8.(f». g) - B(8.(f). 8z(g» - B(8z(f),I.(g»- BU.~(8.(g)))
=
B([8 •. ~](f). g)+
BU. [8 •. ~]( g».
•
Because of 4.2.16 the following definition can he given. 4.2.18 Definition. Let M be a manifold modeled on a COO Banach space M). Then (X, Y I is the unique vector field such that L [X. Y J = and X, Y E ( Lx, L y). This vector field is also denoted Lx Yond is called the Lk tkrivatit¥ 0/ Y willi ~$pect to X, or the Lk brtJcht 0/ X tuUI Y.
ex ""(
Even though this definition is useful for Hilbert manifolds (in particular for finite-dimensional manifolds). it excludes consideration of C' vector fields on Banach manifolds modeled on nonsmooth Banach spaces, such as L' function spaces for p not even. We shall, however, establish an equivalent definition, which makes sense on any Banach manifold and works for C' vector fields. This alternative definition is based on the following result. 4.2.18 Theorem. Let M be as in 4.2.18 and X, Y have (local) flow F,. Then
E
'X,(M) and let X
d dt (F,"'Y) = F,"'(LxY) (at those points where
Proof.
F, is defined).
If t = 0 this formula becomes
ddt
I,-0
F,"'Y= Lx Y
(4)
VECTOR FIELDS AS DIFFERENTIAL OPERA TORS
221
Assuming (4).
d (P,'Y)- .1dF:+.Y=P,' i d.1_ dI ~ ~
.-0
I.-0J;Y=p,"LxY'
Thus Jhe formula in the theorem is equivalent to (4). which is proved in the following way. Both sides of (4) are clearly vector derivations. In view of 4.2.16, it suffices then to prove that both sides are equal when acting on an arbitrary function f e COO( M. F}. Now
: (p'"Y)[f](m)l._o = =
:
I.-o{ df(m).(TF,(m,F._,
0
yo F,)(m)}
:1._op'"(Y[F!,f])(m).
Using 4.2.10 and Leibniz' rule, this becomes
X[Y[f)](m)- Y[X(fJ1(m) ... [X, Y][f](m) . • Note that in the preceding proof we derived the following fact of general utility: if.,: M ..... N is a' diffeomorphism and Y E ~. (N). then for
f e6J(M),
(5)
Since the formula for LxY in Eq. (4) does not use the fact that the norm of E is Coo away from the origin. we can state the following defmition of the Lie derivative on any Banach manifold M .
••2.20 Allermdlve DefInition of Lie bnIcket. If X,Ye~'(M), r;;.1 and X has flow F" the C- I vector field LxY - IX, Y] on M defined by [X,Y)- ddl I
,-0(p,'Y)
is called the Lk deriwltiw of Y with respect to X, or the Lk brrlebt of X andY. . From the point of view of this more general definition, 4.2.19 can be rephrased as follows .
••2.21 ~. Let X, Y e 6X(M), 140; r. Then IX, Y]- LxY is the uniqUe C- I vector field on M satisfying [X. Y][f] == X[Y[fJ] - Y[X(f]] for all f e C+ I(U, F), where U is open in M.
222
VECTOR FIELDS AND DYNAMICAL SYSTEMS
The derivation approach suggests that if X, Y E 6.X.'(M) then IX, Y] might only be C,-l, since [X, Y] maps C+ 1 functions to C- I functions, and differentiates them twice. However 4.2.20 (and the coordinate expression below) show that [X, Y] is in fact c- I. We can now derive the basic properties of the Lie bracket.
4.2.22 Propoeltlon. The bracket IX, Y] on 'X(M), together with the real v«tor space structure of 'X ( M), form a Lk lIIgebra. That is, (i) (,] is R bilinear: (ii) [X, X) - 0 for all X E 'X( M): (iii) [X,[Y, Z)]+(Y,[Z, X))+[Z.[X, Y)) = 0 for all X. Y, Z ( JlICObi ilkntily).
E ~X(M)
The proof is straightforward, applying the brackets in question to an arbitrary function. Unlike 'X(M), the space 'X'(M) is not a Lie algebra since [X. YJE 'X,-I(M) for X. Y E 'X'(M). Note that (i) and (ii) imply that [X. Y] = - [Y. X). for
[X + Y. X+ YJ -0 "" [X. X)+[X. Y)+ [yo X)+ I Y. Y). Also, (iii) may be written in the following suggestive way: Lx[Y. Z) = [LxY, Z) + [y, LxZ):
i.e., Lx is a Lie bracket derivation. Strictly speaking we should not use the same symbol Lx for both definitions of Lxf and LxY. However. the meaning is generally clear from the context. The analog of 4.2.7 on the vector field level is the following.
4.2.23 PropoalUon. (i)
Let ep: M
--+
N be a diffeomorphism and X
E
'X ( M). Then Lx: 'X ( M) --+ ~ ( M) is natural wit" IYspen to p_-1ol'WtU'd by ep. That is. L".xep.Y" ep.LxY, or [ep.X, ep.Y) = ep.[X. Y), or the following diagram commutes: 'J'.
CX(M)--• CX(N)
I
LxI
L".x
CX(M)-- ~(N)
••
VECTOR FIELDS AS DIFFERENTIAL OPERATORS
223
(ii) Lx is """"'" witII rnp«t 10 rntrictiolu. ThaI is. for U c M open, we have [XI U, fIUJ- IX, fHU; or Ihe following diagram commule.f: IV ~(M)----' ~(U)
Lxl ~(M)
Proof.
any Z
we get
!L xlu ~(U)
IV
For (i),letf E "(V), vbe open in N. and tp(m) = n E V. By (5). for E ~x'(M)
fro~
«IP.Z }[f])(n)
= Z [f
° IP)( m).
4.2.21
(IP.[X, f])[f]{n) - [X, f](f°IP]{m)
... X[f(f ° IP ]](m)- f[X(f ° IP ]](m) = X [( IP.Y Hf] ° IP ](m)- f [( IP. X
)(f] ° IP](m)
... (IP.X)[( IP. f)[ f]]( n) - (IP. Y)[ (IP.X)[ f]]( n) ... [IP.X.IP.Y)[f)(n).
Thus IP.[X, fJ- [IP.X,IP.fJ by 4.2.15(ii). (ii}"rollows from the fact that d(1I U) ... d/I u. • let us now compute the local expression for [X. fJ. Let IP: U ... VeE be a chart on M and let the local representatives of X and f be X and Y respectively, so X, Y: V ... E. By 4.2.22, the local representative of IX, fJ is
[X, fJ. Thus,
[X, t][JJ(x) - X[y[JJ](x)- Y[X [JJ](x) ... D( y[j])(x )·X(x)- D( X[i])( x )·Y(x) .. Now f[lJ<x)- D](x)'Y(x) and its derivative may be computed by the product rule. The terms involving the second derivative of ] cancel by symmetry of D 2J(x} and so we are left with
D](x)· (DY{x).·X(x)- DX(x)· Y(x )}.
224
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Thus the local representative of (X. Y} is
DY·X-DX·Y. If M is n-dimensional and the chart '" gives local coordinates (xl •...• x") then this calculation gives the components of IX. Yl as
(6) Part (i) of 4.2.22 has an important generalization to ",-related vector fields. For this. however. we need first the following preparatory proposition.
4.2.24 Propoeltlon. Let "': M -+ N be a C' map of C' manifolds. X E CX,-I(M). and X' E CX,-I(N). Then X - X' iff (X1f)) ° '" - Xlf ° "'lfor all
f
"
E C§'I(V). where V is open in N.
Proof. By dermition, «X11l)o ",)(m) -
d!(",(m»·X'(",(m». By the chain
rule. X(f ° ",)(m) ... d(fo ",)(m)·X(m) .. df( ",(m»·T",,,,(X(m».
If X! X'. then T", ° X - X' ° '" and we have the desired equality. Conversely. if X[f ° "'] - (X'lll)° '" for all f E '!f1(V), and all V open in N, then choosing V to be a chart domain and f the puD-back to V of linear functionals on the model space of N. we conclude that II,,· (X' ° '" )( m) - II,,· (T", ° X)(m), where n = ",(m). (or all II" E Using the Hahn-Banach theorem. we conclude that (X'o ",)(m) = (T", 0 X)(m). for all mE M. • , It is to be noted that under dirferentiability assumptions on the norm of N (as in 4.2.16), the condition "for a1lf E '!f1(V) and all V eN" can be replaced by "for all f E 6J1(N)" by using bump function', This holds in particular for Hilbert (and hence for finite-dimensional) n.illifolds.
r:N.
4.2.25 PropoeHIon. Let "': M -+ N be a C'map of manifolds. X. Y E CX·-I(M). and X',Y'E~:-I(N). If X!.X, and Y!Y'. then
IX. Y] !·[X'. Y1.
·VECTOR FIELDS AS DIFFERENTIAL OPERATORS
225
Proof.' By 4.1.14 it suffices to show that ([X'. Yll1) ° cp = lX. YII cp] for aU I e ~I(Y). where V is open in N. We have 0
([ X'. Y'][/» ° cp'" X'[Y'[/]]· cp - Y'[X'[ I]] cp 0
- X[(Y'[/). cp)- Y[( X'[/). cp) ==
X[Yl/ 0cp)) - Y[xl/ 0.,,)]
==
[X. Y][/·.,,] . •
The analog of 4.2.9 is the following.
4.2.21 Propoeltlon. For every X e ~ (M). Lx ;s a derivation on (Cft( M). ~(M». That is. Lx is R linear on each. and Lx(/ Y) - (I.x/)Y + I(LxY).
' For g E coo(U, E), where U is open in M, we have
Proof.
[X,/y][g] - Lx{Lfyg)- LfyLxB - LxULyg)- fLyLxB - (Lxf}Lyg + fLxLyg- fLyLxB. so [X,/y) - (Lxf}Y + /[X, Y] by 4.2.IS(ii). • Commutation of vector fields is characterized by their nows in the followin& way.
4.2.27 PropoeIUon. Let X, Ye ~r(M), r .. I, and let flows. The loIlowlng are equioalent.
F"G, denote their
(i) lX, y) ... 0; (ii) FiY-,Y;
G:
(iii) X -' X; (iv) F, • G. - G. ° F,. (In (ii)-(ro), equoIity is undentood, as usual, where the expressions are
defined.)
F, • G• .. G, 0 F, iff G. - F, • G. • r, I. which by 4.2.4 is equivalent to y ... FiY; i.e., (iv) is equivalent to (ii). Similarly (iv) is equivalent to (iii). If FiY-Y, then
Proof.
[X,Y)=
dtdl ,-0F,*Y""O.
226
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Conversely. if IX. YJ - LxY "" O. then·
so that p,'Y is constant in t. For t = O. however. its value is Y. so that p,'Y - Y and we have thus showed that (i) and (ii) are equivalent. Similarly . (i) and (iii) are equivalent. • Just as in 4.2.10, the formula for the Lie derivative involving the flow can be used to solve special types of first-order linear n x n systems of partial differential equations. Consider the first-order system: .
f. (XJ(K)aY'(K.t)_YJ(K.t)aXi(K.t») ax' ax}
aaYi(K.t)", (P,,) {
t
J_ I
yl(K.O) "" gi(K) for given smooth functions Xi( K). gi( K) and scalar unknowns yi( K. t ). i = 1•..•• n. where K -= (Xl •. .•• x"). 4.2.28 Proposition. Suppose X= (XI •.... X") hav a complete /1011' F,. Then letting Y == (yl •...• Y") and G = (gl ..... g,,).
y= p,'G is a solution of the foregoing problem (P,,). (See Exercise 4.2C for uniqueness).
ay
at = =
d
dt p,'G = P,'[ x. G) = [F,* X. p,'G)
[X. Y)
since P,' X = X and Y = p,'G. The expression in the problem (P,,) is the ith component of IX. Y). • 4.2.21 Example. ayl
-
at
ay2
Solve the system of partial differential equations: =
ayl
ayl
ay2
ay2
(x+ y ) - -(x+ y) _ _ yl_y2 ax ay
-=(x+y)--(x+ .-)_+yl+y2 at ax . ay yl(X. y.O)
=
x
y2(X. y.O) = y2.
VECTOR FIELDS AND DIFFERENTIAL OPERA TORS
227
The vector field X(x, y) = (x + y, - x - y) has the complete flow F,(x, y) "" «x + y)1 + x, -(x + y)t + y), so that the solution is given by Y(x. y, t) ... F:(x, y2); i.e., { .
yl( x, y, t) ...
«x + y)t + x)( 1- t) - I [y - (x + y)t]2
y2(x, y, I) = t«x
+ Y)I + x)+(t + I)[y - (.t + y)tY. •
In later chapters we will need a flow type formula for the Lie derivative of a time-dependenl vector field. In Section 4.1 we discussed the existence and uniqueness of solutions of a time-dependenl Vl'l'lor field. Let us formalize and recall the basic facts.
4.2.30 DaflnlUon. A C tiIM-depent/Dft vector field i.f a C map X: RxM ..... TM such that X(t,m)ETmM for all (t,m)E:RXM; i.e:, X,E ~.'(M),
lor
where X,(m) = X(t, m). The time-de",_nt flo ... or ft:JOhIlw"opelYl-
F, .• of!l is defined by the requirement thai I .... F, ..,(m) be Ihe integral
curoe of X slarling at m altime t ""' s; i.e.,
d dl F, .• (m) = X(/, ~ .• (m»
and
F. .• (m) = m.
By uniqueness of integral curves we have F, .• F,,, = F,., (replacing the flow property F,+. = F, F.), and F,., = identity. It is customary to ~rite F, - F,.o· If X is time independent, F, .• " F,- •. In general F:X," X,. However, the basic Lie derivative formulae still hold, 0
0
4.2.31 Theorem. Lei X, E ~'(M), r., I for each t and suppose X(/, m) is continuous in (I, m). Then F,.• is of class C' and for f E C+ '(M, F), and Y E 'X'(M). we have
(i) d dlFi..Y= Fi..([X,. Y)) = Fi..(Lx,Y).
(ii)
Proof. The rarst part was proved in Section 4.1. For functions, the proof is a repeat of 4.2.10:
~(Fi..f)(m)- ~(fo F, .• )(m)=df(F, .• (m»· dF,'d!m) - d/(F,.,(m»·X,( F, .• (m» == (Lx,/)( F,.Am»= Fi..(Lx./)(m).
228
VECTOR FIELDS Afl/D DYNAMICAL SYSTEMS
For vector fields, note that from (5), (~.Y)[f1 = ~.(Y[F;,/])
(7)
since F.., = F,~". The result (ii) will be shown to follow from (i), (7). and the next lemma.
4.2.32 Lemma. d
dtF;,/= - X,[~,/]. Proof. Differentiating ferential equation:
F.., F, .• = 0
identity in t. we get the backward dif-
d
dt F.., =
-
TF.., 0 X,.
Thus
d
dt F:"f(m) - - dfU:.,(m »·TF.,,( X,(m»
... - d(f 0 F..,)·X,(m) = - X,(f 0 F..,](m).
9
Thus from (7) and (i). d
dt (~.Y)[f] -= ~.( X, [Y[F:,,/]])- ~.(Y[X,[~,fJ]). By 4.2.21 and (7). this equals (~.[X" Y)[f). • If f and Yare time dependent, then (i) and (ii) read
(a
d -P. dt , ...1 =P. ,.. -atf+ L x/ )
(8)
and
d
dt~'Y= ~s
(ay at + [X,. Y,] ).
(9)
Unlike time-independent vector fields. we generally have
Time-dependent vector fields on M can be made into time-independent ones on a bigger manifold. Let , E ~ (R x M) denote the vector field given
VECTOR FIELDS AND DIFFERENTIAL OPERATORS
229
Figure 4.2.4
by '(s, m) -
(~,
1),Om) e 1(•. m)(R X ~);; T,R X TmM. Define the .suspen-
sion of X by Xe 'X,(R X M) where XU, m) - «t, 1), X(t, m» and observe that X - , + X. Since b: I .... M is an integral curve of X at miff b'( I) ... X(t, b(t» and b(O) ... m. a curve c: J .... R X M is an integral curve of X at (0, m) iff c(t) - (t. b(I». Indeed, if c(1) == (a(t), b(/» then C(/) is an integral curve of X iff C'(I)" (a'(t), b'(/» = X(C(I»; that is a'(t) == I and b'(/) - X(a(/), b(/». Since a(O) - 0. we get aCt) - t. These observatians are
summarized in the following (see Fig. 4.2.4.)
,
4.2.33 P~ltIon. Lei X be a C-time-dependent veclor field on M wilh evolution oPerator F"., The flow F, of the suspension Xe CX'(R X M) is given by F,(s. m) = (t + s. F,+.,.(m». • Proof. In the preceding notations. b(l) = F"o(m). c(t) = F,(O, m) = (I. F"o( m», and so the statement is proved for s = O. In general. note that ~(s. m) = F,+s(O. Fo,.(m» since 1 - F, .. • (0. i;,.•(m» is the integral curve of X. which at 1=0 passes through F,(O.m)=(s.U:.nol·;,.,)(m»=(s.m). Thus F,(s. m) = F,+$(O. Fn. ,(m» = (I + s. (F,+"II ° 1;,. • )(m» = (t + .f. F,+s .• (m». •
BOX 4.2A
AUTOMORPHISMS OF FUNCTION ALGEBRAS
The property of flows corresponding to the derivation property of vector fields is that they are algebra preserving:
P,'(fg) ... (P,'f)( P,'g).
230
VECTOR FIELDS AND DYNAMICAL SYSTEMS
In fact it is obvious that every diffeomorphism induces an algebra automorphism of ~ ( M). The following theorem shows the converse. t 4.2.34 Theorem. Let M be a paracompact second-countable finitedimensional manifold. Let a: ~(M) - ~(M) be an invertible linear mapping that satisfies a(fg)=a(f)a(g) lor all I. g E ~(M). Then there is a unique Coo diffeomorphism fJ!: M - M such that a(f)=fOfJ!.
Remark. A. There is a similar theorem for paracompact second-countable Banach manifolds; here we assume that there are invertible linear maps a.T: COO( M. F) - C oo ( M. F) for each Banach space F such that for any bilinear continuous map B: F x G - H we have
for I ECoo(M. F) and gECOO(M.G). The conclusion is the same: there is a Coo diffeomorphism fJ!: M - M such that
for all F and all I E C oo ( M. F). Alternative to assuming this for all F. one can take F = R and assume that M is modeled on a Banach space that has a norm that is Coo away from the origin. We shall make some additional remarks on the infinite-dimensional case in the course of the proof. B. Some of the ideas about partitions of unity are needed in the proof. Although the present proof is self-contained. the reader may wish to consult Section 5.6 simultaneously. C. In Chapter 5 we shall see that finite-dimensional paracompact manifolds are metrizable. so by \.6.14 they are automatically second countable. Proof. (Uniqueness). We shall first construct a Coo function x: M - R which takes on the values I and 0 at two given points mi' m2 E M. m, tWe tIumk Alan Weinstein for suggesting this theorem. We think this result must be known in the literature. but we could not locate a specific reference except for an analogous result of Mackey (1962) for measurable functions and measurable auto-
morphisms.
VECTOR FIELDS AND DIFFERENTIAL OPERA TORS
23.1
"" m 2 • Choose a chart (U. cp) and mi' such that m 2 1i:. U and such that cp( U) is a ball of radius r, about the origin in E. cp( m, ) = O. Let V C U be the inverse image by cp of the ball of radius r2 < r, and let 8: E -+ R be a Coo-bump function as in Lemma 4.2.13. Then the function x: M -+ R given by .
80 CP.
x= { 0,
U M\U
on on
is clearly Coo and x(m,) = I. x(m 2 ) = O. Now let us assume that cp./ == If·/ for two different diffeomorphisms "'.If of M. for all/ E tj(M). Then there exists a point me M such that cp(m)-If(m) and thus we can find XE~'t<M) such that (X 0 cp)(m) = I,(X 0 If)(m) = 0 contradicting cp·X = If·X. Hence cp == If. The proof of existence is based on the following key lemma. 4.2.35 Lemma. Let M be a (/inite-dimensiona/) parucompact mani" /old and p: 'fJ( M) -+ R be a nonzero algebra homomorphism. Then there is a unique point m E M such that PU) /(m'>. =0
Proof. (Following suggestions of H. Bercovici). Uniqueness is clear. as before. since for m, "" m 2 there exists a bump function / E tj(M) satisfying/(m,) == O. /(m 2 ) = I. To show existence. note first that P(l) = 1. Indeed P(I) = P(12) = P( I )P( 1) so that either P( I) = 0 or P( I) = 1. But P< I) = 0 would imply P is identically zero since PU) = P(l·/) = P(I)·PU). contrary to our hypothesis. Therefore we must have P( I) = I and thus P( c) = c for cER. For mEM, let Ann(m) = {/E tj(M)I/(m) = O}. SecOnd, we claim that it is enough to show that there is an mE M such that kerp={/E~T(M>lPU)=O}==Ann(m). Clearly. if ~U)-/(m) for some m, then kerp = Ann(m). Conversely, if this holds for ~ mE M and / It:. kerp, let c,. PU) and note that / - cE kerPAnn(m), 'SO/(m) == c and thus P
232
VECTOR FIELDS AND DYNAMICAL SYSTEMS
and hence/ e g + / c J; i.e.• / -IJ(M). Similarly. the ideal Ann(m) is maximal since the quotient IJ(M)/Ann(m) is isomorphic to R. Let us assume that k.er fl - Ann( m) for every m e M. By maximality. neither can be included in the other. and hence for every
m e M there exists a relatively compact open neighborhood U", of m and /", e kerll such that /",1 U", > O. Let V", be an open neighborhood of the closure. c1(U",). Since M is paracompact. we can assume that {V",lm eM} is locally finite. Since M is second countable. M can be covered by {V",V eN}. Let ~ = /m and let Xj be bump functions equal to 1 on cl(U",) and v~ishing in M\Vm • If an < IlIn 2sup(x,,(m)/,,2(m)lm eM}). the function I = l::"_la nXnln2 is C"" (since the sum is finite in a neighborhood of every point). I> 0 on M and the series defining I is uniformly convergent. being majorized by E:'_ln-2. If we can show that II can be taken inside the sum. then lIe I) = o. This construction then produces I e ker11./ > 0 and hence 1'"' (I/!)/ e kerll; i.e.• kerll = IJ(M). contradicting the hypothesis fl .. O. To show that II can be taken inside the series sign. it suffices to prove the following "g-estimate:" for any g e IJ(M), III(g)1 '" sup(lg(m~lm eM}. Once this is done, then
~ supl r, a"x .. /"2 " -I
II-
0
as n -+ 00 by uniform convergence and boundedness of all functions involved. Thus 11(/) = E':_lfl(a"x"I,,2). To prove the g-estimate. let " > sup(lg(m~lm e M} so that" ± g "" 0 on M; i.e.• "± g are both invertible functions on M. Since II is an algebra homomorphism. 0-11(" ± g) -" ± lI(g). Thus. ± fl(g) """ for all sup(lg(m~lm eM}. Hence III(g)1 '" sup(lg(m~lm eM}. •
,,>
Aemn. For infinite-dimensional manifolds. the proof of the lemma is almost identical with the following changes: we work with fl: COIl ( M. F) -+ F. absolute values are replaced by norms. second countability is in the hypothesis and the neighborhoods V", are chosen in such a way that 1",1 Vm is a bounded function (which is possible by continuity of 11ft)'
VECTOR FIELDS AND DIFFERENTIAL OPERATOR8
233
Proof of em'... iII·f 71. For each mE M, define the algebra • R by /J",(f) - a(f Xm). Since a is inhomomorphism /J",: . vertible. a(l) .. 0 and !oil i~'; a(1) - a(12) - a( 1)a(1). we have a(1 ).- I. Thus /J",'." 0 for all m E M. By 4.2.35 there exists a unique point, which we call .,,(m)EM such that /J",(f)=/(.,,(m»-(.,,*/Xm). This dermes a map.,,: M - M such that a(f)- rp*/for alii E 'J(M). Since /I is an automorphism, " is bijective and since a(f) - .,,*1 E ~(M),a-I(f) - .".1 E 'J(M) for all I E 'J(M), both rp, rp- I are C'" (take for I any coordinate function multiplied by a bump function to show that in every chart the local representatives of cpo rp - 1 are C"') .
•
The proof of existence in the infinite-dimensional case proceeds in a similar way.
BOX 4.28 THE METHOD OF CHARACTERISTICS
The method used to solve problem (P) also enables one to solve first-order quasi-linear partial differential equations in R". Unlike 4.2.1 J. !he solution will be implicit, not explicit. The equation under consideration in R" is
.;. " I) al. -I... Xi( x 1 ,. ..• x. i-I
I
ax"
y( x 1•...• x. " I) ,
(Q)
where/=/(xl, ... ,x") is the unknown function and Xi,Y.i-J .... ,n are C' real-valued functions on R II + I, r ~ I. As initial condition one takes an (n -I)-dimensional submanifold r in R"+ 1 that is nowhere tangent to the vector field
Thus, if r is given parametrically by X ' ... X i (t l , ... ,t"_I),i==1 ..... n
and
1-=1(11 ..... 1"-1).
234
VECTOR FIELDS AND DYNAMICAL SYSTEMS
this requirement means that the matrix
x· ax·
at;
X" ax" a/.
al art
ax· al,,_.
ax" al"_.
al al"_.
y
has rank n. It is customary to require that the determinant obtained by deleting the last column be '* O. for then, as we shall see. the implicit function theorem gives the solution. The function / is found in the following way. Consider F,. the flow of the vector field r a/ax; + ya / 1 in R" + • and let S be the manifold obtained by sweeping out r by F,. That is. S'" U{F,(f)11 E R}. The condition that xia / axi + ya/al never be tangent to r ensures us that the manifold r is "dragged along" by the flow F, to produce a manifold of dimension n. If S is described by / .. /(x· •...• x") then 1 is the solution of the partial differential equation. Indeed. the tangent space to S contains the vector E7_.X 1a/8xl + Y8/a/; i.e.• this vector is perpendicular to the normal (81/ax· ..... al/ax".-I) to the surface/=/(x· ..... x") and thus (Q) is satisfied. We work parametrically and write the components of F, as Xi=XiU ...... I"_ •• /). i=I ..... n and /=1(1 ...... 1,,-1./). Assuming that
r:.7_. r:.7_.
a
x· ax·
o.
at; ax· al"_.
X" ax" a/. ax" al"_ •
ax· al ax· a/. ax· ar" _.
ax" al ax" a/. ax" ar"_.
one can locally invert to give 1= r( x· .... ,x"), I, = r,( x·, .... x"), i = I ..... n. Substitution into / yields / = /(x· ..... x"). The fundamental assumption in this construction is that the vector field E':_lxla/aXi + ya/al is never tangent to the (n - 1)manifold r. The method breaks down if one uses manifolds r. which are tangent to this vector field at some point. The reason is that at such a point. no complete information about the derivative of 1 in a
VECTOR FIELDS AND DIFFERENTIAL OPERA TORS
235
complementary (n - 1)-dimensional subspace to the characteristic is known. 4.2.38
Examples.
with initial condition one-manifold
Consider the equation in R 2 given by
r=
{(x. y. f)lx ... s. y'" !S2 - S. f = s}. On· this
f
s s- 1
=-1*0
so that the vector field a/ax + fa/ay +3a/afis never tangent to r. Its flow is F,(x. y. f) = (t + x.(3/2)t 2 + fl + y.3t + f) so that the manifold swept out by r along F, is given by x(t. s) = t + s. y(t. s) = (3/2)/ 2 + sl + (I/2)S2 - s. f(/. s) = 3t + s. Eliminating I. s we get
The solution is defined only for I - 2X2 + 4x + 4.1' ~ O.
•
. Another interesting phenomenon occurs when S can no longer be described in terms of the graph of f: e.g.• S "folds over." This corresponds to the formation of shock waves. Further information can be found in Chorin and Marsden (1979), Lax (1973). Guillemin and Sternberg (1977), John (1975), and Smoller (1982).
4.2C, DERIVATIONS ON
C" -1
FUNCTIONS
4.2.37 Theorem (A. Blasst). Let M be a C A , 2 finite-dimensional manifold: where k ~ O. The colleclion of all R linear derivations 8 from ':t h + I ( M ) to 'ff" ( M) is isomorphic to 'X" ( M) as a real t'ector space. The proof of 4.2.16 sho~s that there exists a unique C" vector field X such that 81~k+2(M)= Lxl~"+2( M). So all we have to do is show that 8 and Lx agree on C"+ I functions. Replacing 8 with e- Lx. Proof.
t
Private communication.
236
VECTOR FIELDS AND DYNAMICAL SYSTEMS
we can assume that 8 annihilates al1 C k + 2 functions and we want to show that it also annihilates all C k + 1 functions. As in the proof of 4.2.16, it suffices to work in a chart, so we can assume without loss of generality that M-R" . .For a Cl<+ 1 function / on R" we shall prove that (8/)(0) = 0 which implies that (8/)( p) = 0 for all pER n by translation of the argument of f. Replacing / by the difference between / and it~ Taylor polynomial or order k + I about 0, we can assume /(0) = 0, and the first k + I derivatives vanish at 0, since 8 annihilates any polynomial. We shall prove that / = g + h, where g and h are two Ck+ 1 functions, satisfying gl U - 0 and h IV = 0, where U, V are open sets such that OEcI(U)ncl(V). Then, since 8 is a local operator, 8gIU=O and 8hlV- 0, whence by continuity 8glcl(U) - 0 and 8hlcl(V) = O. Hence (8/)(0) ... (8g)(O) + (8h)(O) - 0 and the theorem will be proved.
4.2.38 Lemma. Let cp: S,,-I c R" -+ R be a Coo function and denote by.,,: R"\{O} -+ S,,-I, .,,(x) = x/llxll the radial projection. Then lor any positive integer r D'( cp 0.,,)( x) = for some C"" functlon.p: S,,-I
-+
(.p 0.,,)( X Vllxll'
L~(R";R). In particular Dk(cp
0
'II")(x)
- O(lIxlI-') as IIxll- O.
Proof. For r - 0 choose cp =.p. For r -I, note that I V D'II"(x)·v= - -DII·II(x)·v+-
IIxll2
II xII ,
and so
But the mapping i(x) = (1/lIxll)DII·II(x) satisfies i(lx) = i(x) for all t > 0 so that it is uniquely determined by 1= iIS,,-I. Hence D( cp • .,,)(x) - (I/II,I"II)(.p o.,,)(x), where .p( y) = Dcp( y)(I y». y E s"-'. Now proceed by induction. • Returning to the proof of the theorem. let I be as before (i.e .. of class C k and Di/(O)'" 0, 0 "'" i "'" k + 1) ·,nd let cp.'" be as in the lemma. From Taylor's formula with! we see that Di/(x) = o(II.I"IIk+ 1-').0"," i "'" k + I. as x -+ 0 ; the product rule and
-ie
+,
VECTOR FIELDS AND DIFFERENTIAL OPERATORS
237
the lemma, DI(f'(fJJO"»(x)=
I:
O(UXUk+I-J)O(Uxlrl)=O(nXUHI-i)
j+l-i
,,»,
so that DIU' (fJJ 0 0 " i " k + I can be continuously extended to 0, by making them vanish at O. Therefore f· (g 0 , , ) is C h I for all R ". Now choose the CfXJ function fJJ in the lemma to be zero on an open set 0, and equal to I on an open set O2 of S"-I,O, noz -=121. Then the continuous extension g of f'(fJJ 0 , , ) to R" is zero on U=,,-'(Ol) and agrees withfon V-".-I(OZ)' Let h-f-g and thus f is the sum of two C k + I functions, each of which vanishes in an open set having 0 in its closure. This completes the proof. • , In infinite dimensions, the proof would require the norm of the model space to be Coo away from the origin and the function'" in the lemma bounded with all derivatives bounded on the unit sphere. Unfortunately, this does not seem feasible under realistic hypotheses. The foregoing proof is related to the method of "blowing-up" a singularity-see for example Takens [1974] and Buchner, Marsden, and Schecter [1982]. There are also difficulties with this method in infinite dimensions in other problems, such as the Morse lemma (see Golubitsky and Marsden [1983] and Buchner, Marsden, and Schecter [ 1983».
4.2.39 Corollary. If M is a C k + I finite-dimensional manifold, the only derivation from Ijk( M) to Ijk( M), where I " k < 00, is zero.
.
Proof. By the theorem, such a derivation is given by a' C k - I vector field X. If X-O, then for somefeljk(M),X[f) is only C k - ' but not Cle. •
We do not know of an example of a derivation not given by a vector field, when the hypotheses of 4.2.37 fail.
BOX 4.2D PRODUCT FORMULAS FOR THE LIE BRACKET
Tltis box is a continuation of Box 4.lA and gives the flow of the Lie bracket [X, Y] in terms of the flows of the vector fields X, Y e 'X(M).
238
VECTOR FIELDS AND DYNAMICAL SYSTEMS
".2.40 Propolltlon. LeI X, Y E ~(M) have flows F, and G,. If B, denotes the flow of(X, Y), then for x EM,
Proof. Let
K.(x) - (G- i
0
F _.j 0 G.j 0 F.j)(x), e> O. The formula
rollows from 4.1.24 if we show that
aae K.(x)1
.-0
=
lX, Y](x)
ror all x E M. This in tum is equivalent to
..!.K:fl
ae
.-0
=
[X. Y}[f]
for any f E C!f(M). By the Lie derivative formula,
By the chain rule. the limit of this as e - 0 + is half the a/ a.fE -derivative of the parenthesis at e = O. Again by the Lie derivative formula, this equals a sum of 16 terms. which reduces to IX. Y][f]. • For example. for n X n matrices ..4 and B, 4.2.40 yields the classical formula e IA .-]= lim (e- A/-!ie--/ ii e A/-!ie·/.,4i)".
"-00 where the commutator is given by (..4. B)-..4B - B..4. To see this, define for any n X n matrix C a vector field Xc E ~(R") by Xc(x) = Cx. Thus Xc has flow F,(x) == e'cx . Note that Xc is linear in C and satisfies (XA • X.)- - XIA ._] as is easily verified. Thus the flow of IX., XA ) is e'IA._] and the formula now follows from 4.2.40. The results or 4.1.25 and 4.2.40 had their historical origins in Lie group theory, where they are known. by the name of exponential /omtUIas. The converse of 4.1.25, namely expressing e,Ae,· as an exponential of some matrix for sufficiently small t is the famous Baker-Campbell-Hausdorff formula (see e.g. Varadarajan (1974, sec.
VECTOR FIELDS AND DIFFERENTIAL OPERATORS
239
2.15). The formulas in 4.1.25 and 4.2.40 have certain generalizations to unbounded operators and are called TrOller product formula., after Trotter (1958). See Chorln et. al. (1978) for further information. Exercl...
4.2A
(i) On R2. let X(x. y) == (x. y; y. - x). Find the flow of X. (ii) Solve the following for f(t. x, y):
af -y~-X af at
ax
£ly
if f(O. x. y) ... ysin x. 4.2B (i) Let X and Y be vectors fields on M with complete flows F, and G,• respectively. If [X. Y) = 0, show that X + Y has flow H, "" F, 0 G,. Is this true if X and Yare time dependent? (ii) Show that if [X, Y) = 0 for all Y E <:\ ( M). then X = O. ( Hint: From the local formula conclude first that X is constant: then take for Y linear vector fields and apply the Hahn..:.Banach theorem. In infinite dimensions, assume the conditions hold locally or that the model spaces are ex». 4.2C Show that. under suitable hypotheses, the solution f(x, t) = g( F,(x» of problem (P) given in 4.2.11 is unique. (Jlint: Consider the function E(t) = fRol/,(x, t)- 12(X' 1)1 2 dx, where II and 12 are two solutions. Show that dE/dt ~ aE for a suitahle constant a and conclude by Gronwall's inequality that E = O. The "suitable hypotheses" are conditions that enable integration by parts tQ be performed in the computation of dE/dt.) Adapt this proof to get uniqueness of the solution in 4.2.28. 4.2D Let X. Y E 'X( M) have flows F, and G" respectively. Show that
[X'Y]-=/~tl s
1•• -0
(Hint: The flow of p,'Y is s ..... F, 0 G.
4.2E 4.2F
(F,oG,oF , ), 0
F _ I')
Show that SO(n) is paralleIizable. See Exercise 3.5S for a proof that SO(n) is a manifold. (Hint: SO(n) is a group.) Solve the following system of partial differential equations. ayl
-
at
ay2
ayl
ayl
ax
ay
yl- y2
ay2
ay2
I
= (x + y)-.- +(4x -2y)- -
Tt = (x + y)
ax +(4x -2y) ay -4Y +2Y
2 •
240
VECTOR FIELDS AND DYNAMICAL SYSTEMS
with initial conditions yl(X. y.O) = x
+y
(Hint: The flow of (x + y.4x -2y) is
4.20 Consider the following equation for I(x, t) in dit1er1{ence lorm:
ill + ~(H(f)) =0 ilt ilx where H is a given function of f. Show that the characteristics are given by .k - H'( I). What does the transversality condition discussed in Box 4.2B become in this case? 4.2H Let M. N be manifolds with N modeled on a Banach space which has a C" norm away from the origin. Show that cp: M -+ N is C" iff f 0 cp: M -.R is C· for all/ E '!f"(N). 4.21 Develop a product formula like that in Box 4.1A for the flow of X + Y for time-dependent vector fields. (Hint: You will have to consider time-ordered products.)
4.3 AN INTRODUCTION TO THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS We have seen quite a bit of theoretical development concerning the interplay between the two aspects of vector fields. namely as differential operators and as ordinary differential equations. It is appropriate now to look a little more closely at what flows look like geometrically. Much of the work in this section holds for infinite-dimensional as well as finite-dimensional manifolds. The reader who knows or is willing to learn some spectral theory from functional analysis can make the generalization. This section is intended to link up the theory of this hook with I.'ourses in ordinary differential equations that the reader may have taken. The section will be most beneficial if it is read with such a course in mind. We begin by introducing some of the most basic terminology regarding the stability of fixed points.
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
241 '
4.3.1 Definition. Let X be a C l vector field on 1II1 n-man;fold M. A point mo is called a cmictll po;"t (also called a s;"plar point or an ('qui/ibri,,", poiItt) of X if X( mo)" O. The Iilteari'{.lItion of X at a critical point mo is the linear map X'( mo): T",.M - T",.M defined by X'(m o )'''= dd (TF,(mo)·.,)1 t /-0 where F, is the flow of X. The eigenvalues of X'( m 0) are called the clu"uctemtic exponents of X at mo.
Some remarks will clarify this definition. First of all. F, leaves mo fixed: F,(mo)=m o' since c(t)=m o is the unique integral curve through mo' Conversely. it is obvious that if F,( mo) = mo for all I. then is a critical point. Thus TF,(mo) is a linear map of TmoM to itself and so its t derivative at O. producing another linear map of Tm.M to itself. makes sense. '
mo
Computationally. this definition is not so convenient. The following is more useful. 4.3.2 PropoalUon. Let mo be a critical point of X (/lid let (V.,,) he a chart on M with cp(m o ) - Xo e R". Let x -= (xl •...• x") denote cOlJrdinateJ in R" and XI(XI ..... X") ..... X"(x· ..... x") the component,l' of Ihe local repr.esentalive oJ X. Then the matrix of X'( mol in these coordinates is
Proof.
This follows from the equations d
XI( F,(x» = dt F,'(x) after differentiating in x and setting x equations in Lemma 4.1.9) •
= XO'
t = 0, (Sec also the linearized
'
Another. equivalent procedure for linearizing. which generalizes to sections of arbitrary vector bundles. is outlined in E~ercise 4.3A. The name "characteristic exponent" arises as follows. We have the linear differential equation
242
VECTOR FIELDS AND DYNAMICAL SYSTEMS
and so TWlo F.I = e,x·(m.) •
Here the exponential is defined. for example. by a power series. The actual computation of these exponentials is learned in differential equations courses, using the Jordan canonical form. (See Hirsch-Smale (1974). for instance.) In particular. if ~I"" '~n are the characteristic exponents of X at mo. the eigenvalues of T....F, are e'''' ..... e'''·. The characteristic exponents will be related to the following notion of stability of a critical point.
4.3.3 DeflnlUon. Let mo be a critical point of X. Then mo is sl'" (or Lillpulrov stabk) if for any neiKhborhood U 01 mo. there is a neighborhood V of mo such that if m E V. then m is + complete and F,(m) e U for all t ~ O. [See FiK. 4.3.I(a).) (ii) mo is IUY"'PIOIically stable if there is a neighborhood V of mo such that if me V. then m is + complete. F,(V)C F,.(V) if t > sand (i)
lim F,(V) 1-
+ 00
=
{mo}
[i.e .• for any neighborhood U of mo. there is a T such that F,( V) c U if t ~ T). [See Fig. 4.3.l(b).)
It is obvious that asymptotic stability implies stability. The harmonic oscillator x = - x giving a flow in the plane shows that stability need not imply asymptotic stability [Fig. 4.3.I(c»). The following result of Liapunov gives a basic criterion for stability.
.v
x
(a) Stable
(b) Asymptotically stable
Figura 4.3.1
(c) Harmonic oscillator
THe QUALITATIVe THeORY OF DYNAMICAL SYSTeMS
243
4.3.4 Theorem. Suppose X is C' and mo is a critical point of X. Assume the characteristic exponents of mo have strictly negative real parts. Then mo is asymj)lotically stable. [In a similar way, if Re(l£i) > 0, mo is asymptotically unstable, i.e., asymptotically stable as t -+ - 00.) Proof. We can assume M = E = R n and that mo = O. Let - E> r = max(Re;\., ..... Re;\.n). Then we claim there is a norm 11'11 on E in which
lIe,x'(OIIl
~
e-·' .
If X'(O) is diagonalizable (e.g.• has distinct eigenvalues) a~ a complex.matrix. this is easy, for we can let II II be thp. sup norm associated with a basis of eigenvectms. If X'(O) is not diagonalizable. we can approximate it by one and get the same conclusion. (If the reader is familiar with the spectral radius formula. choose IlIe m , x '(OI(x)1I1 IIxll = sup ,"I • m;o 0 e /;0
0
where 111'111 is any norm.) Write A = X'(O) = DX(O). From the local existence theory, there is an r-ball about 0 for which the time of existence is uniform if the initial condition Xo lies in this ball. Let R(x) = X(x)- DX(O)·x.
Findr2 ~ r such that IIxll ~ r2 implies IIR(x)!1 ~ allxll. where a = E/2. Let B be the open rd2 ball about O. We shall show that if Xo E B, then the integral curve starting at Xo remains in Band -+ 0 exponentially as t -+ + 00. This will prove the result. Let x(t) be the integral curve of X starting at Xo' Suppose x( t) remains in B for 0 ~ t < T. The equation x(t}=x o + [X(x(s»ds
o
=xo+ ll.fx(s)+R(x(s))]d~
o
gives, by the variation of constants formula (Exercise 4.1 E).
and so
244
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Letting/(t) ... e"lIx(I~I, I( I)
"
II Xo II + a 10 I/( S ) tis,
and so, by Gronwall's inequality,
I( I) " IIxolle"'. Thus Hence x(l) E B,O" 1 < T, so as in 4.1.19, x(l) may be indefinitely extended, in 1 and the foregoing estimate holds. • A critical point mo is called hyperbolic or elemenlary if none of its characteristic exponents has zero real part. A generalization of Liapunov's theorem shows that near a hyperbolic critical point the flow looks like that of its linearization. (See Hartman (1973, Ch. 9] and Nelson (1969, Ch. 3), for proofs and discussions.) In the plane, the possible hyperbolic flows near a critical point are shown in Fig. 4.3.2. (Remember that for real systems, the characteristic exponents occur in complex conjugate pairs.) In higher dimensions, we have similar portraits for hyperbolic critical points. Here is the idea. For a critical point, mo, let In(mo) denote the insel, . defined by
and similarly, the
011'8.' is defined by
In the case of a linear system,
.r= Ax, where A has no eigenvalue on the imaginary axis (so the ongm is a hyperbolic critical point), In(O) is the generalized eigenspace of the eigenvalues with negative real parts, while Qut(O) is the generalized eigenspace corresponding to the eigenvalues with positive real parts. Clearly, these are complementary subspaces. The dimension of the inset, S( A ,0) = dimln(O) is called the slability index of the critical point. Now for a nonlinear system
m= K(m),
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
245
1',
(e)
(d)
Figure 4.3.2
suppose mo is a hyperbolic critical point. The linearization theorem alluded to above slates that the phase portrait or X near mo is topologically conjugate to the phase portrait or the linear system
.t= Ax near the origin, where A = X'(mo), the linearized vector field at mo. This means there is a homeomorphism or the two domains, preserving the oriented trajectories of the respective flows. Thus in this nonlinear hyperbolic case, the inset and outset are CO submanifolds. Another important thtw:'m of dynamical systems theory, the stahle manifold theorem (Smale [1 '" • says that in addition, these are smooth injectively immersffi ~lIb rnanli .. lds, intersecting transversally at the critical point mo. See I \
246
VECTOR FIELDS AND DYNAMICAL SYSTEMS
•
\
FlguI'84.3.3
for an illustration showing part of the inset and outset near the critical point. It follows from these important results that there are (up to topological conjugacy) only a few essentially different phase portraits, near hyperbolic critical points. These are classified by the dimension of their insets. called the stability index.
s(X. mol = dimln(mo) as in the linear case. The word index comes up in this context with another meaning. If m is a critical point of a vector field X. the topological index of m is + I if the number of eigenvalues (counting multiplicities) with negative real part is· even and is - I if it is odd. Let this index be denoted I( X. m). so that J( X. m) = (- nSfX • m). The Poincare-Hopf index theorem states that if M is ' compact and X only has (isolated) hyperbolic critical points. then
L
J(X, m) = X(M)
mis a critical point or X
where X(M) is the Euler characteristic of M. (For isolated nonhyperbolic
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
247
critical points the index is also defined but requires the development of degree theory for its definition-a kind of generalized winding number; see Box 7.5A). For a proof and discussion, see Guillemin and Pollack [1974, p. 133).
We now illustrate these basic concepts about critical points with some classical applications.
4.3.5 Exampl.. A. The simple pendulum with linear damping is defined by the second-order equation
x + eX + k sin x =
0
(c>O).
This is equivalent to the planar dynamical system v { x= v=-cv-ksinx The phase portrait is shown in Fig. 4.3.4. The stable focus at the origin
Figure 4.3.4
248
VECTOR FIELDS AND DYNAMICAL SYSTEMS
represents the motionless, hanging pendulum. The saddle point at (k7T,O) corresponds to the motionless bob, balanced at the top of its swing. B. Another classical equation models the buckling column (see Stoker [1950, ch. 3, sec. 10)):
or equivalently. the planar dynamical system
x-v co a,x a 3 x x-------m m m 3
with the phase portrait shown in Fig. 4.3.5. This has two stable foci on the horizontal axis, denoted m, and m2' corresponding to the column buckling. (due to a heavy weight on the top) to either side. The saddle point at the origin corresponds to the unstable equilibrium of the straight, unbuckled column.
~-== Figure 4.3.5
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
249
::: ::::::::::::. :a:::::::::::::::::: :::::::::::::: ................................................. ... ........... .................... ............. .
................... ............. . ,
..... , ..... .
, • • • • • • • • • • 1 •.
:::::::::.::::::::::
,Figure 4.3••
:Note that in this ph:!, ,ait, some initial conditions tend toward one stable focus. while some tend toward the other. The two tendencies are divided by the curve, In(O,O), the inset of the saddle at the origin. This is called the ~eparatrix, as it separates the domain into the two disjoint open sets, In(mo) and In(m,). The stable foci are called ell/ractors, and their insets are called their basins. See Fig. 4.3.6 for the special case x + x - x + Xl = O. This is a special case of a general theory, which is increasingly important in dynamical systems applications. The attractors are regarded as the principal features of the phase portrait; the size of their basins measures the probability of observing the attractor, and the separatrices help find them. A Another basic ingredient in the qualitative theory is the notion of a closed orbit (also called a limit cycle). 4.3.8 DeflnlUon. An orMr
/I
vector field X is called cltn«l when
y( t) is not a fixed point and Iltere u U T > 0 such thaI y( I + T) = y( t) for all t. The of y, In(y), is the set of points me M such that F,(m) - y as t - + 00 (i.e., the distance between F,(m) and the (compact) set {y(t)I0.,.;; t
wet
250
VECTOR FIELDS AND DYNAMICAL SYSTEMS
4iI T} tends to zero as t ..... 00.) Likewise, the 0IItHt, Out(y). is the set of points tending to y as t ..... - 00.
4.3.7 Example. One of the earliest occurrences of an attractive closed orbit in an important application is found in Baron Rayleigh's model for the violin string (see Rayleigh (1887, vol. I, sec. 68a)),
or equivalently,
with the phase portrait shown in Fig. 4.3.7. This has an unstable focus at the origin, with an attractive closed orbit around it. That is, the closed orbit y is a limiting set for every point in its basin (or inset) In(y). which is an open set of the domain. In fact the entire plane (excepting the origin) comprises the basin of this closed orbit. Thus every trajectory tends asymptotically to the limit cycle y and winds around closer and closer to it. Meanwhile this closed orbit is a periodic function of time, in the sense of Definition 4.3.6. Thus the eventual (asymptotic) behavior of every trajectory (other than the unstable constant trajectory at the origin) is periodic; it is an oscillation.
Figure 4.3.7
THE QUALITA TlVE THEORY OF DYNAMICAL SYSTEMS.
251
This picture thus models the sustained oscillation of the violin string. under the innu~nce of the moving bow. Related systems occur in electrical engineering under the name Van der Pol equation. • We now proceed toward the analog of Liapunov's theorem for the stability of closed orbits. To do this we need to introduce Poincare maps and characteristic multipliers. 4.3.8 DeflnlUon. Let X be a vector field on a manifold M. A local trYIIUWnaI :rection of X at m E M is Cl suhmanifold SCM of codimensioll one· with me S and for all s E S. X(s) is not {·ontained ill 7: .'''. Let X be a vector field on a manifold M with integral F: L'i\ c M X R --+ M. y a closed orbit of X with period T, and Sa llJ!:al trllnlTl'r.ml section of X at me y. A Poincare ""'" of y is a mapping 8: Wo -+ WI where:
Wo. WI C S are open neighborhoods of me S. and 8 is a diffeomorphism; (PM 2) there is a continuoUs function 6: Wo --+ R such that for all s E Woo (s. T - 6(s» E 6i\. and 8(s) '"" F(s. T- 6(s»; and finally. (PM 3) ift E]O. T - 6(s)[. then F(s. t)~ Wo(see Fig. 4.3.8).
(PM I)
Figure 4.3.8
252
VECTOR FIELDS AND DYNAMICAL SYSTEMS
4.3.. Theorem (Emtmee IIIIIllllliqwlleu 0/ PoinCtlN ""'I's), (i) If X is a vector field on M, and y is a closed orbit of X, then there exists a Poincare ' mop ofy, (ii) If 9: Wo .... WI is a Poincare map of y (in a local transversal section Sat m e y) and 9' also (in S' at m' e y), then 9 and 9' are locally conjugate. That 18, there are open neighborhoods W1 0/ me S, Wi of m' e S', and a diffeomorphism H: W1 .... Wi, such that W2 c Wo n WI' Wi C Wo n WI and the diagram
commutes.
Proof.
(i) At any point m e y we have X( m) .. O. so there 'exists a now box chart (U, cp) at m with cp(U) ... V x Ie R,,-I xR, Then S X{o}) is a local transversal section at m. If F: 6j)x c M x R .... M is the integral of X. 6j)x is open, so we may suppose U x [ - ", ., ) c OJ)x' where., is the period of y. As ~(m) ... m e U and ~ is a homeomorphism, Uo = ~- I( U)n U is an open neighborhood of m e M with ~(uo) cU. Let Wo'" S n Uo and Wz ~(Wo)' Then Wz is a local transversal section at me M and F.: Wo .... W2 is a diffeomorphism (see Fig. 4.3.9). Now if U1 - ~(Uo)' then we may regard Uo and U2 as open submanifolds of the vector bundle V x R (by identification using,,) and then F.:
_,,-lev
s IJ
FIgu... 4.3.'
u
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
253
Vo ..... V 2 is a diffeomorphism mapping fibers. into fibers. as !p identifies orbits with fibers. and F. preserves orbits. Thus W2 is a section of an open subbundle. More precisely. if 'IT: V X j ..... V and p: V x·1 -I> I are the projection maps. then the composite mapping
has a tangent that is an isomorphism at each point, and so by the inverse mapping theorem. it is a diffeomorphism onto an open submanifold. Let WI be its image. and 8 the composite mapping. We now show that 8: Wo ..... WI is a Poincare map. Obviously (PM I) is satisfied. For (PM 2). we identify V and V x I by means of !p to simplify notations. Then 'IT: W2 ..... WI is a diffeomorphism. and its inverse ('lT1 W1 ) - I: WI -I> W2 C WI xR is a section corresponding to a smooth function 0: WI ..... R. In fact. 0 is defined implicitly by
=
('lTF.(wo),PF.(wo)-I)(lIh»
=
('lTF.( wo),O)
=
8( wo).
Finally, (PM 3) i~ nhvious as (V.!P) is a flow box. (ii) The prOl . ,Irdensome because of the notational complexity in the definition of local conjugacv. so we will be satisfied to prove this uniqueness under additional simplIl)'ing hypOtheses that lead to global conjugacy (identified by italics). The general case will be left to the reader. We consider first the special case m "" m'. Then we choose a flow box chart (V.!P) at m. and assume SUS' c V. and that Sand S' intersect each . orbit arc in Vat most once, and that they intersect exactly the same sets of orbits. (These three conditions may always be obtained by shrinking Sand S'.) Then let W2 = S, W:i = S'. and H: W2 -I> W; the bijection given by the orbits in V. As in (i). this is easily seen to be a diffeomorphism. and Ho8=8'oH. Finally, suppose m .- m'. Then Fu( m) = m' for some a e )0. T(. and as 6D x is open there is a neighborhood V of m such that V X {a} C "Dx ' Then
254
VECTOR FIELDS AND DYNAMICAL SYSTEMS
F,,(U () S) -. s" is a local transversal section of X at m' E y. and H = Fa effects a conjugacy between a and a" - Fa 0 a 0 Fa- , on S". By the preceding paragraph. a" and a' are locally co~ugate. but conjugacy is an equivalence relation. This completes the argument. • If 'Y is a closed orbit of X E 'X (M) and mE y. the behavior of nearby orbits is given by a Poincare map a on a local transversal section Sal m. Clearly Tma E L(T",S. TmS) is a linear approximation 10 8 al m. By uniqueness of a up to local conjugacy. Tm·8' is similar to Tm8. for any other Poincare map 8' on a local transversal section at m' E y. Therefore. ' the eigenvalues of T",8 are independent of m E 'Y and the particular section S atm.
".3.10 DeftnlUon. If'Y is a closed orbit of X E 'X(M). the characteristic y are the eigenvalues of T",8. for any Poincare map e at
llUlltiplie" of X at any mE y.
Another linear approximation to the flow near y is given by Tm FT E L(T",M. T",M) if mE y and 'I' is the period of y. Note that fi(X(m» = X(m). so TlftF. always has an eigenvalue I corresponding to the eigenvector X( m). The (n - I) remaining eigenvalues (if dim( M) = n) are in fact the . characteristic multipliers of X at y.
Propoeltlon. If y is a closed orbit of X E 'X ( M) of period 'I' and cy is the set of characteristic multipliers of X at y. then cy U{l} is the set of eigenvalues of TmF.. for any m E y.
".3.11
Proof. We can work in a chart modeled on R n and assume m = O. Let V be the span of X(m) so R n = TmSe V. Write the flow F,(x. y)'" (F,'(x. y). f;2(X. y». By definition. we have
and
Thus the matrix of T", F. is of the form
where A .. D,F.2(m). From this it follows that the spectrum of TmF. is' {I)Ucy'
•
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
255
If the characteristic exponents of an equilibrium point lie (strictly) in the left half-plane. we know from Liapunov's theorem that the equilibrium is stable. For closed orbits we introduce stability by means of the following definition. 4.3.12 DeflnlUon. Let X be a vector field on a manifold M and y a closed orbit of X. An orbit F,(m o ) is said to. w_ tOwtlrd y if mo is + complete and for an}' localtran.rver,ral S to X at me y there is a to su('h that 'F,Il( mo) E S and successive applications of the Poincare map yield a sequence of points that converges to m. See Fig. 4.3,10. '
Figure 4.3.10
4.3.13 Theorem. Let y be a closed orbit of X E ',,\ ( M) and let the characteri~tic multipliers of y lie strictly inside the unit circle. Then there is a neighborhood U of y such that for any mo E U, the orbit through ""0 winds toward y. The proof of 4.3.13 relies on the following lemma.
4.3.14 Lemma. Let f; S ..... S be a smooth map on a manifold S with f( so) = So for some so' Let the spectrum of lie inside the unit circle. Then there is a neighborhood U of So such that if s e U, fer) E U and I"(s)"'" So as n ..... IX>, where I" - f 0 f 0 • • • 0 fen times).
T,J
The lemma is proved in the same way as 4.3.4. The Iinear'transformation T,.freplaces e ....(11 in the argument. Compitctness of S· then aUows the argument to be extended from the Poin<:ari: section to a neighborhood of the closed orbit.
256
VECTOR FIELDS AND DYNAMICAL SYSTEMS
4.3.15 DeflnlUon. 1/ X E ~(M) and y is a closed orbit 0/ X, y is called "yper6oIic i/ none 0/ the characteristic multipliers 0/ X at y has modulus I. The local qualitative behavior near a hyperbolic closed orbit is especially simple. For example, they are isolated (see Abraham-Robbin (1961, Ch.5]). The local qualitative behavior near an hyperbolic closed orbit, y, may be visualized with the aid of the Poincare map. S: Wo c S .... WI C S, as shown in Fig. 4.3.8. The qualitative behavior of this map. under iterations. determines the asymptotic behavior of the trajectories near y. Let m E y be the base point of the section, and s E S. Then In(y), the inset of y, intersects S in the inset of m under the iteration of S. That is, s E In( y) if the trajectory F,(s) winds toward y, and this is equivalent to saying that SACS) tends to mo as k .... + 00. The inset and outset of mo E S are classified by linear algebra, as there is an analogue of the linearization theorem for maps at hyperbolic critical points. The linearization theorem lor maps says that there is a CO coordinate chart on S, in which the local representative of S is a linear map. Recall that in the hyperbolic case. the spectrum of this linear isomorphism avoids the unit circle. The eigenvalues inside the unit circle determine the generalized eigenspace of contraction. This is the inset of mo E S. under the iterates of S. The eigenvalues outside the unit circle similarly determine the outset of mo E S. Although this argument provides only local CO submanifolds, the global stable manifold theorem improves this: the inset and outset of a fixed point of a diffeomorphism are smooth, injectively immersed subrnanifolds meeting transversally at mo. Returning to closed orbits, the inset and outset of y C M may be visualized by choosing a section S", at every point me y. The inset and outset of y in m intersect each section S", in submanifolds of Sm. meeting transversally at me y. In fact, In(y) is a cylinder overy, that is, • bundle of injectively immersed disks. So, likewise, is Out( y). And these two cylinders intersect transversally in y, as shown in Fig. 4.3.11. These bundles need not be trivial. Another argument is sometimes used to study the inset and outset of a closed orbit, in place of the Poincare section technique described before, and is originally due to Smale (1961). The flow F, leaves the closed orbit invariant. A special coordinate chart may be found in a neighborhood of '1. The neighborhood is a disk bundle over y, and the flow.. is a bundle map. On each fiber, F, is a linear map of the form Z,e R " wht.:re Z, is a constant, and R is a linear map. Thus, if s "'" (m, f) is a point in the chart, the local representative of F, is given by the expression F,(m, J)= (m"Z,e R , . / )
called the Floquet normal/orm. This is the linearization theorem lor closed
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
257
Figure 4.3.11
orbits. A related result. the F/oquellheorem. eliminates the dependence of Z, on I. by making a further (time-dependent) change of coordinates (see Hartman [1973. ch. 4. sec. 6). or Abraham and Robbin (1967)). Finally. linear algebra applied to the linear map R in the exponent of the Floquet normal form. establishes the CO structure of the inset and outset of y. To get an overall picture of a dynamical system in which all critical elements (critical points and closed orbits) are hyperbolic. we try to draw or visualize the insets and outsets of each. Those'with open insets are attraclors. and their open insets are their basins. The domain is divided into basins by the separatrices. which includes the insets of all the nonattractive (saddle-type) critical elements (and possibly other. more complicated limit' sets, called chaotic atlractors. not described here.) We conclude with an example of sufficient complexity, which has been at the center of dynamical system theory for over a century. 4.3.18 Example. The simple pendulum equation (4.3.5) may be "simplified" by approximating sin x by two terms of its MacLaurin expansion. The resulting system is a model for a nonlinear spring with Imear damping.
x=v.
258
VECTOR FIELDS AND DYNAMICAL SYSTEMS
Adding a periodic forcing term, we have
x-v, k
b- - cv - kx + 3xl + FooswI.
This time-dependent system in the plane is transformed into an autonomous system in a solid ring by adding an angular variable proportional to the time, , - wI. Thus
x= v;
v- - cv -
kx
+
iXl +
Fcos(J;
IJ= w. Although this was introduced by Baron Rayleigh to study the resonance of tuning forks, piano strings, and so on, in his classic book Theory of Sound (1877J, this system is generally named after Duffing who obtained the first important results (see Duffing [1908] and Stoker (1950».
Figure 4.3.12
THE QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
259
Depending on the values of the three parameters (c; k, F) various phase portraits are obtained. One of these is shown in Fig. 4.3.12, adapted from the experiments of Hayashi (1964]. There are three closed orbits: two' attracting, one of saddle type. The inset of the saddle is a cylinder topologically, but the whole cylinder revolves around the saddle-type closed orbit. This cylinder is the separatrix between the two basins. For other parameter values the dynamics can be chaotic (see for example. Holmes (1979a,b] and Ueda (1980». Exercl...
4.3A"
(i)
Let E -> M be a vector bundle and mn EM an element of the zero section. Show that TmoE is isomorphic to TmnM E9 Em" in a natural, chart independent way. (ii) If E: M -> E is a section of E, and E(m o) = 0, define E'(mo): TmoM -> Emo to be the projection of TE< m n ) to Em". Write out E'( mo) relative to coordinates. (iii)
Show that if X is a vector field. then X'( mo) defined this way coincides with Definition 4.3.1.
4.3B
Prove that the equation 1J+2klJ- qsin(J = O(q > 0, k > 0) has a saddle'point at (J = 0, IJ= O.
4.3C
Consider the differential equations ;= ar' - br. 6= I using polar coordinates in the plane. (i) Determine those a, b for which this system has an attractive periodic orbit. (ii) Calculate the eigenvalues of this system at the origin for various a, b.
4.3D Floquet's theorem in differential equations deals with the fundamental matrix solution of a linear homogeneous system x= A(t)x, where A is a T-periodic matrix. The fundamental solution G(t) satisfies G= AG and G(O) = I. Floquet's theorem states that G(t) has the form P(t)e·' where P is I-periodic and B is constant. Look up the proof in Hale (1969) or Hartman (1973). (ii) Relate this result to the stability analysis of periodic orbits, as outlined in the text. (i)
4.3E
Let X E 'X (M), cp: M -> N be a diffeomorphism and Y = cp. X. Show that (i) mE M is a critical point of X iff cp(m) is a critical point of Y and the characteristic exponents are the same for each. (ii) y c M is a closed orbit of X iff cp( y) is a closed orbit of Yand their characteristic multipliers are the same.
260
4.3F
VECTOR FIELDS AND DYNAMICAL SYSTEMS
The energy for a symmetric heavy top is
pJ
+ J + MglcosO where I, J> 0, b, Po/!' and Mgl> 0 are constants. The dynamics of the top is described by the differential equations 0= aHlflO. p, = - aHlaO. (i) Show that 0 = O. p, = 0 is a saddle point if 0 < Po/! < 2{Mgli (a slow top) (ii) Verify that cosO = 1- ysech2({1Fi 12). where y = 2- b 21P and P= 2MgljI, describes both the outset and inset of this saddle point. (This is called a homoclinic orbit.) (iii) Is8 ... 0, p, =- 0 stable if Po/! > 2..jMgll? (Hint: Use the fact that H is constant along the trajectories.) 4.4 FAOBENIUS' THEOREM AND FOLIATIONS
The three main pillars supporting differential topology and calculus on manifolds are the implicit function theorem, the existence theorem for ordinary differential equations, and Frobenius' theorem. which we discuss brieny here. First some definitions:
*
4.4.1 Definition. Let M be a mani/old and let E c TM be a subbundle 0/ its langent bundle. (So E is a vector bundle over M and is a submani/old 0/ TM by means 0/ a vector bundle chart 0/ E.) We call E a dutributiorl (or a p/IIM /kill) on M.
We say E is invoIutiw if/or any two vector /ields X and Y defined on \open sets 0/ M and which take values in E,[X. YJ takes values in E as well. (ii) We say E is iII,egrabk if for any mo E M there is a (local) submanifold N eM. called a (local) iIItegrvl numi/oill of E at mo containing mo whose tangent bundle is exactly E restricted to N. (i)
The situation is shown in Fig. 4.4.1.
*According to Lawson (1977). the theorem or Frobenius is due to A. Clebsch and F. Deahna.
FROBENIUS' THEOREM AND FOLIATIONS
281
E"'II
Figure 4.4.1
4.4.~
Examples A. Any subbundle E of TM with one-dimensional fibers is involutive; E is also integrable. and this is seen in the following way. Using local bundle charts for TM at mo e M with the lIubbundle property for E, we can find in an open neighborhood of mo. a vector field that never vanishes and has values in E. Its local integral curves through mo have as their tangent bundles, E restricted to these curves. The situation is nicer if the'vector field can be found globally and has no zeros. Then through any point of the manifold there is exactly one maximal integral curve of the vector field. and this integral curve never reduces to a point. We shall return to this crxample later. B. LCt I: M ~ N be a submersion and consider the bundle ker TI c TM. This bundle is involutive since for any X. Y e <X( M). where X, Y take values in ker TI, TIl X, Y) == 0 by' 4.2.25. The bundle is also integrable since for any mo e M, the submanifoldr l(f(mo» has the restriction of ker TI to itself as its tangent bundle (see Section 3.5). C. Let T" be the n-dimensional torus, n ~ 2. Let I ~ k ~ nand consider E-«vl .... ,v,,)eTT"lvHI == .. • = v" = O}. This distribution is involutive and integrable; the integral manifold through (t I ' " .• In) is T k x (/HI'· ... /,,). D. E =
TM is involutive and integrable; the integral submanifold through any point is M itself. E. An example of a noninvolutive distribution is as follows. Let M == S0(3), the rotation group (see Exercise 3.5S). The tangent space at
262
VECTOR FIELDS AND DYNAMICAL SYSTEMS
I - identity consists of the 3 X 3 skew symmetric matrices. Let
E,- {AeT,SO(3)A-
[~
o o -p
a two-dimensional subspace. For Q E S0(3), let
Then
is a distribution but is not involutive. In fact. one computes that the two vector fields with p "'" I, q = 0 and p = O. q = 1 have a bracket that does not lie in E. (Some additional background in the theory of Lie groups makes this example natural). • Frobenius' theorem asserts that the two conditions in 4.4. J are equivalent.
4.4.3 Frobenlu.' Theorem. only if it is integrable.
A subbundle E of TM is involutive if and
Proof. The" ir' part is obvious since the bracket of two vector fields on a. manifold N is again a vector field on N. If E has one-dimensional fibers, Frobenius' tneorem amounts to the existence theorem for differential equations. The proof of the "only ir' part actually proceeds along these lines. By choosing a vector bundle chart, one is reduced to this local situation: E is a model space for the fibers of E, F is a complementary space, and U x VeE x F is an open neighborhood of (0,0), so U x V is a local model for M. We have a map
f: UXV-L(E,F)
such that the fiber of E over (x, y) is E(J<,Y)"'" {(".j(x. Y)''')I"E E) c E x F,
and we can assume we are working near (0.0) and f(O, 0) = identity.
FROBENIUS' THEOREM AND FOLIA TlONS
263
The condition of involutivity is eaSy to work out. One computes the bracket of two vector fields of the form
X(x. y) - ( .. (x. y)./(x. y)· ..
e... y».
Y(x. y) = (b(x. y)./(x. y)·b( .... y». Using the local formula for [X, Y). one finds that E is integrable iff at a poini (x. y)
D/·( .. ./· .. )·b= D/·(b./·b),"
(I)
for all II, bEE. Consider these two time-dependent vector fields on U X V:
X/ex. y) = (0. I( IX. y )·x) and
Y,(x. y) = (x. 1(lx. y )·tx). These are chosen with the form of E(JC •.V) in mind. so Yo is simple. but especially so that
dY, [ X/,.Y, ] + dt = O.
(2)
Condition (2) is easily verified (it does not use (I)). From it, we gel d
dt P,'Y, - 0 where F, is the flow of X" Fo = identity. Since X,(O,O) .. 0, we can assume F, is defined for 0 ... I ... 1. Thus
Since YI(x, y) = (x, I(x. y)x) and Yo(X, y) - (X,O). we ~pect FI to be a maps E(JC •.v) to E X{O} and be local diffeomorphism of U X V such that the required coordinate change to complete the proof. . Let N= FI(U X (Yo}), We shall complete the proof by showing that if (x, y) EN, then
r.
Clearly, since the first component of X, is zero,
F,(x, y). - (x, cp(t, x, y»
2tU
VECTOR FIELDS AND DYNAMICAL SYSTEMS
for a map .,(1, x, y) e F. A typical tangent vector to F,(U X{Yo}) is clearly (." D...,(I, x, Yo)·.,),
.,e E.
Thus the proof is complete if we establish this identity:
0 ...,(1, x, Yo)'" ·'/('X, .,(1, x, Yo»·." which forI-I shows that the tanaent space to F,(U X (Yo}) at (x,.,(x, Yo,I» equals E( ... y,' Indeed, they are equal at 1- 0 and
a
a, .,(1, x, Yo) = I(IX,.,(I, x, Yo»'x so
a
at 0 ...,(1, x, Yo)'" -I0.l( lx, .,(1, x, Yo»''''x
+ 0d(IX,.,(I, x, Yo»' (0...,(1, x, Yo)''')'x + 1(IX,.,(t,x, Yo»''''
Also,
(3)
a
a, tl( tx, .,(1, x, Yo»'" "" I( tx, .,(1, x, Yo»'"
+ to.l(tx,.,(t, x, Yo»'x'" + tDd(IX,.,(t, x, Yo»' (J(tx, .,(t, x, Yo»'x ) . ., (4) For x, Yo' ., fIXed, let
+(t) - O".,(t, x, Yo ) . ., - II( lx, .,(1, x, Yo»·., so that IjI(O) = O. Subtracting (3) and (4) and using (1) we find thai
Hence by Gronwall's inequality, or by uniqueness of solutions of differential equations, +(1) - 0, and the result follows. •
The method of using the time-one map of a time-dependent flow 10 provide the appropriate coordinate change is useful in a number of situations. Sometimes these are called "Lie transforms." We shall use this
FROBENIUS~
THEOREM AND FOLIATIONS
265
method again in Chapter 6 to prove the Poincare lemma and in Chapter 8 to prove the Darboux theorem. For a direct proof of the Frobenius' theorem from the implicit function theorem using manifolds of maps in the spirit of Box 4.IC. see Penot (1970). The Frobenius theorem is intimately connected to the global concept of loliations. Roughly speakirig, the local manifolds N obtained berore can be glued together to form a "nicely stacked" family of submanifolds filling out M (see Fig. 4.4.1 or Example 4.4.2A).
y
--+---x
figure 4.4.2
4.4.4 DeflnlUon. Let M be a manifold and 4» = {eo}.. E If a partition of M into disjoint connected sets. 4» is called a /oliation if each point 0/ M has a chart (V, cp), cp: V -+ V' X V' c E $ F sueh that for each the connected V' X{c~}, components (V e .. )' of V f .. are given by cp«V where c~ E F are constants for each II E.A and fl. Sueh charts are called /oIitItItJ (or distinpUW) by ell. The dimsuion (respectively, etJtIinwIuion ) 0/ .tile /oliation 4» is the dimension of E (respectively, F). See Fig. 4.4.2. .
n
e. n e.>') -
n
en
Note that each leaf is a connected immersed submanifold. In general, this immersion is not an embedding; i.e., the induced topology on eo from M does not necessarily coincide with the topology of ea (the leaf ea may accumulate on itself, for example). A differentiable structure on ea is induced by the foliated charts in the following manner. If (V, cp), cp: V -+ V' X V' c E$F is a foliated chart on M, and x: E$F -+ E is the canonical. projection, then X 0 cp restricted to (U defines a chart of
n eo>'
e...
4.4.5 Ex....pI.. A. The trivial /oliation of a manifold M has only one leaf. M itself. It has codimension zero. If M is finite dimensional both M and the leaf have
266
VECTOR FIELDS AND DYNAMICAL SYSTEMS
the same dimension. Conversely, On a finite-dimensional manifold M a foliation of dimension equal to dim(M) is the trivial foliation. B. The discrete foliation of a manifold M is the only zero-dimensional foliation; its leaves are all points of M. If M is finite dimensional, the dimension of M is the codimension of this foliation. C. A vector field X that never vanishes on M determines a foliation; its leaves are the maximal integral curves of the vector field X. The fact that this is a foliation is the straightening out theorem (see Section 4.1). D. Let f: M - N be a submersion. It defines a foliation on M (of codimension equal to dim( N) if dim( N) is finite) by the collection of all connected components of I( n) when n varies throughout N. The fact that this is a foliation is Theorem 3.5.5. In particular EeF is foliated hy the family {E x {f}}, e F· •
r
We next define the tangent bundle to a foliation.
4.4.6 ProposlUon. Let M be a manifold and ell = {e.. }.. e A a foliation on M. The set T( M, ell) = u "e A U me "Tmf. .. is a subbundle of TM called the ""'gmt InuuIk to the foliation. The quotient bundle JI(eII) = TMjT( M, ell), is called the "OrmaJ InuuIk to the foliation ell. Proof. Let (U, fJI), fJI: U - U' x V' - E e F be a foliated chart. Since TufJI(T): .. ) = E X (O) for every u E U () we have TfJI(TU () T( M, = (U' x V') x (E x {O». Thus the standard tangent bundle charts induced by foliated charts of M have the subbundle property and naturally induce vector bundle charts by mapping v'" E T",(M, ell) to (fJI(m), T",fJI(v",» E (U' X V')x(E x (O». •
ell»
e.. ,
Now we can reformulate Frobenius' theorem in terms of foliations.
4.4.7 Theorem. lent: (i)
(ii) (iii)
Let E be a subbundle of TM. The following are equiva-
there exists a foliation ell on M such that E E is integrable; E is involutive.
=
T(M,eII);
Proof. The equivalence of (ii) and (iii) was proved in 4.4.3. Let (i) hold. Working with a foliated chart, E is integrable by 4.4.{j, the integral submanifolds being the leaves of ell. Thus (ii) holds. Finally, we need to show that (ii) implies (i). Consider on M the family of (local) integral manifolds of E. each equipped with its own submanifold topology. It is straightforward to verify that the family of finite intersections of open subsets of these local integral
FROBEN/US' THEOREM AND FOLIATIONS
267
submanifolds defines a topology on M, finer in general than the original one. Let {E .. }ae A be its connected components. Then, denoting by (fa n V)fj, the connected components of Ea n V in V, we have by definition that E(une.)fl=(E restricted to (Vne,,)fj) equals ~«Vne .. )fj). Let (.,.- I( V), y,), V c M be a vector bundle chart of TM with the subbundle property for E, and let!p: V -+ VI be the induced chart on the hase. This means, shrinking V if necessary, that!p: V -+ V' X V' c E(fJF and y,(.,.- I( V) nE}=(V'XV')X(EX{O}). Thus !p«Vne .. )fj)=u'X{c~} and so M is foliated by 'the family ell == (Ea}ae A' Because E(u() I'.}" = T«V n C,,)fj). we also have T(M,eII) = E . • There is an important global topological condition that integrable subbundles must satisfy that was discovered by Bou 11970]. The result, called the Bott vanishing theorem can be found, along with related results. hy readers with background in algebraic topology, in Lawson [1977]. Let R denote the following equivalence relation in a manifold M with a given foliation ell: xRy if x, y belong to the same leaf of ell, It is of interest to know whether M / R is a manifold. Foliations for which R is a regular equivalence relation are called regular foliations. (See Section 3.5 for a discussion of regular equivalence' relations.) The following is a useful criterion.
4.4.8 Propoaltlon. Let ell be a foliation on a manifold M and R the equivalence relation in M determined by ell. R is regular iff for every' me M there exists a local submanifold Em of M such that Em intersects every leaf in at most one point (or nowhere) and TmEmal Tnr( M, ell) = Tm M. (Sometimes Em is called a slice or a local cross-section for the foliation,)
Proof. Assume that R is regular and let ",: M -+ M/ R be the ca!lonical projection. For Em choose the submanifold using the following construction. Since ", is a submersion, in appropriate charts (V,!p). (V, y,). where !p: V -+ U' x V' and y,: V -+ V'. the local representative of",. "'",>/-: U' X V' -+ V' is the projeCtion onto the second factor (3.5.5) and every leaf",- I( v) C V. v e V is represented in these charts as V' x {v,}, where (,' = y,( v). Thus if Em = y,- I({O}X V'), we see that Em satisfies the two required conditions. The converse is obtained by reversing the argument. •
To get a feeling for the foregoing condition we will study the linear flow on the torus.
4.4.9 Example. On the two-torus T 2 consider the glohal now F: R X T 2 -+ T2 defined by F(t.(sl' S2»'" (s le 2,"', S2e2",a,) for a fixed number lie
268
VECTOR FIELDS AND DYNAMICAL SYSTEMS
[0.1[. By 4.4.5C this defines a foliation of T2. If a E Q. notice that every integral curve is closed and that all integral curves have the same periqd. The condition of the previous theorem is easily verified and we conclude that in this case the equivalence relation R is regular; T 2/ R = s I. If a is irrational the situation. however. is completely different. Let cp(t) '= (e 2w ;/. e 2w ;a/) denote the integral curve through (I. I). The following argument shows that cl( cp(R» = T 2; that is. cp(R) is den.ve in T 2. Let p(e 2"'''.e 2'''Y)E T2; then for all mE Z.
where y = ax + z. It suffices to show that C = {e 2 ,,;m,. E Slim E Z} is dense in SI because then there is a sequence m, E Z such that e 2";m,,. converges to e 2 .. ;t. Hence. cp(x + mil) converges to p. If for each k E Z+ we divide SI into k arcs of length 2f1/k. then. because {e 2..;"'·ES l lm-I.2..... k+.I} are distinct. for some 1 < n, < m, < k + 1. e 2";m,,. and e 2f1 ;ro,,. belong to the same arc. Therefore. le 2,,;m.,. - e1 ..;ro'-1 < 2f1/k. which implies le 2",P.- -II < 2f1/k. where Pte - m" - n". Because
U {e 2 ..tas E Slls E (jp".(j + I)p,,]) = S... jEZ.
every arc of length less than 2'IT/k contains some e 2,,;jP •• which proves c1(C)=SI. Thus any submanifold Em.mET2. not coinciding with the integral curve through m will have to intersect cp(R) infinitely many times; the condition in the previous theorem is violated and so R is not regular. • Even though the concept of foliations seems encompassing enough to study "nice" partitions of a manifold into submanifolds. there are important situations when foliations are unsatisfactory because they are not regular or the leaves jump in dimension from point to point. Consider for example. R2 as a union of circles. Here the origin is a singularity. As this example suggests. one would like to relax the condition that M / R be a manifold. provided that M/R turns out to be a union of manifolds that fit "nicely" together. Stratifications. a concept generalizing foliations. turn out to be the natural tool to describe the topology of orbit spaces of comp~ct Lie group actions (see. for instance. Bredon [1972]. Burghelea. Albu. and Ratiu [1975]. Fischer [1970]. and Bourguignon [1975]). We shall limit ourselves here only to the definition in the finite-dimensional case and some simple remarks. 4.4.10 Definition. ut M be a locally compact topological space. A
/ktI,itM of M is a partition of M into manifolds {M";.,, , called
s,ra';lI,ra,,,.
FROBENIUS' THEOREM AND FOLIATIONS
269
-YlllisfYing the following conditions: (S I)
(S2)
(S3) (S4)
M. are manifolds of constant dimension; they are .whmanifoldv of M if M is it.~elf a manifold; . ' The fami(v {M;} of connected components of all the M is a locally finite partition of M; i.e.,for every m E M. there exists an open nei!{hborhood U of m in M intersecting only finile(v many Mff. If M; ncl( Mff) .. 0 for (a. a)" (b, Pl. then M: eMf and dime MQ)'
From the definition it follows that if
M; ncl( Mil)
T
0 and if mE
M:;
c c1(Mff) has an open neighborhood U in the topology of M;; such that U c M: ncl(Mf). then necessarily M;; eMf and thus dim(Mu ) < dime M~). To see this, it is enough to note that the given hypothesis makes M;; ncl( Mf)
M:.
open, in Since it is also closed (by definition of the relative topology) and M: is connected. it must equal M: itself, whence M(~ c c1( Mf) and by (S3), M:c Mf and dim(M
Exercl... 4.4A
4.48
4.4C
Let M be a manifold such that TM = E1al ... alEp • where E, is involutive. Show that t~ere are local charts ( U, q> ). q>: U ~ E I e ... eE, such that E, is described by the equations ~j - O. j "" i. In R4 consider the family of surfaces given by .\: + y2 + Z2 -t 2 = const. Show that these surfaces are the leaves of a foliation that is not regular. What part of R4 should be thrown out so as to obtain a regular foliation? Let f: M ..... N be a C'" map' and • a foliation on N. The map f is said to be transversal to ., denoted f7f'l. if for every me M, T",f(T",M) +1{(m)(N,.)=1{(m)N and (Tmf)-I(1{<",)(N,.» splits in TmM. Show that if (f.. }.. e ... are the leaves of., the connected components of f-I(f .. ) are leaves of a foliation (denoted by f*(.» on M, and if • has finite codimension in N, then so does the foliation/*(.) on M and the two codimensions coincide.
270
VECTOR FIELDS AND DYNAMICAL SYSTEMS
4.4D (Bourbaki (197ID. Let M be a manifold and denote by M' the manifold with underlying set M but with a different differentiable structure. Show that the collection of connected components of M' defines a foliation of M iff for every me M. there exists an open set U in M. m e U. a manifold N. and a submersion p: U -+ N such that the submanifold p-I(n) of U is open in M' for all n E N. (Hint: For the "if' part use 3.3.5 and for the "only if" part use Exercise 3.2F to define a manifold structure on the leaves; the charts of the second structure are (U () a )lI-+ U'.)
e
CHAPTER
5
Tensors
In the previous chapter we studied vector fields and functions on manifolds. Now these objects are generalized to tensor fields. which are sections of special vector bundles that generalize the tangent bundle. This study is continued in the next chapter when we discuss differential forms. which are tensors with special symmetry properties. One of the objectives of this chapter is to extend the pull-back and Lie derivative operations from functions and vector fields to tensor fields. ' 5.1
TENSORS ON LINEAR SPACES
Prepatory to putting tensors on manifolds. we must first understand them on vector spaces. This subject is an extension of linear algebra sometimes called .. multilinear algebra." Ultimately our constructions win be done on each fiber of a vector bundle producing a new vector bundle. The case of the tangent bundle win be the most important. As in Chapter 2. E. F •... denote Banach spaces and Lk(E••..•• Ek: F) denotes the vector space of continuous k-multilinear maps of E. x ... X Ek to F. The special case L(E.R) is denoted E*. the dual space of E. If ' I- (~ ••...• ~,,) is an ordered basis of E. there is a unique ordered _basis of E*. the dUal basis ,. ... (~I •...• ~"). such that ~J(e,)=8I;. where 81;=1 if j = i and 0 otherwise. Furthermore. for each ~ e E.
271
272
TENSORS
and for each a e E·, n
a=
E a(~/)~;' i-I
Employing the summation convention whereby summation is implied when an index is repeated on upper and lower levels, these expressions become ~ _ ~i(~)~1 and a - a(~I)~;' . We may map E into E" -= L(E·,R) by associating with each ~ e E,~*·e E*·, given by ~··(a)= a(~) for allae E*. If E has finite dimension, then E·· and E have the same dimension and as the map ~ .... ~.* is one to one, it is an isomorphism. In infinite dimensions. recall that E is reflexive when ~ ..... ~.* is an isomorphism. .
5.1.1 DeflnlUon. For a vector space E we put 1'.'(E)=L'-I··(E· •...• E*. E •...• E; R) (r copies of E· and s copies of E). Elements of 1'.'(E) are called 'MSOI'S ... Eo COII''''''''''''''t of order r and of order s; or simply. 01 typ. (~). Given '. e 1'.:'(E) and'2 e 1'.:'(E) the '~lISor protblc, of'l and'2 i.' the tensor "~'2 e r.:'++.7(E) defined by
COWl"'"
'1~'2( PI •...• P',. yl •...• y". II" ··.1.,. RI.··· .R,,)
='1 (pi, .... p". II ....•1., )'2 (yl •... , Y". '1.··· .R.,) where pl. yj e E*
and~"j
e E.
Replacing R by a space F gives r.'(E; F), the F-valued tensors of typl." The tensor product now requires a bilinear form Oil the value space for its definition. For R-valued tensors•• is associative. bilinear and continuous; it is not commutative. We also have the special cases Td(E)'" E**. Tlo(E)- E*. T2o(E)- L(E; E*). and TII(E)- L(E; E··) and make the convention that Too( E; F) ... F. (~).
5.1.2 Proposition. Let {e;};e I be an algebraic basis of E and {~j}jeJ an algebraic basis of E*; i.e .• every element of E (resp. E·) is a finite linear combination of elements of {e;}/e I (resp. (ej}jeJ)' Then an algebraic basis of r.'(E) is given by
In particular, if dim(E) = n, then r.'(E) has the structure of an n,+s-dimensional vector space.
TENSORS ON LINEAR SPACES
273
We must show that the elements. ® ... ® •.I, ®.iI® ... ®.J. of . 'I T,'(E) are linearly independent and span T,'(E). Suppose a finite sum ti""/j ....•/ ® .. , ®.j ®.it® ... ®.J, = O. Then apply this to (."" .... . • k"./:,.~·,,~/,> using tite identification .,(.') = .i(.,) to give t ", .. A",." = n, Next, we easily check that for I e T,'( E) we have a finite sum I == I( ." •... , Proo,' ,,-
.1,• 5j.9 • ••• ,5j, • ). ® ... ® •.I. ®.J.® . , . ®.i, • 51.
..
_
The coefficients t l ""1 1""j, ... ,(.;" ••••• '·.eit .... ,e,.) are called the components of I relative to i. .
5.1.3 Exampl.. A. If' is a (~) tensor on'E then' has components
tij = I(e j • •, ) . an n x n matrix. This is the usual way of associating a bilinear form with a matrix. For instance. in R 2 the bilinear form
,(x, y) = Ax,y, + Bx,yz + CX 2 Y, + D:(~r, (where x = (x,. x 2 ) and J' = Cv,. Y2» is associated to the 2 x 2 matrix
(~ ~). B. If' is a (~) tensor on E., it makes sense to say that' is symmetric; i.e.• t(., •• 2) = '(.2'.')' This is equivalent to saying that the matrix ,'/ is symmetric, Symmetric (~) tensors I can be recovered from their quadratic form Q(e) = 1( ••• ) by 1(., •• 2) = HQ(., + .2)- Q(., -e2»)' the polarIzation identi~v. If E = R 2 and t has the matrix
(~ ~)
(n
then Q(.l') .... Ax~ + 2Bx,x 2 + Cx~. Symmetric tensors are thus closely related to quadratic forms and arise. for example. in mechanics as moment of inertia tensors and stress tensors. C. In general. a symmetric (~) tensor is defined by the condition
t( a' , .... a') = I( a"(1) ..... a"e,) for all permutations of (J of {I ..... s}. and all elements a' ..... a· e E·. On~ may associate to I a homogeneous polynomial of degree k. P( a) ... ,( a •. , .. a) and as in the case s == 2. P and, determine each other. A similar definition holds for (~) tensors. It is clear that a tensor is symmetric iff all its components in an arbitrary basis are symmetric.
274
TENSORS
D. An inner product (.) on E is a symmetric (~) tensor. Its matrix is often written gij - (_/'_j)' Thus gij is symmetric and positive definite. The inverse matrix is written gij. Eo The space L"(E, ..... E,,; F) is isometric to L"(E,,(,) ..... Eo (,,); F1 for any permutation a of {I •...• k}. the isometry being given by A ..... A, A('.. (1) ..... ' ..(k» - A(,' ... ·.'k)· Thus if , E 1'.'(E; F). the tensor' can be regarded in (,;,) ways as an (r + s )-multilinear F-valued map. For example. if t E Tl( E). the standard way is to regard it as a 3-linear map t: E* x E* x E -+ R. There are two more ways to interpret this map. however. namely as E* x E x E* -+ R and as E x E* x E* -+ R. In finite dimensions. where one writes the tensors in components. this distinction is important and is reflected in the index positions. Thus the three different tensors described above are written
F. In classical mechanics one encounters the notion of a dyadic (cc. Goldstein (1980». A dyadic is the formal sum of a finite number of dyads; a dyad being a pair of vectors ',.'2 ERl written in a specific order in the form "'2' The action of a dyad on a pair of vectors. called the double dot product of two dyads is defined by
','2: ","2 = ("'''')('2'''2)' where . stands for the usual dot product in Rl. In this way dyads and dyadics are nothing else but (~) type tensors on R3; i.e.• ','2 E T20(R 3), by identifying (R 3)* with R3. G. Higher order tensors arise in elasticity and Riemannian geometry. In elasticity the stress tensor is a symmetric 2-tensor and the elasticity tensor is a fourth-order tensor (see Marsden and Hughes [1983]). In Riemannian geometry the metric tensor is a symmetric 2-tensor and the curvature tensor is a fourth-order tensor (see Spivak [1979]). A The classical operations of tensor algebra are the interior product and the contraction. These are defined as follows. The inlerior product of a vector vEE (resp. form P E E*) with a tensor t E 1'.'( E; F) is a ~,) (resp. (' • ') type F-valued tensor defined by
L
(iot)( P' ..... P'. v, •.... v,_,) =
1(14' ... .• P'. v. v, .... ,v,_ ,)
(i")( P' ... ·.P'-'. v, ..... v,) "" '(14.14' .... ,14'- '. v" .... v.).
TENSORS ON LINEAR SPACES
275
Clearly;,,: 1'.'(E; F) -+ 1'.'-I(E; F). ;-: 1'.'(E; F) -+ 1'.' I( E; F) are linear. continuous maps. as are v ..... ·;".p ..... ,'fS. If F ... R and dim(E)=n these operations take the following form in components. If e A Crespo e k ) denotes the kth basis (resp. dual basis) element of E. we have
By 5.1.2 these formulas and linearity enable us tn compute anv interior product. The contraction of the k th j::Ontravariant with the Ith covariant index. or for short. the (k./)-conlraclion. is a family of linear continuous maps C/: 1'.'(E) -+ 1'.'_-II(E) defined for any pair of natural numbers r. s ~ I by
eN VI® ... ®V,®pl® ... ®p') -P'(Vk}V I ® .... ®t3k ® .. · ®V,®pl® ... ®9'®", ®P'. means that the symbol underneath is omitted. If dim(E) = nand E E* denote a basis and the dual basis. the expression for the contraction operation in components takes the form where
e; E E. e j
The Kronecker delta is the tensor 3 E TII(E) defined by 8(a, e) = aCe). If E is finite dimensional. 3 corresponds to the identity I E L(E; E) under the canonico:al isomorphism TII(E) .. L(E; E). where we have identified E** with E. Relative to any basis. the components of 3 are the usual Kronecker symbols 8j; i.e., 3 = 8je;®e j • Suppose E is a finite:dimensional real inner product space with a basis el, ... ,e" and corresponding dual basis el ..... e" in E*. Using the inner product. with matrix denoted by gil' we get isomorphisms b: E -+ E* ; e ...... (e . . )
and its inverse . 0f The matnx·
b'IS gij; I.e., •
#:E*-+E.
276
TENSORS
. and of • is gil; i.e.•
where e i and (Xj are the components of e and Cl. respectively. We call b the index lowering operator and I the index raising operator. These operators can be applied to tensors to produce new ones. For example if t is a (~) tensor we can define an associated tensor I of type ( : ) by
I(e. Cl) ... ,(e. cxI). The components are
i/ =
gJktilc
(as usual. sum on k).
In the classical literature one writes t/ for gJktik' and this is indeed a convenient notation in calculations. However. contrary to the impression one may get from the classical theory of Cartesian tensors, , and I are different tensors. Confusion over this point can lead to trouble. 5.1.4 Exam.,.... Let E be a finite-dimensional real vector space with basis el ..... e" and dual basis el •...• e". A. If t E T I2(E) and e = eie,. then
Thus. the components of ill' are ePt k',. The interior product of the same tensor with Cl- Clpe P takes the form
B.
If' E Tl(E). the (2. I)-contraction is given by
C. An important particular example of contraction is the trace of a ( : ) tensor. Namely. if , E Til ( E). then
tr(,} = CII(t)
= ti,.
TENSORS ON LINEAR SPACES
277 '
D. The components of the tensor associated to g hy raising the second index are gil = gikgj/< = gJkgk; = 6 J,. E. Let e ®eJ ®e k ®e'®e"/ • t E '7'3(E) '2 • t = ,;jk 1m, Then t has quite a few associated tensors. depending on which index is lowered or raised. For example
and so on. F. The positioning of the indices in the components of associated tensors is important. For example if t E T2o( E). we saw earlier that 1/' = gJk',k' However Ii; = gJkl k ,. which is in general different from Ii' when tis not symmetric. For example, if ~ = R3 with gij = 6/, and the nine components of , in the standard basis are '12 = I. 121 = - 1.1" = 0 for all other pairs i. j. then l,i = Iii' Iii = 'ji' so that 112 = 112 = I while 121 = 121 = - I. G. A convenient abuse of language often occurs in the physics literature when considering traces of tensors that are not of type ( :) but which can be brought to this type by lowering and raising indices. i.e.• tensors of type (~) and (~). The trace of a (~) tensor is thus defined to be the trace of the associated (:) tensor; i.e.• if t = lije;®ej • then tr(t) ... Iii'" giklik. The question naturally arises whether we get the same answer by lowering the first index instead of the second. i.e., if we consider Ii;' By symmetry of gii we have 111 - gkllik,.. 1/. so that the definition of the trace is independent of which index is lowered. Similarly. if IE T20(E). trCt) = 1/; - g,kl'k ... (kk' In particular tr(l') == gi 1 == glkg;k - dime E). '" Now we tum to the effect of linear transformations on dual spaces. If cpEL(E.F), the Iranspose of cpo denoted cp*EL(F*.E*) is defined by cp*(p)·e = p(cp(e». where PE F* and e E E. Let us analyze the matrices of cp and cp*. As customary in linear algebra, ve<;tors in a given basis are represented by a column whose entries are the components of the vector. Let", E L(E. F) and let e= (el ..... en ) and i=(/I ...../m) be ordered bases of E and F respectively. Put cp(e;) =
278
TENSORS
A"; fa. (We use a diffetent dummy index for the F-index to avoid confusion). This defines the matrix of 4p; A = (A";). Thus. for" = t"e, E E the components of 4p(v) are given by cp(v)" = A";";. Hence. thinking of" and cp(v) as column vectors, this formula shows that cp(,,) is computed hy multip(l'illg v on the kft by A. the matrix of 4p. as in elementary linear algebra. Thus the upper index is the row index, while the lower index is the column index. Consequently. 4p(~I) represents the ith column of the matrix of 4p. Let us now investigate the matrix of 4p* E L ( r. E*). If rand are the dual ordered bases of i and J respectively. then cp*(r)·~;=- r'cp(~I)"" A";6"" - A GI, i.e.• 4p*( AG;~; and thus the ith component of cp*( Is A";. Hence 4p.( is the ath row of A. Consequently the matrix 0/ cp* is the transpose of the matrix 0/ cpo If P- p" E r then cp*( P) - p"cp*( PQAQ;~I. which says that the ith component of 4p·(P) equals cp·(P); = /l"A";. Thinking of elements in the dual as rows whose entries are their components in the dual basis. the foregoing calculation says that cp*( P) is computed by multiplying P on the rigllt by A. the matrix of cpo again in agreement with linear algebra. Now we tum to the effect of linear transformation on tensors. We start with an induced map that acts "forward" like 4p.
r
r) -
r)
r
r·A"d,,r) r) -
5.1.5 Definition. If 4p E L(E. F) is an isomorphism, define the push-lor'ttffIrfI of 4p, T,'4p = 4p. E L(T/(E). T,'(F» by 4p.'(p', ...• P',
I" ... ,/,) = t(4p*(P'), ... ,4p*(P'),cp-'( I,),· .. ,cp-'( /,»)
where IE T,'(E), P' •.... P' E
r. and I"
...•/, E F.
We leave the verification that CP. is continuous to the reader. Note that T{'4p - (4p - ') •• which maps" forward" like cpo If E and F are finite dimensional. then 1'0'(£) - £. To'(F) - F and we identify 4p with To'4p. The next proposition asserts that the push-forward operation is compatible with compositions and the tensor product. 5.1.8
P~ltIon.
Let cp: E .... F and IjI: F-+ G be isomorphisms. Then
(i) (+04p)._IjI.04p •. (ii) (iii)
(iv)
II i: E
-+
£ is
th~
identity. then so is i.: T,'(E) .... T,'(E);
cp.: T,'(E)-+ T,'(F) is an isomorphism, and(4p.)-' = (cp-'). If I, E T,:'(E) and '2 E T,:Z( E). then 4p.(I,®t 2 ) = CP.(t, )®4p.(t2)
TENSORS ON LINEAR SPACES
Proof.
279
For (i),
".(tp.t)(yl, ... ,Y'.81' ...• 8.) .
== tp.t( ".(y1) •...• ".(y,).,,-1(81 ) ..... "-1(8.))
where yl •...• y' E G·. 81 ..... 8. E G. and t E T.'( E), We have used the fact that the transposes and inverses satisfy (" ° tp ). = tp. ° ". and (" ° tp) - I = tp - I ° ,,- I, which the reader can easily check. Part (ii) is an. immediate consequence of the definition and the fact that i· = ; and ;- I =;. Finally, for (iii) we have tp.o(tp-I). =;., the identity on T.'(F), by (i) and (ii). Similarly, (tp-I).0tp. = i. the identity on T.'(E). Hence (iii) follows. Finally (iv) is a straightforward consequence of the definitions. . , Since (tp-I). maps "backward" it is called the pull·back of tp and is denoted tp.. The next proposition gives a connection with component notation.
5.1.7 ProposlUon. Let tp E L(E. F) be an isomorphism of finite dimensional vector spaces. Let A"i denote the matrix of tp in the ordered ba.fes e of E and J of F, i.e. tp(.,) = A"J". Denote by (B i,,) tire matrix of cp- I. i.e. tp' I( /,,) _ B i"./. Then (B',,) is the intlerse matrix of ( A",): B i" A", = 6'/, Let t e T,'(E) with components t~' "Il'" 'J. relatille tn I and q ~: '1;'( F) with .('omponents q"'" '~'''' '''''. relative to f. Then the components /If cp. t relative to / and of tp., relative to I are given respectively by
·Proof. We have .i = tp-I(tp(.i»'" tp-I(A"J,,) = A"/tp-I( I,,) == A~IBi..1' whence Bi"A"/ = 6i / for all i, j. Similarly. one shows that A"/B i" - 3"". so
280
TENSORS
- t{ AU,
.',
,~.....
I
AU,,It;. .i, Bl, h5," • ,
I
I
••••• Bi •
• h--,· I
•
)
To prove the second relation. we need the matrix of (cp ')* e I. ( E*. We have
so that (cp -. )*(e') = case. •
Biu
r).
r. Now proceed with the proof as in the previous
Note that the matrix of (cp 1)* e L (E*, r ) is the transpose of the inverse of the matrix of ._ The assumption that cp be an isomorphism for •• to exist is quite restrictive but clearly cannot be weakened. However, one might ask if instead of "push-forward," the "pull-back" operation is considered, this restrictive assumption can be dropped. This is possible when working with covariant tensors, even when cp E L(E, F) is arbitrary.
5.1.8 Definition. If. E L(E, F) (not necessarily an isomorphism). define the pull-back cp*e L(T.,o(F), r.°(E» by cp*,( e l , ... ,e.) "" ,( cp( e.), ... ,cp( e.
n,
The next proposition asserts that cp* is compatible with compositions and the tensor product. Its proof is almost identical to that of 5.1.6 and is left as an exercise for the reader.
5.1.9 Propo.ltlon. Let cp e L (E; F ),1jI e L (F; G). Then (1jI 0.)* = cp*o 1jI*. (ii) If;: E .... E is the identity, then so is ;* e 1.( 0 ( E). 0 ( E». (iii) If cp is an isomorphism. then so is cp* and cp* = (cp • )*. (iv) If'l E r.~( F) and t2 e r.~( F), ,hen cp°(t.®t 2) = (cpo,. )®( cp*1 2 ). (i)
r.
r.
TENSORS ON LINEAR SPACES
281
'Finally the components of 1f*1 are given by the following. 5.1.10 propotltlon. LeI E and F be finile-dimensionl!' vec!or sPlfces and. cp E L(E, F). For ordered bases i= (e l , .. . ,en ) of E and 1= (/I .... ./m) of F. suppose that cp( e i ) = AQ'/Q' and leI t E T.0 ( F) have components I",h, relative 10 j. Then Ihe componenls of cp*t relalive 10 i are given hI' ( m*l) .... ,,"'j, - , h",,", Ah,J,
PI'fJOf.
(1f*1)".,,'"
···A",
I,'
(If·,)(eft ... ·.e,,)
=
t{ If(e,,) •...• If(e,,))
=
t{ Ah'J,/", ..... A\,/",)
=
I{ Ih, .....I", )A'\" . A\
=
'h
LA"'· .. A'" j. • 11
I'· "',
Examples. A. On R 2 with the standard basis {e l • e 2 }. let t E + 2e l ®e 2 - e2®e l + 3e2®e2 and let If E 1_
To2(R2) be given by 1= el®e
r~
:].
since A has an inverse matrix given by B = [ ~ I -; I ]. According to 5.1.7. t~e components of i = CP*I relative to the standard hasls of R 2 are given by iii=Biu 8 j "I U " ,
with 811=1,821=812=-1
and
8?2=2
so that ,-12 _ 8 118 21,11 + 8 118 22,12 + 8 128 21,21 + 812822,22
- t· ( -\ ).\ + I· 2· 2 + ( - 1)( - I)· ( - I) + ( - 1)·2·3 , =-4
= (-1)·1·1+( -IH -1)·2+2·1·( - 1)+2·( - 1)·3 = ,-11 -
=
-7
.:·,n l /
I
+ 8 118 12 ,12 + 8 12 8 11,21 + R12R12t22
1 . I . I + I . ( - 1)·2 + ( - I)· J.( - I) + ( - I ) -( - 1)3
=4
282
TENSORS
and
- (-1)·( -1)·1 +( -1)·2·2+2·( -1).( -1)+2·2·3 -II.
Thus, cp., - 4el~el -4el~e2 -7e2~el + Ile2~e2. B. Let ''''el~e2-2e2~e2ETII(R2) and consider the same map cp E L(R2. R2) as in part A above. We could compute the components of CP.' relative to the standard basis of R 2 using the formula in 5.1.7 as before. An alternative way is to proceed directly using S.I.6(iv). that is. the fact that CP. is compatible with tensor products. Thus
C.
Let t - -
2el~e2
E T2o(R 2 ) and cp E L(R\ R 2) be given hy the
matrix A - [~ ~ I ~]. Again we shal\ compute cp., E T2o(~3) by using the fact that cp. is compatible with tensor products and that the matrix of •• E L(R2. R1) is the transpose of A. Recall that cp·(e i ) is the ith row, since matrices act on the right on covectors. Denote by /1. /2, /1 the standard dual basis of R1. Then cp·(e l ) ... /1 + 2/1 and ••(e 2 ) _ - /2 + /3. so that cp.(,) ... -2 •• (e l )®cp.(e 2) = -2( /1 +2/3)~( _/2 + /3) = _2/1~/2 -2/1~/3 +4/3~/2 _4/3~/3 . •
Exercl...
S.IA Compute the interior product of the tensor' = el~el~e2 + 3e2~e2~ e l with e'" - e l +2e2 and u= 2e l + e 2. What are the (1,1) and (2.1) contractions of ,., S.IB Compute all associated tensors of , ... e lee 2 ee 2 + 2e2~el~e2 e2eelee' with respect to the standard metric of R2.
TENSOR BUNDLES AND TENSOR FIELDS
5.lC
Let
1=
283
2e l ®e l - e2®e l + 3e l ®e 2 and cp E L(R 2 .R 2 ). '" E L(R3.R2)
[!
be given by the matrices I :]. [~ ~ -;1]. Compute: . tr(I). cp*(I). ",*(1). trcp*(I). tr",*(t). cp*'. and all associated tensors of t. cp*(t). ",*(t) and cp*(1) with respect to the corresponding standard inner products in R2 and R). 5.ID Let dim(E)-n and dim(E)=m. Show that T,'(E: E) is an mn'+'· dimensional real vector space by exhibiting a basis. 5.2
TENSOR BUNDLES AND TENSOR FIELDS
We now extend the tensor algebra to local vector bundles. and then to vector bun.dles. For VeE (open) recall that V X E is a local vector ·bundle. Then V X 1','(E) is also a local vector bundle in view of 5.1.2. Suppose cP: V X E -. V' X E' is a local vector bundle mapping and is an isomorphism on each fiber; that is. CPu = cpl{u}X EE L(E. E') is an isomorphism. Also. let CPo denote the restriction of cP to the zero section. Then cp induces a mapping of the local tensor bundles as follows. 5.2.1 Definition. If cp: V X E -. v' X F' is a /0('(1/ vector bundle mapping such that for each u E V. CPu is an isomorphism. let If *: V X T.'( E) -co V' X T.'( F') be defined by
where t
E
1','( E).
Before proceeding. we shall pause to recall some useful facts concerning linear isomorphisms. 5.2.2 ProposlUon. Let GL(E. E) denote the set of linear isomorphisms from E to E. Then GL(E. E) C L(E. E) is open.
This was proved in 2.5.4. Let us also recall 2.5.5. 5.2.3 ProposlUon. Let (i': L(E. E) -. L(E-. E*); cp ..... cp* 'and GL(E. E) -co GL(E. E); cp ..... cp-I. Then (£ and ~ are of class coo and
Smoothness of (£ is clear since it is linear.
~:
284
TENSORS
5.2.4 ProposlUon. If cP: V x F --+ v' x F' is a local vector bundle map and CPu is an isomorphism for all u E V. then cp.: V X T,' ( F ) --+ V' X T,'( F') is a local vector bundle map and ( CPu). = ( cp.) u is em isomorphism for aI/liE V. MoreOt'er. if cP is a local vector bundle isomorphism thell so is CP •• Proof. That CP. is an isomorphism on fibers follows from 5.1.6 (iii) and the last assertion follows from the former. By 5.2.1 we need only establish that (CPu). = (cp.)u is of class C". Now. CPu is a smooth function of u. and. by 5.2.3. CP: and cP'; I are smooth functions of u. The Cartesian product of smooth functions is easily seen to be smooth and (CPu). is a multilinear mapping on a Cartesian product of smooth functions (this is not linearity in cp). Hence from the product rule ( CPu). is smooth. •
This smoothness can be verified also for finite-dimensional bundles by using the standard bases in the tensor spaces as local bundle charts and proving that the components (cp.t)i, .. \ "-j, are COO functions. We have the following commutative diagram. which says that CP. preseroes fibers: '1'. V X T,'( F) - - V' x T,'( F')
wI
1w'
v'
V
5.2.5 Definition. Let w: E --+ B be a vector bundle with E" = w . I(b) the fiber over bEB. Define T,'(E)=U"eBT,'(E,,) and 'IT;: T,'(E)--+B by .,,;(e)'" b if e E T,'(E,,). Furthermore,for a subset A of B. we put 1','( E)IA'" U /t8 ... T,'(E,,).lf"": E'--+ B' is another vector bundle and (cp. CPo): E --+ E' is a veclor bundle mopping with cp" = cpl E" an isomorphism for all b E B. leI CP.: 1','( E) --+ 1','( E') be defined by cp.1 1','( E,,) = (cp,,) •. Now suppose that (EI V, cp) is an admissible local bundle chart of w, where V c B is an open set. Then the mapping cp.IIT,'( E)I V] is obviously a bijection onto a local bundle. and thus is a local bundle chart. Further, (cp.),,;" (cp,,). is a linear isomorphism. so this chart preserves the linear structure of each fiber, which in this case is given in advance. We shall call such a chart a nalural chart of 1','( E). .
5.2.8 Theorem. If w: E --+ B is a veclor bundle. then the set of all natural charts of w:: 1','( E) --+ B is a vector bundle atlas. Proof. Axiom (VB 1) ping natural charts.
j,
(Ihvious. For md "' •. I
I •. n '\
'IUppose we have two overlap-
. 'i",,·ity. let them have the same
TENSOR BUNDLES AND TENSOR FIELDS
285
domain. Then a = '" 0 fP - I is a local vector bundle isomorphism. and hy 5.1.6. "'.~(fP.)-1 = a •• a local vector bundle isomorphism by 5.2.4 • This atlas of natural charts. the natural atlas of "'.r. generates a vector bundle structure. and it is easily seen that the resulting vector bundle is Hausdorff. and all fibers are isomorphic Banadlahlc: spaces. Hereafter. -r,'( E) will denote all of thi.f structure. 5.2.7 ProposlUon. If f: E -+ E' is a vector bundle map that is an isomorphism on each fiber. then f.: -r,'( E) -+ -r,'( E') is also a vector bundle map that is an isomorphism on each fiber. Proof. Let (U. fP) be an admissible vector bundle chart of E. and let (V. ",). he one of E' so that I( U ) c V and f"o/- = '" 0 f 0 fP I is a local vector bundle mapping. Then using the natural atlas. we see that (/ .)"' •. ~. = (/"'~,).. •
5.2.8 Proposition. Suppose I: E -+ E' and g: E' ..... E" are vector bundle maps that are isomorphisms on each fi~er. Then .WI is 1{ f. and 0
(i) (g 0 f). = g. 0 f •. (ii) If i: E -+ E is the identity. then i.: -r,'( E) -+ T:( E) is the identity. (iii) If f: E ...... E' is a vector bundle isomorphism. then so i.f I and (/.)
1=
(/- I) •.
Proof. For (i) we ell8mine representatives of (1{ 0 f). and g. 0 f •. These representatives are the same in view of 5.1.6. Part (ii) is clear from the definition. and (iii) follows from (i) and (ii) by the same method as in 5.1.6 .
We now specialize to the important case where "': E vector bundle of a manifold.
-+
•
B is the tangent
5.2.9 Definition. Let M be a manifoid and 1".11: TM -+ M its tangent bundle. We call r.'(M) = -r,'( TM) the ~clo" b"ndle 0/ tensors contravariant 0'" r IIlIII covariant orde,. s; or simp~)' 0/ type (.~). A Iso Tt( M) is called the cotangent bundle 0/ M and is denoted by 1":': PM --+ M.
Since E can be embedded in E**, we may thereby embed TM as a closed submanifold in TOI( M). If M is finite dimensional we may identify TM = Tol( M). The zero section of r.'( M) may be identified with M. Recall that a section of Ii bundle assigns to each base point b a vector in the fiber over b. In the case of -r,'( M) these vectors are called tensors. Also. the addition and scalar multiplication of sections takes place within . each fiber. The COO sections of "': E -+ B were denoted r OO ( "'). or r OO ( E).
186
TENSORS
Jlecall that '5'( M) denotes the set of mappings from M into A that are of
class COO (the standard local manifold structure being used on A) together ~th its structure as a ring; namely. f + g. cf. fg for f. g E ~(M). c E A are given by (f + g)(x) =- f(x)+ g(x). (cf)(x) = c(f(x» and (fg)(x) = f(x)· g(x). Finally. recall that a vector field on M is an element of ':X (M) = r"'(TM).
5J.10 DefinlUon. A lellSor field of type (~) on a manifold M is a Coo section of T,'( M). We denote by '5.'( M) the set f"'(T,r( M» together with its (infinite-dimensional) real vector space structure. A coveclor field or dille,,",titIf OIfI-Io"" is an element of *( M) = 5 10 ( M ). IffE'5'(M) and IE5.'(M). let fl: M-+T,r(M): m ...... f(m)t(m). If obo XjE 'X(M). i-I ... .• s. and a i E *(M). j = I •... ,r, let
ex
ex
'(al •...• a'. XI ..... X.): M .... A; m ...... I(m}{ al(m), .... X.(m»). (falso t' E 5/( M). let I®I': M .... T,'++/(M); m ...... l(m)®I'(m). 5.2.11 ProposlUon. I(CXI .....
X,) E
With f. t. Xj' a i • and I' as in 5.2.10. fl E '5:( M).
~(M), and I®I' E ~1:.:~·.
proof. The differentiability is evident in each case from the product rule in l(lCal representation. •
Note that for the tangent bundle TM. a natural chart is obtained by taking TIJI. where IJI is an admissible chart of M. This in turn induces a chart (f,,), .. T,'IJI for T,rM. We shall call these the natural charts of T,'M. Now we turn to the expression of tensor fields in local coordinates. p-ecallthat a/ax; - (TIJI)-I(e i ), forlJl: U .... U'c An a chart on M, is a basis of~(U). The vector field a/ax; corresponds to the derivationf ...... ilf/ax'. Since dx i( a/ax}) = axil ax] = 6j, we see that dx j is the dual basis of l at every point of U, i.e., that dx i = 1JI*(e i ). where el ..... e n is the dual ))8Sis to el .... •e n • Let
a;ax
i, a/ax" , .•. , il/ax I .) E 6r(U) t i""i,jl"'j. .... I(dx i ,, ... ,dx ., J .
fben applying 5.16(iv) at every point yields
",him is the coordinate expression of an G) tensor field. To discuss the behavior of these components relative to a change of c;oordinates. assume that Xl: U ..... R. i-I ... . ,n is a different coordinate
TENSOR BUNDLES AND TENSOR FIELDS,
287
system. We can write a/ax/-a/a/ax i , since hoth are bases of 'X(U), Applying both sides to x" yields a/=axi/ax i i,e., (a/ax i ) (axi/ax/Xa/ax i ). Thus the dx i as dual bases change with the inverse of the Jacobian matrix ax i / axi; i.e., dx i = (ax i/ ax) ) dx'. Hence writing t in both coordinate systems and isolating equal terms gives the following change of coordinate formula for the components:
_""
ax '" , 'axl,
ax Ie, axh ax" a.t l,
(Ix it
I ,'" '/ ... / = - - ... - - . - - ... --t" iJ.~/.
'j,
i,"'i.
This formula is known as the len,wriality aiterion. A set of n'U functions li' .. ··/l, ...},/or each coordinate syslem on Ihe open sel U of M locally define an (:) tensor field iff a change of coordinates has the aforemenlioned effect on them. This statement is clear since at every point· it assures that li''''/l''''j,(u) are the components of an (~) tensor in T"U and conversely. The algebraic operations on tensors, such as contraction, inner products and traces, all carry over, fiberwise, to tensor fields. For example, if . 1m E TII(T",M) is the Kronecker delta, then I: M .... TII(M); m ..... 1m is obviously C"", and I E ~II(M) is called the Kronecker delta. Similarly, a tensor, field of type (~) or (~) is called symmetric, if it is symme,tric at every point. A basic example of a symmetric covariant tensor field is the following.
a
5.2.12 Definition. A weak pseudo-RierntlMitur IMtric on manifold Mis E T20(M) Ihal is symmetric and weakly nondegenerate, i.e., such that al each m EM, ,( m )( 0"" wm) = 0 for all "'m E Tm M implies q" - O. A' (strong) pseudo-R;'lIUIlfIIitur ""'ric is a 2-tensor that. in addition is (strongly) ~rtlteforall me M; i.e., the map om .... '(m)(o"'.·) isa (Banachable space) isomorphism of TmM onto T!M. A weak (re,fp. strong) pseudoRiemannian metric is called weak (resp. strong) Riema",," if in addition ,(mX om' "... ) > 0 for all q" E T.M, "m" O. .
alensor g
A strong Riemannian manifold is necessarily modeled on a Hilbertaable space; i.e., the model space has an equivalent norm arising from an inner product. Also, for fmite-dimensional manifolds weiak and strong metrics coincide: indeed T.M and T:,M have the same dimension and so a one-to-one map of T",M to T!M must be an isomorphism. It is possible to . have weak metrics on a Banach' or Hilbert manifold that are not strong. For example. the L2 inner product on M - CO aO, I),R) is a weak metric that is not .trong. For a similar Hilbert space example, see Exercise S.2e. Any Hilbert space is a Riemannian manifold with constant metric equal to the inner product. A symmetric bilinear (weakly) nondegenerate
286
TENSORS
Recall that iff( M) denotes the set of mappings from Minto R that are of class Coo (the standard local manifold structure being used on R) together with its structure as a ring; namely, I + g. cl. Ig for I, g E ~(M), c E Rare given by (f + g)(x) ... I(x)+ g(x). (cf)(x) = c(f(x» and (fg)(x) = I(x)' g(x). Finally, recall that a vector lield on M is an element of ~X(M)'" foo(TM).
5.2.10 DefInition. A tftUor lield 01 type (~) on a manilold M is a Coo section 01 T,'( M). We denote by ~T.'( M) the set f""( T,'( M» together with its (infinite-dimensional) real vector space structure. A covector lield or di/fefWItitll OM-form ;s an element 01 <X *( M) = '5 10 ( M ). II IE iff(M) and t E '5.'(M), let It: M -- T:(M): m ...... I(m)t(m). II also Xi E <X(M), i-I, .. . ,s, and a i E <X *(M). j = I •... . r. let
I( a l , ... ,a', XI , .. . ,X.): M II also t' E '5/( M), let tet': M
5.2.11
Propoaltlon.
I( a l •.. .• X,)
E
-- R; m ...... t(m)( a l ( m ) •.. .• X.(m»). -+
T,'++/(M): m ...... t(m)el'(m).
With I. t. Xi' a i • and t' as in 5.2.10. It E ~j/( M).
~(M). and tet' E~. ::~'.
The dirrerentiability is evident in each case from the product rule in local representation. •
Proof.
Note that for the tangent hundle TM. a natural chart is obtained by taking TfJ!. where fJ! is an admissible chart of M. This in turn induces a chart (TfJ!)." T,'fJ! for T,'M. We shall call these the natural charts of T,'M. Now we tum to the expression of tensor fields in local coordinates. Recall that a/ax i - (TfJ!)-I(ei ). for fJ!: U -oU'c R W a chart on M. is a basis of <X(U). The vector field a/ax i corresponds to the derivation I""" al/ax/. Since t/x i ( aI axi) = axil axi = 8 ii • we see that dxi is the dual basis of aI ax; at every point of U. i.e.• that t/x i = fJ!*( e i ). where e I .... • e lt is the dual basis to el .... ,ew' Let
Then applying 5.I6(iv) at every point yields
which is the coordinate expression of an (;) tensor field. To discuss the behavior of these components relative to a change of coordinates, assume that Xi: U -0 R. i ... I •...• n is a different coordinate
TENSOR BUNDLES AND TENSOR FIELDS,
287
system. We can write alax ' - a/alax j , since hoth are bases of ~.(U). Applying both sides to Xk yields a/ = ax ilax; i.e., (alax l ) ... (axilaxi)(alax i ). Thus the dx; as dual bases change with the inverse of the Jacobian matrix ax j I ax;; i.e., dx i = (axil ax ') dx '. Hence writing t in both coordinate systems and isolating equal terms gives the following change of coordinate formula for the components:
"11
. 'I.
This formula is known as the tensoriality criterilln. A set of n'+J junctions ti""ij, ...,.!or each coordinate system on the open set U of M locally define an (:) tensor field iff a change of coordinates has the aforementioned effect on them. This statement is clear since at every point it assures that ti""\"'j,(u) are the components of an (;) tensor in T,p and conversely. The algebraic operations on tensors, such as contraction, inner products and traces, all carry over, fiberwise, to tensor fields. For example, if . 1m E TII(T",M) is the Kronecker delta, then I: M - TII(M); m ...... I", is obviously Coo, and I E 5 11( M) is called the Kronecker delta. Similarly, a tensor, field of type (~) or (~) is called symmetric. if it is symme.tric at every point. A basic example of a symmetric covariant tensor field is thc= following.
a
5.2.12 Definition. A weak psellllo-RiI"""",iIut metric on manifold M is tensor g E T20(M) that is symmetric and weakly nondegenerate, i.e., such that at each m EM, g( m )( """ HI",) = 0 for all HI", E T", M implies "", = o. A' (strong) pseudo- Rientlllllfilut metric is a 2-tensor that. in addition is (strongly) IfDIfIIqeMfYlte for all m EM; i. e .• the map "m ..... I( m )( f'",. ' ) is a ( Banachable space) isomorphism of T",M onto T:,M. A weak (resp. strong) pS('IJdoRiemannian metric is called weak (resp. strong) Riemannilut if in addition I( m)( "",. "",) > 0 for all ,,'" E T", M. "", .. o.
ci
A strong Riemannian manifold is necessarily modeled on a Hilbertizable space; i.e., the model space has an equivalent norm arising from an inner product. Also. for finite-dimensional manifolds weak and strong metrics coincide: indeed T",M and T:,M have the same dimension and so a one-to-one map of T",M to T:,M must be an isomorphism. It is possible to , have weak metrics on a Banach' or Hilbert manifold that are not strong. For example. the L 2 inner product on M = CO «(0, I J, R) is a weak metric that is not ~trong. For a similar Hilbert space example, see Exercise S.2C. Any Hilbert space is a Riemannian manifold with constant metric equal to the inner product. A symmetric bilinear (weakly) nondegenerate
288
TENSORS
two-form on any Banach space provides an example of a (weak) pseudoRiemannian constant metric. The most famous pseudo-Riemannian manifold that occurs in the theory of special relativity is R4 with the MinkiJwski pseudo -Riemannian metric
.
g(x)( v. w) =
Vlw l
+ V 2 W 2 + V 3 w3 -
V4 W4•
where x. v. K' E R4. As in the algebraic context of Section 5.1. pseudo-Riemannian metrics (and for that matter any strongly nondegenerate bilinear tensor) can be used to define associated tensors. Thus the maps I. b become vector bundle isomorphisms over the identity~: TM ... T* M.': T* M ... TM • • - b I. where v!. = g( m)( vm .·). In particular. they induce isomorphisms of the spaces of sections b: ~"\ (M) ... ~~ *( M). #: ~l: *( M) -+ ~X (M). In finite dimensions this becomes the operation of raising and lowering indices. Thus formulas like the ones in Example 5.1.4E should be read pointwise in this context. There is. however. a particular index raising operation that requires special attention. 5.2.13 DetlnHlon. LeI M he a pseudo-Riemannian n-manifold with metric g. For f E ~(M). grad f = (df)# E ~l: (M) is called the grrulimt off. To find the expression of grad f in local coordinates, note that if gij
then for
a
X_XI_. ax'
we have
Xb( Y) - g( X. Y) - X'Yie(
a~i' a~j ).
g(
=
and
a
Y=Y'-. ax'
a~;' a~j ) .. Xiyjg,}.
i.e.
If II E ~X *( M j has the coordinate expression II = a; dxi. we have CJ!. = a;gi}a I ax) where gij is the inverse of the matrix gij' Thus for 11:= df. the
local expression of the gradient is gradf = gijJ.L
~
ax} iJx'
If M
=
or (gradfr
=
gi} af. ax}
R n with standard Euclidean metric g" = S", this formula becomes g.rad f =
af ~:
il:t' il:t'
i.e..
g.rad f =
(~ . . . .. : ~n ). il.\
(l.'
the familiar expression of the gradient from vector calculus.
TENSOR SUNDLES AND TENSOR FIELDS
289
Now we turn to the effect of mappings and diffeomorphisms on tensors. 5.2.14 DefinlUon. If fJi: M ~ N i.f a diffeomorphi.fm and t E T,'( M), let fJi.t - (TfJi). ° to fJi ',the push-forward of t by fJi. If t E T,'( N). the puN-back of t by fJi is given by fJi*' = (fJi - ').t. 5.2.15 PropoalUon. If fJi: M ~ N is a diffeomorphism. and t E '?i.'( M). then (i) fJi*t E '?i,'(N); (ii) fJi.: '?i,'(M) ~ '?i,'(N) is a linear isomorphism. (iii) (fJi. y,). - fJi*o y,.; and (iv) fJi.(t®t') - fJi*'®fJi*t'. where t E ~'i:(M) and t'E~/(M).
Proof. (i) The differentiability is evident from the composite mapping theorem. together with 5.2.4. The other three statements are proved fiberwise. where they are consequences of 5.1.6. • . As in the algebraic context. the pull-back of covariant tensors is defined even for maps that are not diffeomorphisms. Globalizing 5.1.8 we . get the following. If fJi: M ~ Nand t E ~'i.o(N). then fJi*l, the puN-back
5.2.16 Definition. of I by fJi is defined by
for
mE
M. v" .... v,
E
TmM.
The next proposition is similar the proof of 5.1.9.
!(l
5.2.15 and is pfllved by globalizing .
5.2.17 ProposlUon. If fJi: M ~ N is Coo and I E '?is o( N), then (i) fJi*1 E '?i,o(M), (ii) fJi*: '?i,o(N) ~ '?i,o(M) is a linear map; (iii) (y, ofJi)* = fJi*o y,* for y,: N ~ P; (iv) if cp is a diffeomorphism ,hen cp* is an isomorphism with inverse CP •• and (v) if 'I E 5s~( N). '2 E '!j,~(N). fJi*( ',®t 2 ) = (fJi*', )®( fJi*/ 2 ).
For finite-dimensional manifolds the coordinate expressions of the pull-back and push-forward can be read directly from 5.1.7 and 5.1.10. taking .into account that Tcp is given locally by the J&cobian matrix. This yields the following. . 5.2.18 PropoSition. LeI M and N he finite-dimensional mallifold~. cp: M ~ N a C' map and denote by .~.I = cpl(X I..... x P ) the local expression of cp relative to charts where p = dime M) and j = I .... , n = dime N).
290
TENSORS
(i)
If 'E~/(M) and cp is a diffeomorphism. the coordinates of the cp • tare
push~forward
If' E ':i/(N) and cp is a diffeomorphism, the coordinates of the pull-back cp.' are
(ii) If' E '.'i, o( N) and 11': M ~ N is arhitrarr. the ('()(}rdinates of the pull-hack cp.' are
Notice the similarity between the formulas for coordinate change and pun-back. The situation is similar to the passive and active interpretation of similarity transformations PAP-I in linear algebra. Of course it is important not to confuse the two. 5.2.19 Exampl... A. Let cp: R2--+R2 be defined by cp(x.y)=(x+ 2y. y) and lett = 3x (a/ax)®dy +(a/ay)®dy E '.'i11(R 2 ). Then the matrix of cp. on vector fields is (
;~ ) = [~ ~ ]
and on forms is
(;;:) =
[~
t]- [6 -2] I . I
=
In other words. cp.(
a: ) :x. =
cp.(dx) = dx - 2 dy. and
CP.(~v)=dy.
Noting that cp -I(X. y) = (x - 2y. y). we gel
CP.' = 3( x - 2y )cp.( :x )®cp.( ~r) + cp.( ;~ ) ®q>.( eld =
a
(aax iil) II ®dl" + - a ®d)'. ®dl" = (3x -6r +2)ly' . ax' ay
3(.t-2r)- ®dy+ 2- + .
ax
TENSOR BUNDLES AND TENSOR FIELDS
291
B. With the same mapping and tensor. we compute ",*1. Since
",*(..!.) =..!., ax ax ",*(dx) =dx +2dy, and ",*(dy)
=
dy.
we have
a ®dry + (- a iJ = 3(x, + 2.1')2 - + - a) ®dry = (3x +61' -2)-'®dv + - a ®d~'. ax ax ay . iJx" a.I' , C. Let "': R J ..... R 2 • ",(x.y.z)=(2x+z,xyz) and 1:=(u+2v)du® dU+(U)2du®dv E T2o(R 2 ). Since ",*(du) = 2dx + dz and ",* (00) = yzdx + xzdy + xydz, we have ",*, = (2x
+ z +2xyz)(2 dx + dz)®(l dx + dz)
+(2x + z)2(2dx =
+ dZ)®(yzdx + xzdy + xydz)
2[4x +2z +4xyz +(2x + Z)2 yz] dx®dx +2(2x + z ('adx®dy
+2[2x + z +2xyz +(2x + z )2xy] dx®dz + xz(2x + z)2 dz®dy + [2x
+ [4x + 2z +4xyz + (2x + z )2] dx®d:(
+ z +2xyz + xy(2x + z )2] dz®dz.
D. If "': M - N represenis the deformation of an elastic body and. is a Riemannian metric on N, tben C - fJI*1l is called the Cauchy - Green tensor: in coordinates
Thus, C
~asures
how fJI deforms lengths and angles.
•
Finally, we describe an alternative approach to tensor fields. Suppose '?t(M) is defined as before, and ~(M) either our way or some equivalent way. With the "scalar multiplica\ion"
IX
becomes an ~(M)-module. That is, ~ (M) is essentially a vector space over ~(M), but the "scalars" ~(M) form only a commutative ring with identity,
292
TENSORS
rather than a field. as
Ilf may not exist. even if f '* O. We may thus define
L"f(M'(~X (M).'S( M» = P( M)
the
~1 ( M)
linear mappings. and similarly '1:;( M) == L;~~)( P( M ) •...• ~(M); 5( M»
the -:t( M) multilinear mappings. From 5.2.10. we have a natural mapping '!r,'( M) ~ '1:;( M) which is lj( M) linear.
5.2.20 ProposlUon. Let M be a finite-dimensional manifold or be modeled on a Banach space with norm C'JO aw~y from the origin. Then '?T:( M) is isomorphic to the lj(M) multilinear maps from .P( M) X ... X ~x ( M) into lj(M). regarded as lj(M) modules or as real vector spaces. In particular. ~ *( M) is isomorphic to P( M). Proof. Consider the map '?T,'( M) ~ L,,( M,(.(*( M), ...• ~X (M); \1 ( M» given by l(al •...• a'.XI •...• X.)(m)=/(m)(al(m) •...• X.(m». This map is clearly -:t( M) linear. To show it is an isomorphism. given such a multilinear map I. define I by I( m)( a l ( m)•...• XJ ( m» = I( a l •...• X,)( m). To show this is well defined we must show that. for each 11, E Tm( M). there is an X E '.\ ( M) such that X( m) ... q.,. and similarly for dual vectors. Let (V. cp) be a chart at m and let T",cp(q.,) == (cp(m), G,). Define Y E ~X.(V') by Y(u) == (u'. q;) on a neighborhood VI of cp(m). Extend Y to V' so Y is zero outside V2 • where cl(VI)c V2 • cI(V2 )c V'. by means of a bump function. Define X by X" - Y on V. and X - 0 outside V. Then X( m) "" q.,. The construction is similar for dual vectors. Also. I(m) so defined is Coo; indeed. using the chart CPt the local representative of , is Coo by Box 3.3A since I induces a Coo map M X T,'( M) ~ R (by the composite function theorem). which is (r + s)linear at every m E M. If M is finite dimensional this last step of the proof can be considerably simplified as follows. In the chart cp with coordinates (x' •.... x").,=ti,·\ .. ,.(illilx")®··. ®illilx"®dx"®'" ®dx" and all' components of , are C'" by hypothesis. •
The preceding proposition can be clearly generalized to the C A situation. One can get around the use of a smooth norm on the model space if one assumes that the multilinear maps are localizable. that is. are defined on ~*(V)X ... x 'X.(V) with values in 'ff(V) for any open set U in a way compatible with restriction to V. We shall take this point of view in the next section. The direct sum '!reM) of the '?T:< M). including '?Too( M) = lj( M). is a real vector space with ® -product. called I" . ' , . " . , J!f<,ebra of M. and if cp: M ~ N is a diffeomorphism. CP.: '?T (M)" i'" .llgebra isomorphism.
THE LIE DERIVA TlVE: ALGEBRAIC APPROACH
293
The construction of ~1 ( M) and the properties discussed in this section can be generalized to vector bundle valued (~) tensors (resp. tensor fields), i.e., elements (resp. sections) of L(T*Me ... eT*MeTMe ... eTM, E), the vector bundle of vector bundle maps from T*Me ... eTM (with r factors of T* M and s factors of TM) to the vector bundle E, which cover the identity map of the base M. Exercl...
S.2A
Lei cp: R 2 \{O} -oR 2 \{O} be defined by cp(x. y) = OX2 + tv 2 • - xy)' and let t = xiJ / iJx@dx@dy + yiJ / fly@dl'@dy. Show that cp is a diffeomorphism and compute CP.t. cp*t. Endow R 2 with the standard and Riemannian metric. Compute the associated tensors of t. cp., as well as their (1. I) and (1.2) contractions. What is the trace of the interior product of t with iJ / + xiJ / fir? Let cp: R 2 -oR\ cp(x.y)=(y.x.y+x 2 ) he thc deformation of an elastic body. Compute the Cauchy-Green tensor and its trace. Let H be the set of real sequences {a n },.-I.2. such that IIll,,1I 2 = Ln2a~ <00. Show that H is a Hilhert spal·C. Show that g(a.h)= La~bn is a weak Riemannian metric on H that i~ nnt a strong metric.
CP.'.
ax
S.28 S.2C
5.3 THE LIE DERIVATIVE: ALGEBRAIC APPROACH This section extends the Lie derivative Lx from vector fields and functions to the full tensor algebra. We shall do so in two ways. This section does this algebraically and in the next section. it is done in terms of the flow of X. The two approaches will be shown to be equivalent. . We shall demand certain properties of Lx such as: if t is a tensor field of type (~). so is Lx', and Lx should be a derivation for tensor products and contractions. . First of all. how should he defined on covector fields? If Y is a vector field and Cl is a covector f' i ! 1hen the contraction Cl' Y is a function, so Lx (Cl' Y) and Lx Yare alread 'HIed. (See Section 4.2). Jiowever. if we would like to have the derivation pwperty for contractions. namely
"x
this forces us to define LxCl by
(LxCl)'Y= Lx(Cl·Y)-Cl·(LxY) for all vector fields Y. Since this defines an ~'f(M)-linear map. LxCl is a well-defined covector field. The extension to general tensors now proceeds inductively in the same spirit.
294
TENSORS
5.3.1 DefInition. A diflnwrlilll operator on the full tensor algebra '5 ( M) of a manifold M is a collection (O;( U)} of maps of '5.'( U) into itself for each r and s ~ 0 and each open set U C M. any of which we denote merely 0 ( the r, s and U are to be inferred from the context). such that (DOl)
0 is a lemor Iknvat;on i.e. 0 is R-linear and if
IE '5:( M). Cl I•...• Cl, E ~X *( M) and Xl •...• X .• E ~X (M). then
O(I(ClI, ... ,Cl,. XI •...• X.))
=
(OI)(ClI •...• Cl,. XI.···.X.)
,
+
$
E I(ClI,···.OCl/ •...• Cl"
XI.···,X.)
+
E I(ClI •...• Cl,. XI.···,OX" •...• X.,).
k-I
j-I
We sometimes refer to this by saying that 0 COIIUnIIln with COIIlrtlC-
tHwu. (D02)
0 is local. or is IIIltIIral with rnpect 10 rntrict;OIU. That is. for U eVe M open sets. and t E ~'(V) (Ot)IU= O(tIU) E c:r:(U)
or the following diagram commutes:
'5/(V) -
IV
01
~'(U)
10 , ~T:(U)
5/( V) IV
Note that we do not demand that 0 be natural with respect to push-forward by difreomorphisms. The reason is that it is not needed for the following unique extension theorem, and indeed. the latter can be used to extend the covariant derivative. which is not natural with respect to diffeomorphisms. .
5.3.2 Theorem. Suppose for each open set U c M we have maps Eu: Iff( U) ..... Iff( U) and Fu: <.X. ( U ) ..... X ( U). which are (R linear) tensor derivations and natural with respect to restrictions. That is
f. g E '!leU);
THE LIE DERivA TIVE: ALGEBRAIC APPROACH
295
For/ E 6J(M). Eu(/I V) = (EM!) I V; Fu(/®X)- (Eu!)®X + /®FuX; (iv) for X E 'X(M).Fu(XI V) = (FMX)I V.
(ti) (iii)
Then there is a unique differential operator D on ~ ( M) that coincides· with EIJ on ~T( V) and with Fu on ~l: (V). Proof. Since D must be a tensor derivation. define D .on 'X *( V) by (Da)·X=D(a·X)-a·(DX)=Eu(a·X)-a·FuX for all XE~t(V). Thi~ shows that D exists and is unique on 'X *( V) (by the Hahn- Banach theorem). The rest of the theorem follows by requesting (DOl). namely. define Du on ~'(V) by
(Dut)(a, ..... a,. X, ..... X.) = EIJ(t(a, ..... a,. X, ..... X,)
,
- L
,-, t(a, ..... DaJ ..... a,.X, ..... X~) ..
- L
t(a, ..... a,. X, ..... FuX" .... ,X.).
k-I
If V is any open subset of V, by (ii) and (iv) it follows that Dv(tl V)(Dut)W. This enables us to define D on 6J(M) hy (Dt)(m) = (Dut)(m), . where V is any open subset of M containing m. Since D/. is unique. so is D. and so (D02) is satisfied by the construction of D. •
5.3.3 Corollary.
We have D(t,®t 2 ) = Dt,®t 2 + t,®Dt 2
(i) and
D8 = O. where 8 is Kronecker's delta.
(ii)
Proof. (i) is a direct application of (DOl). For (ii) let aE 'X*(U)'and X E 'X(V) where V is an arbitrary chart domain. Then
(D8)(a. X) == D(8(a.X»-8(Da, X)-8(a, DX) =
D(a·X)-Da·X-a·DX-O.
Again the Hahn-Banach theorem assures that Da = 0 on V, and thus by (D02), Da ==
o. •
296
TENSORS
Taking Ev and Fv to be LXI v we see that the hypotheses of theorem 5.3.2 are satisfied. Hence we can define a differential operator as follows.
5.3.4 Definition. If X E ~'X ( M). we let Lx be the unique differential operator on '!i( M). called the Lie tkrivative wit" IYS~ct to X. such that Lx coincides with Lx as given in 4.2.6 and 4.2.20. 5.3.5 Propoaltlon. Let cp: M -. N be a diffeomorphism and X a vector field on M. Then Lx is ""tUM Moit" res~ct to pusIt-/OrwtlN by cp; that is, L •• xcp.t .. cp.Lxt for t E '!i,'( M). or the following diagram commutes; ~T:( M)
Lx
'1'.
~'i:( N)
!
5:( M)
1Lrp.x
"'.
.
~:(
N)
Proof. For an open set V eM define D: ~1:(U) -. '~:(V) by Dt = cp·L",.xl v( CP.t), where we use the same symbol cp for cpl V. By naturality on '5"(V) and <.X(V). D coincides with LXIV on '5"(V) and 'X.(V). Next. we show that D is a differential operator. For (001). we use the fact that
which follows from the definitions. Then for X. XI'" .• X, E a l ..... a, E '.\ *( V). D( t( a l ..... a,. Xl" .. • X.))
=
':X (V) and
°
cp*Lrp.x (cp.( t( 1" ... a,. XI'" .. X.»)
, +
L
(cp.t)(cp.al.· ... Lrp.xcp.o, ..... cp.a,.·j\I ... ·.cp.X.)
i-I J
+
E (cp.t)(cp.al ..... cp.a,.cp.XI ..... Lrp.xcp.XA ..... cp.X.)]. k -I
THE LIE DERIVATIVE: ALGEBRAIC APPROACH
297
By (DOl) for Lx. this becomes
, (01)(°1 •...• 0,. XI' ...• X.)+ ~ 1(°1•...• 0°, .... ,0" Xl ... ·'~.• ) ) -I .f
+ ~ t(al, ...• a"
XI'· .. ·OXk .. · .. X.)
A -I
since cp. - (cp I). by 5.2.14. For (D02), let
IE
'.'i, r( M) and write
[by (1)02) for Lx] =O(tIU)
The result now follows by 5.3.2. • The reader can check, by using the same reasoning, that a differential .operator that is natural with respect to diffeomorphisms on functions and vector fields is natural on all tensors. Lt;t us now compute the local formula for Lxt where I is a ,tensor field of type (~). Let cp: U c M ~ VeE he a local chart and let X and 1 he the principal parts of the local representatives, CP.X and cp.1 respectively. Thus X: V ..... E and;: V ..... r,'( t:). Recall from Section 4.2 that the local formulas . for the Lie derivatives of functions and vector fields are: (Lxff(x)
=
Dj(x)·X(x)
(I) ,
and (LxY nx) = DY(x )·X(x)- DX( \). Y( r)
(2)
In finite dimensions these become Lxf=
x,!..L ax'
and (2')
298
TENSORS
Let us first find the local expression for Lxa where a is a one-form. By 5.3.5. the local reprc:.sentative of Lxa is CI>.(Lxa) = L.".xCl>*a, which we write as Lj&' where X and &. are the principal parts of the local representatives. so X: V ..... E and &.: V ..... E*. Let vEE be fixed and regarded as a constant vector field. Then as Lx is a tensor derivation, Li(&"V)'" (Li&)'''+ &·(Liv).
By (I) and (2) this becomes D(&· v)·X- (Lia)'v - &. (DX· v).
Thus (Li&')'V= (D&"X)·v+&.·(DX·v).
In the expression (D&"X)'v, D&.·X means the derivative of &. in the direction X; the resulting element of E* is then applied to v. Thus we can write Li&'= D&.·X+ &.·DX.
(3)
In finite dimensions, the corresponding coordinate expression is ( L a) v' x,
=
aa;. ax'
ax'. 'ax;'
-XlV' + a.--v'·
i.e.,
.aa;
ax}
( L a).=X'-+ax, axi , ax;
(3')
Now let t be of type (~), so i: V ..... L(E*, ... ,E*, E, ... ,E; R). Let a l , ... ,a' be (constant) elements of E* and VI" ..• v, (constant) elements of E. Then again by the derivation property.
Lx (1( a l •...• a'. VI'"
.• V,)] = (Lil)' (a l •...• a'. VI'" .• V.)
, +
~-( a I •...• L ; a."I.···.v, ' ) LJt ia •...• i-I J
+
E ~al •...• a'.vl.···.Li"i.· ..• V,). i-I
Now using the local formula (I )-(3) for the Lie derivatives of functions.
299 '
THE LIE DERIVATIVE: ALGEBRAIC APPROACH
vector fields. and one-forms. we get
(DLf)· (ex l•...• ex'. Vi •••• ,Vs ) = (Lii)( ex l •...• ex'. t'I ..... 1', )
,
+ 1: l(ex l..... ex;·1)X ..... ex'.v1..... 0.) i-I
.r
+
E ~exl •...• ex'.VI ..... -
DX·o; ..... O,).
i-I
Therefore.
,
- E l(ex l..... exi ·DX..... ex'.v 1..... b,) i-I
s
+
E i(exl ..... ex'.I'I ....• DX.O, ..... V,). i-I
In components. this reads
(L
x' );'''';'
i,"'i, =
X.
a t i,"·i';, ax.
"'j,
_ axi,I t 1i''''i,i "'}' _
ax
"
ax t , "m}" ".; + --. m
aX"
".,"
"
(
. d'Ices) aII upper In
. d'Ices) + ( aIIIower In
(4)
We deduced the component formulas for Lx' in the case of a finit~ dimensional manifold as corollaries of the general Banach manifold formulas. Because of their importance. we shall deduce them &gain in a different manner. without appealing to 5.3.5. Let
. . . -~ a ... '=1"""' ,
II'''}.
axi,
a·
"t:r',( U )
~_~dxJI ••• ~dxJ'E~1
axi,
where U is a chart domain on M. If X=
x"a/ax'.
•
.
the tensor derivation
300
TENSORS
property can be used to compute Lx" For this we recall that
and that
by the general formula for the bracket components. The formula for
Lx(dx') is found in the following way. The relation
8\ = dx*(a/ax i )
implies by (001) that
- (L.(dx'») (- a) + dx' (ax' - -. - a ). ax' ax' ax' Thus
(LX(dx"»)(~) -dx'( ax' ~) _ ax'.
ax'
ax' ax'
ax'
io Lx(dx')-(ax'/ax')dx'.
Now ODe simply applies (DOl) and collecta terms to pt the same local formula for Lx' found in (4). Note espcciaIly that
LI
hi
(.L)_I ax J
and
L..!...(dxJ)-I,
foraD i,j.
la'
U.I~
A. Let
,- x :, edxedy + , Compute
L.,.
:, edyedy and X -
:x + x :"
THE LIE DERIVATIVE: ALGEBRAIC APPROACH
301
Solution. Method 1. Note that
LJtI-LiI •• _a_I-I•. ~I+L.at
(i)
. III'
lit
i/o
if ..•
Li-I=L~{X~®dX®dY+ )'~®dY®d)'} a" i/x ay i1y
(ii)
= L:!.(X~ ax fJ y
®dX®dv)+L~ ()'~ ®dY®dr) . I/x . ay "
=
11_ ®dx®ti)' +0 ,Ir '
Now note Lxi-
1/1'
~ =O.L.i- ~ = - L~(X~) = fJy '.I' ax 'ax ay
L,,~dx ar
= O. and
-{I' ~a)' +X.O}
=
-~. ay
L,,!_dy = dx. I/r
Thus (iii)
L i! , = L '.1.1'
=
.iI '.1..
{x~ ®dx®dl' + y~ ®dy®dv} fJy . ay "
(0+0+0+ x :v ®dX®dX) a ®dy®dy +0+ y-a a ®dx®d)' + )'-~ a ) + ( x-a Y y - . oy ®dy®dx .
Thus. substituting (ii) and (iii) into (i). we find
a a a, + x-a y ®dy®dy + y-a )' ®dx®dy + .v-a y ®dy®dx
a
a
= (y + I) fJy ®dx®dy + x ay ®dx®d.\
a
a
+ x ay ®dy®~v + y ay ®dy®dx. Method 2. Osing component notation. I is a tensor of type (~) 'whose nonzero components are t 2 '2 = x and t 222 = y. The components of X are
302
TENSORS
x' -I and X 2 ""' x. Thus, by the component formula (4), ; ,ax; ; ax m (L xl ) ;jk'" X" - Ia e t jle - t jle --, + t ",Ie - - . ax ax ax'
;
+ t jp
ax p Ie' ax
The nonzero components are
(LXI)\, = 0-0+0+ x = x; (LXI)222'" x -0+0+0 = x; (LXI)\, = 0-0+0+ y = y. and hence
a
a
LxI = (y + I) ay ®dx®dy + x iJy ®dx®dx iJ
+ x ay
®dy®dy
+y
iJ ay ®dy®dx.
The two methods thus give the same answer. It is useful to understand both methods since they both occur in the literature, and depending on' the circumstances, one may be easier than the other. B. In Riemannian geometry vector fields X satisfying Lx g = 0 are called Killing vector fields; their geometric significance will become clear in the next section. For now, let us compute the system of equations that the components of a Killing vector field must satisfy by the method described in the previous example. If X = xla I ax;, and g == g;jdx; ®dxi, then
Note that Lxg is still a symmetric (~) tensor, as it must be. Hence X is a Killing vector field iff its components satisfy in every chart the following
THE LIE DEfflVA TlVE: ALGEBRAIC APPROACH
303
system of n partial differential equations, called Killing '.~ equations
ag;j ax" ax" x"--" + g"j--' + g;,,--. = o. ax ax' ax' C. In the theory of elasticity, if. represents the displacement vector field, the expression L.g is called the slrain tensor. As we shall see in the next section, this is related to the Cauchy-Green tensor C = .,,*g by linearization of the deformation .". p. Let us show that Lx does not necessarily commute with the formation of. associated tensors; e.g.~ that (Lx'),," (/'xl),J' where I'" Ija/ax'edx' e TII(M) and 1-I;jdx'edx J is the associated tensor with components IiI = g;"I·r We have from (4)
.
(L I)' x j
al;
ax;
ax·
x·_J ax" - t·j-ax"- + I'Ie -ax-j •
and so
But also from (4)
ax' ax" x"-al;j I t + I'j-' + ,;,,--. ax ax' ax' • a (gill ,j )+ g,le' " ax' + gill.-, ax A ..... x -. ax ax' ax'
(Lxf)/j'"
j-'
il , "a/~ " ax' +g"I.-, ax".. =x"ag -"I ax j + X w"-ax" +g,,,1 jax' ax'
Thus. in order to have equality it is necessary and sufficient that il , " ax' X "ag -"lj+g,,,1 j-I
ax
"ax' =0 ax +gjfl j--" ax
for all pairs of indices (i, j), which is a nontrivial system of n 2 linear partial differential equations for g;)" If X is a Killing vector field. then
ax' ax' ,agu, g,,, axl + gIl ax" .. - x ax .
304
TENSORS
which substituted in the preceding equation, gives zero. The converse statement is proved along the same lines. In other words, a necessary and sufficient condition that Lx commute with the formation of associated tensors is that X be a Killing vector field for the pseudo-Riemannian metric g. •
As usual, the development of Lx extends from tensor fields to F-valued tensor fields. Exercl...
S.3A Let 1- xya/axedx + yiJ/ayedx + a/axedy e TII(R ~). If cp: R2 -. R Z is defined by cp(x, y) = (x + y. xy). show that cp is a diffeomorphism and compute tr(I), cp.'. cp.l. LxI. Lxcp·'. L.,..xl and tr(L •• xcp.l) for X ... ya lay + x 2 alay. S.38 Verify explicitly that Lx'*(Lxt)' where' denotes the associated tensor with both indices lowered. for X and I in Exercise 5.3A. 5.3C Compute the coordinate expressions for the Killing equations in R 3 in standard. cylindrical, and spherical coordinates. What are the Killing vector fields in R "? 5.4 THE LIE DERIVATIVE: DYNAMIC APPROACH
.
We now turn to the dynamic interpretation of the Lie derivative. In Section 4.2 it was shown that Lx acting on an element of ~'t( M) or '.'x (M). respectively, is nothing but the time derivative at zero of that element of ~if(M) or ~(M) dragged along by the flow of X. The same situation holds for general' tensor fields. Given I e '!f, r ( M) and X E ~X ( M). we can find a curve at I( m) in the fiber over III hy using the flow of X. The derivativ~ of this curve is the Lie derivative. (In spirit. the flow plays the same role for Lie derivation that parallel translation plays in covariant differentiation.) More precisely, for m E M and a vector field X on M let (U. a, F) be a now box at m. For each A e III - ]- a, a[ we can form the diffeomorphism F" ... FI U x{A}: U -. U" == F,,(U). For I E ~,'( M), define the map T(m): I" -. T:T..,(M): A... Fr('1 U,,)(m). Then the A-derivative at zero of T(m) should equal Lx'(m)(see Fig 5.4.1). Note: We use A for the time variable to avoid notational confusion with the tensor'.
5.4.1 Theorem. Let XE~·Xk(M).tE'!T.'(M) be of class C~ and F" be the flow of X. Then
THE LIE DERIVA TlVE: DYNAMIC APPROACH
305
integral curve of X M
FIgu ... 5.4.1
Proof.
It suffices to show that
d~ I 1\
~
_ ()
Ftt = Lx t .
Indeed. if this is proved then
d~ F:t = 1\
I
dd F:+,.t 1-',.-0
=
I
dd F,.*Ftt 1-',.-0
For smoothness of T(m), we have T(m)(~) = (n~ I(m».t( FA(m». Hence we need verify only that (TF'; I(m». is a smollth function of ~. Consider F: U XI" - M. which is smooth in (u. ~). Then TF: TV x ( I x R) - M is .also smooth. For smoothness of TF". note Tf~ - TFI TF X{~}X {O}. Then from 5.2.4, we see that (TF'; 1)* is smooth in ~. Since Fo is the identity, it is clear that T(m) is a curve at t(m). Now define 9x :
(cp*t)( ~*(lI, ...• CP*(I'. cP* Xl •...• cp* X,) = cp*( t{ (II •.... a'. Xl •... • X,)) for cP a diffeomorphism. Hence 9x is a differential operator. It remains to
306
TENSORS
show that ex coincides with Lx on <5(M) and ~ ( M). For f X E 'X(M). we have
~'J ( M). and
E
and
by 4.2.10 and 4.2.19. respectively. Thus by 5.3.2 and 5.3.4. 0xl
=
1... 1 for all
t E <:i(M).
•
This theorem can also be verified in finite dimensions by a straightforward coordinate computation. See Exercise 5.4A. The identity in this theorem relating flows and Lie derivatives is often taken as the definition of the Lie derivative (see Exercise 5.4C). 5.4.2 Corollary. If t ThaI is. t = Ftt.
E
<:i (M). L xt = 0 iff t is constant along the flow of
x.
As an application of 5.4.1. let us generalize the naturality of Lx with respect to diffeomorphisms. As remarked in section 5.2. the pull-back of covariant tensor fields makes sense even when the mapping is not a diffeomorphism. It is thus natural to ask whether there is some analogue of 5.3.5 for pull-backs with no invertibility assumption on the mapping ",. Of course. the best one can hope for, since vector fields can be operated upon only by diffeomorphisms. is to replace the pair X. '" _ X by a pair of !p-related vector fields. 5.4.3
Propo.ltlon.
Let "':
M ..... N
be
Coo. X
E
'X (M). Y
E
~(N). X!: Yand tE <:is o(N). Then ",-(Lyt) = Lx",-t. Proof. Recall from 4.2.4 that X!: Y iff Gil. ° '" = '" ° FA' where F/I. and Gil. are the flows of X and Y. respectively. Thus by 5.4.1.
Lx",·t=.!!..1 Ft",·t=.!!..1 (",oFA)·t= ~I (Gil. ° ",)·t d" 11.-0 d" 11.-0 d" /1.-0
.!!..I d"
11.-0
",·G:t = ",-
.!!..I d"
G:t = ",.( Lyt). • /1.-0
As for functions and vector fields. the Lie derivative can be generalized to include time-dependent vector fields.
THE LIE DERIVATIVE: DYNAMIC APPROACH
307
5.4.4 Theorem. Let XA E ~X k (M), k ~ I for each;\, E R and suppose that X(;\.. m) is continuous in ;\.. m. Then if FA.,. is the evolutiol/ operator for XA• we have
where t
E
Proof.
~'1"( M) is of dass C k •
As before. it is enough to prove this at;\.
= JL
where FA• A = identity.
for then
Now as in 5.4.1, ex.' = (d/d;\')f"A.,.tI A _,. is a differential operator that coincides with Lx. on ~'t and ~'\ by 4.2.31. Thus by 5.3.2, ex. = Lx. on all. tensors. • We caution again, as in Section 4.2. that it is not generally true for time-dependent vector fields that the right hand-side in 5.4.4 is Lx.f"A.,.t. Let us generalize the relationship between Lie derivatives and' flows one more step. Call a smooth map t: R X M -+ T.'( M) satisfying tA(m) = t( >'. m) E (Tm M)~ a time-dependent tensor field. Theorem 5.4.4 generalizes to this context as follows.
5.4.5 Theorem. Let tA and XA be C k time-dependent tensor and vec·tor fields. k ~ I and denote by FA.,. the et'olution operator (If X,. Then
Proof.
By the product rule for derivatives and 5.4.4 we get d -d i F! t-d~ A-A. A.,. A - d"A
I
A-A"
Pt
A.,.
An
+F!
A•. ,.
dtA _.d>'
I
A -Au
308
TENSORS
5.4.8 Examples A. If g is a pseudo-Riemannian metric on M. the .Killing equations are Lxg ... 0 (see Example 5.3.68). 8y 5.4.2 this says that F!g = g. where F" is the now of X. i.e., that the now of X consists of isometries. B. In elasticity the vanishing of the strain tensor means. by Example A. that the body moves as a rigid body.
5.4A
Verify Theorem 5.4.1 by a coordinate computation as follows. Let F,,(x) = (yIp,. x), .. . ,y"(A. x» so that ilyi/ ilA = Xi( y) and iI y'/ ax' satisfy the variational equation
Then write
(.._ ).,......,....!II -a,·' . . . a,·· - - ... --I' • fix ' fix' fI, , fI, . 1~,
I
!II
I
iJxll
iJx)'
!II.
!II
•
i"';
.Jt- ..J.
DilfeRDtiate this in A at A - 0 aDd obtain the coordinate cxprasioo (4) 01 SectioG S.3 for Lx" S.4B Carry out abe proof outlined in Exercise S.4A for time-depeodent wc:tor fields. S.4C Star1iDa with Theon:m 5.4.1 as the definition of L II ,. dJeck ....t Lx .Ii.... (DOl). (1)02) ud the propertieI (i)-(iv) 01 S.3.1S.4D Let C be. contraction operator mappina 5'.'(M) to ~~i '(JI). Use both 5.4.1 ud (DOl) to show that L II ( 0) - C(LII')' S.4E
F.xteod theorem 5.4.1 to F-valued teDsors.
S.4F
(lNfomttl/iOft LmtmG). Two tensor fields " aDd " +'2 of the same type OIl • compact manifold AI are called ~I if then is • diffeomorphism .,: AI ... M such that "-', - " + 'a. Assume that for o.. A .. )•• wc:tor facld X). on M can be found that satisfies LxJl1
Pro\Ie that I, and
+ '2)+1 2 • 0
'I + '2 are equivalent as follows: Let 'Pl be the now
of Xl with 4fIo -Id. Now calculate :", 'P%(',
+ A'2)'
NOI.: the method in this argument is caUed the method of U. trtI1&r/omu. It has been used already in the proof of Frobenius
PARTITIONS OF UNITY
309
theorem (section 4.4) and we shall see it again in 6.4.14 and 8.1.2. The method is also used in the theory of normal forms (cf. Takens [1914] and Guckenheimer and Holmes [1983]).
5.5 PARTITIONS OF UNITY A partition of unity is a technical device that is often used to piece smooth local tensor rields together to form a smooth global tensor field. Partitions of unity will be especially useful for studying integration. but in this section they are used to study when a manifold admits a Riemannian metric.
5.5.1 Definition. If t is a tensor field on a manifold M. the carrier of t is the set of m E M for which t( m) * O. and is denoted carr t. The .fuppon of t. denoted suppt. equals the closure of carr t. Also. we s~v t has compact support if suppt is compact in M. An open set U eM is called a C' carrier if there ex/st.I' an f E ~'f'( M). such that f ~ 0 and U = carr f. A collection of subsets {Ca} of a manifold M (or. more generalll'. a topological space) is called toeaUy linite if for each 111 E M. there is a neighborhood U of m such that U () Ca = 0 eX('ept for finitc/I' many indices a: 5.5.2 ,Definition. A partition 01 lIIIity on a manifold M i.t a collection
{(1I,. gil). where (i)
(ii) g; E '!F( M). gj( m) ~ 0 for all m EM. and supp g, c 1I, for all i; (iii) For each mE M. Ljg;(m) = I. [By (i). this is a finite ovum.) If if = {( Va. CPa)} is an atlas on M. a partition 01 unity ,fubordinate to ti'is a partition of unity {( 1I,. g,)} such that each open set 1I, is a subset of some chart domain Val;)' If for any atlas if there exists a partition of unity subordinate to it. we say M admits part;tions olllll;ty.
Occasionally one works with C k partitions of unity. They are defined in the same way except g, are only required to be C* rather than C'''. Partitions of unity are used for a basic patching constnlction. which may be explained as follows.
5.5.3 Proposition. Let M be a manifold with an atlas ti = {( VB' CPa)} SO CPa: Va -+ V~ c E is a chart. Let ta be a C k tensor field. k ~ I. of fixed type (~) defined on V~ for each a. and assume that there' exists a partition of unity
310
TENSORS
{( lI;, gi)} subordinate to (i. Let t be defined by t(m) == Lg/CP:(I)ta(/)(m), /
a /lnite .fum at each me M. Then t is a
e· tensor field of type (;) on M.
Proof. Since {lI;} is locally finite, the sum at every point is a finite sum, and thus t(m) is a type (~) tensor for every mE M. Also, t is e* since the local representative of' in the chart (VO(i),CPa(i) is Lj(gioCP.;(J)tO(j)' the summation taken over all indices j such that Valil n Va(j) '* 12l; by local finiteness the number of these j is finite. •
It should be remarked that t is not unique; it clearly depends on the choices of the indices a(i) such that lI; c VO(i) and on the functions gi' As we shall see later, under suitable hypotheses. one can always construct partitions of unity; again the construction is not unique. The same construction (and proof) can be used to patch together local sections of a vector bundle into a global section when the base is a manifold admitting partitions of unity subordinate to any open covering. To discuss the existence of partitions of unity and consequences thereof, we need some topological preliminaries.
5.5.4 DeftnlUon. Let S be a topological space. A covering {Va} of S is called a ",iMrrtettt of a covering {V,} if for every VA there is a V, sudI that Va c V,. A topological space is called paracomptld if every open covering of S has a locally finite refinement of open sets, and S is Hausdorff. 5.5.5 Propoaltlon. Second-countable, locally compact Hausdorff spaces are paracompact.
By second countability and local compactness of S, there exists a sequence 0., ... ,0", ... , of open sets with c1(0,,) compact and u nENO" ""' S. Let V" - 0. U .•• U 0", n -1,2,'" and put U,- V,. Since (V,,) is an open covering of S and c1(V.) is compact, cl(U,)c v,. u ... U v, .. Put V2 ... v" U ... U V, ; then c1(V2 ) is compact. Proceed inductively to show that S is the countable union of open sets V" such that c1(U,,) is compact and c1(V")cV,,n If Wo is a covering of S by open sets, and K,,=c1(V,,)\V,,_., then we can cover K" by a finite number of open sets, each of which is contained in some W" n V,,+.' and is disjoint from c1(V,,_2)' The union of such collections yields the desired refinement of {Wa}' • Proof.
Another class of paracompact spaces are the metrizabJe spaces (see 5.5.15 in Box 5.5A). In particular, Banach spaces are paracompact.
PARTITIONS OF UNITY
311
5.5.8 Propoaltlon. Every paracompact space is normal. Proof. W~ first show that if A is closed and u E S\A. there are disjoint neiahborhoods of u and A (relularity). For each v E A leI UU ' be disjoint neiahborhoods of u and v. Let W" be a locally finite refinement of the covering Vv ' S\A. and V - U W". the union over those a with W" n A "" 0. A neighborhood Uo of u meets. a finite number of Wa' Let U denote the intersection of Uo and the corresponding UV ' Then V and U are the required neighborhoods. The case for two closed sets proceeds somewhat similarly, so we leave the details for the reader. •
v..
Later \:Ve shall give general theorems on the existence of partitions of unity. However, there is a simple case that is commonly used, so we present it first.
5.5.7 Theorem. Let M he a second-countable Then M admits partitions 01 unity.
(Hau.~dorlf)
n-manilold.
Proof. The proof of 5.5.5 shows the following. Let M be an n-manifold and {W,,} be an open covering. Then there is a locally finite refinement consisting of charts CV;, cp;) such that cp, (V.) is the disk of radius 3. and such that cp/-·(D.(O» cover M, where D.(O) is the unit disk, centered at the origin in the model space. Now let (£. be an atlas on M and let {( cp;)} be a locally finite refinement with these properties. From 4.2.13, there is a nonzero function hi E <j(M) whose support lies in V. and hI;" O. Let
v..
h;(u) g;(u) = L;hi(u)
(the sum is finite). These are the required functions.
•
If (V.. ) is an open covering of M, we can always find an atlaS
cl' ... {( u,. CPi)} such that (u,) is a refinement of (V.. ) since the atlases generate the topology. Thus. if M admits partitions of unity. we can find partitions of unity subordinate to any open covering. . , The case of CO.partitions of unity differs drastically from the smooth case. Since we are primarily interested in this latter case. we su1llllUtrize the topologicai· situation, without giving the proofs: If S is a Hausdorff space, the following are equivalent:
S is normal. (ii) For any two closed nonempty disjoint sets A, B there is a continuous function I: S -+ (0, I) such that I( A) - 0 and 'f( B) - I (Urysohn's lemma). (i)
312
TENSORS
(iii) For any closed set A c S and continuous function f: A --+ rlI. h), there is a continuous extension j: S --+ [a. h) of f (Tietze extension theorem). A Hawdorff space is paracompact iff it admits a CO partition of unity subordinate to any open covering.
Indeed, if {( U;. g;)} is a continuous partition of unity subordinate to the given open covering {Va}' then by definition, {U;} is an open locally finite refinement. The converse-the ·existence of partitions of unity-is the hard part; the proof of this and of the equivalences of (i) with (ii) and (iii) can be found for instance in Kelley (1975) and Choquet (1969: sec. 6). These results are important for the rich supply of continuous functions they provide; We shall not use these topological theorems in the rest of the book, but we do want their smooth versions on manifolds. Note that if M is a manifold admitting partitions of unity subordinate to any open covering. then M is paracompact, and thus normal by 5.5.6. This already enables us to generalize (ii) and (iii) to the smooth (or C k ) situation.
5.5.8 Propoeltlon. Let M be a manifold admitting smooth (or C") partitions of unity. If A and B are closed disjoint sets then. there exists a smooth (or Cit) /unction /: M ... [0.1] such that f(A) - 0 and /(B) -I.
Proof. As we saw. the condition on M implies that M is normal and thus there is an atlas {(Va' CPa)} such that V n A '* 0 implies Va n 8 = 0. Let R
(II;. g;} be a subordinate C k partition on unity andf =
LS;. where the sum is over those i for which It; n B '* 0. Then f is C k • is one on 8 and zero on A .
•
Next we prove an extension theorem, which is a smooth analogue of (ii).
5.5.9 PropoalUon. Let M be a manifold admitting partitions of unityj and let '11': E ..... M be a vector bundle with base M. Suppose 0: A ... E is a C" section defined on the closed set A (i.e .• every point a E A hils a neighborhood Va and a C" section G,,: V" ..... E extending G). Then G can be extended to a C" global section i: M ..... E. In particular, iff: A ..... F is a k function defined on the closed set A, where F is a Banach space. then there is a Cit extension j: M ..... F; iffis bounded by a constant R, i.e .• llf(a~I" Rfor all a E A, then so is j.
c
Consider the open covering {V". M\A la E A} of M. with Va given by the definition of smoothness on the closed set A. Let {(U;, g;)} be a C" partition of unity subordinate to this open covering and define G;: U; ..... E, by Gj ... G" IU; for all U; contained in some Va and OJ 0 on all U; disjoint Proof.
=
PARTITIONS OF UNITY
313
from all Va. a EA. Then g,o,: If; -+ E is a C A section over If; and since supp(g;o;)csupP(g;)Clf;, it can be extended in a C A manner to M by putting it equal to zero on M\lf;. Thus 8;0,: M -+ E is a C"-section of 'If: E -+ M and hence a= 1:;g;o; is a C k section; note that the sum is finite in a neighborhood of every point mE M. Finally, if a EA.
a( a ) =
L 8, ( a ) 0, ( a ) = ( L g; ( a ) ) °(a ) = °(tl ) • ,
"
i.e.•
alA =
0.
The second part of the theorem is a particular case of the one just proved by considering the trivial bundle M x F -+ M and the section defined by o(m)= (m.J(m». Finally. the bounded ness statement follows from the given construction. keepIng in mind that all the g, have values in [0.1]. •
°
Before discussing general questions on the existence of partitions of unity on Banach manifolds. we will discuss the existence of Riemannian metrics. Recall that a Riemannian metric on a Hausdorff manifold M is a tensor field g E 5 2o( M) such that for all m E M, g( m) is symmetric and positive definite. Our goal is to find topological conditions on an n-manifold that are necessary and sufficient to ensure the existence of Riemannian metrics. The proof of the necessary conditions will be simplified by first showing that any Riemannian manifold is a metric space. For this, define form,nEM, d(m. n) = inf{l( y )Iy: [0.11-+ M is a c l curve with y(O)
Here I(y) is the length of
tlJ~-
=
m. y( I) = n}:
curve y. defined by ,
where Y(t) = dy/dt is the tangent vector at y(t) to the curve y and == [gY(I)(y(t), 1'(1»)1/2 is its length .
. IIY(t~1
. 5.5.10 Proposition. d is a metric on each connected ('I)mponent oj M whose metric topology is the original topology oj M. Proof. Clearly d(m. m) = O. d(m. n) = d(n, m). and d(m. p)" d(m"n)+ d(n, p) by definition. Next we will verify that d(m. n) > 0 whenever m * n .. Let m E V c M where (U, qJ) is a chart and suppose qJ( V) = u'. c E. Then for any. u E V. g(uXv. V)I/2, defined for vE TuM. is a norm on TuM.
314
TENSORS
This is equivalent to the norm on E, under the linear isomorphism TmfJ!. Thus, if f is the local expression for g, then f( u') defines an inner product on E, yielding equivalent norms for all u' E U'. Using continuity of g and choosing U' to be an open disk in E, we can conclude that the norms f( u')1/2 and f( m')1/2, where m' -= fJ!( m) satisfy: af( m,)1/2 ~ f( U,)1/2 ~ bf(m,)1/2 for all u' E U', where a and b are positive constants. Thus, if 71: [0, I] ..... U' is a C l curve, then
~a
f,(m')(
,,(I). ,,(/»1/2 dl
I
I
~a,(m') ( 10 ,,(/)dl.fo ,,(t)dl
)1/2
~ ai( m')( 71(1 )-1/(0).1/( I )-1/(0» 1/2
Here we have used the following property of the Bochner integral: II[f(/) dtll
~ t//f(t)//dt.
valid for any norm on E (see the remarks following 2.2.7). Now let y: [0. I) ..... M be a C l curve. yeO) = m. y(l) = n. m E U.( u. ' O. The equivalence of the original topology of M and of the metric topology defined by d is clear if one notices that they are equivalent in every chart domain U, which in turn is implied by their equivalence in cp( U). • Notice that the preceding proposition holds in infinite dimensions. 5.5.11 PropoaIUon. A connected Hausdorff n-manifold admits a Riemannian metric if and only if it is second countable. Hence for Hausdorff n-manifolds (not necessarily connected) paracompaclness and metrizabilily are equivalent.
PARTITIONS OF UNITY
315
Proof. If M is second countable, it admits partitions of unity by 5.5.7. Then the patching construction 5.5.3 gives a Riemannian metric on M hy choosing in every chart the standard inner product in R". Conversely, assume M is Riemannian. By 5.5.10 it is a metric space, which is locally compact and first countable. being locally homeomorphic to R". By \.6.14, it is second countable. •
The main theorem on the existence of partitions of unity in the general case is as follows.
5.5.12 Theorem. Any second-countable or paraCtlmptll"l manifold moJt!/ed on a.separable Banach space with a C A norm away from 'he origin tldmils C k partitions of unity. In particular paracompact (or second ("OItntahie) manifolds modeled o~ separable Hilbert spaces admit eX) parlitions flf unity. 5.5.13 Corollary. Paracompact «()r second countable) manifolds modeled in separable real Hilbert spaces admit Riemannian metrics. Theorem 5.5.12 will be proved in the following two boxes. There are Hausdorff nonparacompact n-manifolds. These manifolds are necessarily nonmetrizable and do not admit partitions of unity. The standard example of a one-dimensional nonparacompact Hausdorff manifold is the "long line." In dimensions 2 and 3 such manifolds are constructed from the Priifer manifolds. Since nonparacompact manifolds occur rarely in applications, we refer the reader to Spivak [1979, vol I, Appendix A] for the aforementioned examples. .
BOX S.U PARTITIONS OF UNITY: REDUCTION TO THE LOCAL CASE
Let us begin with some topological preliminaries. Let S be a paracompact space. If {UII} is an open covering of S. it can be refined to a locally finite open covering (WIl ). The first lemma below will show that we can shrink this covering further to get another one, {Va} such that cl( Va) C Wa with the same indexing set. A technical device used in the proof is the concept 'of a well-ordered set. An ordered set A in which any two elements can be ' compared is well-ordered if every subset has a smallest element. The axiom qf choice is equivalent to Zermelo's lemma: "Every set can be well-ordered." See Appendix A.
5.5.14 Lemma (Sltrinlcing i.emma). Let S be a normal space and {Wa)"eA a locally finite open covering of S. Then there exists a locally
316
TENSORS
finite open refinement {Va}a. A (with the same indexinR set) ,\'uch that cI(Ve)C We'
Proof. Well-order the indexing set A and call its smallest element a o. The set Co = S\ V .. > "OW.. is closed. so by normality there exists an open set v..o such that Co C c1( Va.) C Wa.. If Vy is defined for all y < a. put Ca - S\( V y < aVy V V y > eWy») and by normality find Va such that C.. C c1(V.. )C W... The collection {V.. }.. eA' is the desired locally finite refinement of (W.. }.. e A' provided we can show it covers S. Let s E S. By local finiteness of {W,,}. of he\ongs only to a finite collection W.. ,..... W.... If fJ,., max(al ..... a n ) then clearly s E Wy for all y > fJ. so that if in addition sEVy for all y < fJ. then s E Cp C Vp, i.e. s E Vp. •
5.5.15 Lemma (A. H. Stone). compact.
Every pseudometric space is para-
Proof. Let {V.. }.. EA be an open covering of the pseudometric space S with distance function d. Put Vn."-={xEV,,ld(x.S\Vn)~1/2n}. By the triangle inequality we have d(U,•. ". S\U. I I.,,)? 1/2 n -1/2" I I = 1/2 n t I. Well-order the indexing set A and let u,~ n = U•. ,,\ V II < II V. f l . p . If y,6 E A. we have Vn~y C S\V. t I,3' if Y < 6, or u,~" c S\ Vn+I.y. if 8
Let us now tum to the question of the existence of partitions of unity subordinate to any open covering. 5.5.16 Proposition (R. Palais). Let M be a paracompact manifold modeled on the Banach space E. The following are equivalent. M admits C" partitions of unity. Any open covering of M admits a locally finite refinement by C" carriers; (iii) For any open sets 1, 2 , such that c1(01) cO2 , there exists a C" carrier V such that 0 1 eVe 02' (iv) Every chart domain of M admits C" partitions of unity subordinate to any open covering. (i) (ii)
°°
PARTITIONS OF UNITY (v)
317
E admits C· partitions 01 unity subordinate to any open covering olE.
Proof. ((i) implies (ii». If {( ll" g,)} is a C"-partitioll of unity suhordinate to an open covering, then clearly carr K, form a locally finite refinement of the covering by C k carriers. «ii) implies (iii». Let {V,,}ae A be a locally finite refinement of the open covering (02 ' S\c1( O.)} by C" carriers and denote hy 10 E CS'* (M), the function for which carr 10. = Va' Let B = {a E A IVa C 0 2 }' Put V = nile BVfJ./ = E"e sill and remark that 0. eVe O2 , carr I = V. «iii) implies (iv». Let U he any chart domain (If M. Then V is diffeomorphic to an open set in E which is a metric space. so is paracompact by Stone's theore~ 5.5.15. Let {Un} he an arbitrary open covering of V and {VfJ} be a locally finite refinement. By the shrinking lemma we may assume that c1( VfJ) c V. Again by the shrinking lemma refine further to a locally finite covering {WfJ} such that c1( W,I ) C "~. But by (iii) there ex.ists a C·-carrier Oil such that W" c 0" C VfJ. and so {Oil} is a locally finite refinement of {VfJ} by CA-carriers. whose correspqnding functions we denote by IfJ. Thus I = E"I" is a C k map and {(VfJ./fJ/f)} is a C" partition of unity subordinate to {Un}' ((iv) implies (v». Consider now any open covering {Vn}n' • of E and let (V. q» be an arbitrary chart of M. Refine first the covering of E by taking the intersections of all its elements with all translates of q>(V). Since E is paracompact. refine again to a locally finite open covering {Vp}. The inverse images by translations and cr of these open sets are subsets of V, hence chart domains, and thus by (iv) they admit partitions of unity subordinate to any coveri~g. Thus every Yp admits a C" partition of unity subordinate to any open covering, for example to {VIl n Vala E A}; call it {gf}. Then g = Ei.llgf is a C" map and the double-index.ed set of functions gf/g forms a C" partition of unity of E. «v) implies (iii». If E admits C" partitions of unity subordinate to any open covering. then so does every open subset by the (already proved) implication of (i) =0 (iii) applied to M = E. which is paracompact by 5.5.13. Thus if (V, q» is a chart on M. V admits partitions of unity, since q>( V) does. Finally, we show (iii) implies (i). Choosing a locally finite atlas, this proof repeats the one given in the last part of (iv) ,. (v). • As an application of this proposition we get the following.
5.5.17 Proposition. filions of unity.
El.'erl' paracompact n-manilold admits C<XJ-par-
318
TENSORS
1'roof.
By 5.5.16(ii) and (v) it suffices to show that every open set in
R" is a Coo carrier. Any open set U is a countable union of open disks D1• By 4.2.13, D/- carr/;, for some Coo function /;: R" -+ R. Put M, - sup(JID"/;(%~11 %E R", k.; i} and let 00
/- E /;/2 IM/. I-I
By Exercise 2.4J / is a Coo function for which carr/ ... U clearly holds.
•
In particular, second-countable n-manifolds admit partitions of unity, recovering 5.5.7. .
BOX AlB PARTInONS OF UNITY: THE LOCAL CASE Theorem 5.5.15 reduces the problem of the existence of partitions of unity to the local one, namely finding partitions of unity in Banach spaces. This problem has heen studied by Bonic and Frampton (1966) for separable Banach spaces.
5.5.1' PropoalUon CR. /lottie tntd J. FlYIIIIfJtOtl/I966J). separable Banach space. The /oIlowing are equivalent.
Lei E be a
(i) Any open set 0/ E is a C k carrier. (ii) E admits CIt partitions 0/ unity subordinate to any open· covering
(iii)
o/E; There exists a bounded nonempty CAt carrier in E.
Proof. By 5.5.16, (i) and (ii) are equivalent since E is paracompact by 5.5.15. It remains to be shown that (iii) implies (i), since clearly (ii) implies (iii). This proceeds in several steps. First, we show that any neighborhood contains a Cit carrier. Let U be any open set and let carr / c Dr(O) be the bounded carrier given by (iii), / E C"(E), /;;. O. Let e E U, fix eo E carr/, and choose r> 0 such that D.(e) c U. Define g E Ck(E), g ~Oby
K>O,
where K remains to be determined from the condition that carr g c +lIeoIO/r, this
D,(e). An easy computation shows that if K> (r
PARTITIONS OF UNITY
319
inclusion is verified. Since e E carr g. carr g is an open neighborhood of e. Second. we show that any open set can be covered by a countable locally finite family of C" carriers. By the first step. the open set V can be covered by a family of C" carriers. By Lindelors theorem. 1.1.6. V == U "V" where Vn is a C" carrier. the union being over the positive integers. We need to find a refinement of this covering by C" carriers .. Let/"E'C"(E) be such that carr In = V". Define Vn={eEEl/n(e» O. h(e) < lin for all i < n}. Oearly VI = VI and inductively Vn = v,. n n'; < ,,!; I(]_ oo.l/n[). By the composite function theorem the inverse image of a C" carrier is a C" carrier. so that /, I(]_ oo.l/n() is a C" carrier. since ]00.1/ n[ is a ~" carrier in R (see the proof of 5.5.17). Finite intersections of C" carriers is a C" carrier (just take the product of the functions in question) so that Vn is also a C~ carrier. Clearly Vii c v". We shall prove that {Vn } is a locally finite open covering of U. Let. e V. If. e Vn for all n. then clearly e e VI = VI' If not, then there exists a.smallest no such that e E v" .. Then /,(e) = () fori < no and thus e e V" - (e e EI/".(e) > 0, h(e) < Iino for all i < no}. Thus {Vn} covers V. This open covering is also locally finite for if e E V" and N is such that I(e) > liN. then the neighborhood{u E VII.,(e) > 1/ N) has empty intersections with all Vm for m > N. Third we show that the open set V is a C"-carrier. By the second step. V - U "V". with Vn a locally finite open covering of V by C" carriers. Then I = Enl" is C". i(e) ~ 0 for all e E E and carr 1= V. • The separability assumption was used only in showing that (iii) implies (i). There is no general theorem known to us for nonseparable Banach spaces. Also. it is not known in general whether Banach spaces admit bounded C" carriers. for Ie ~ I. However. we have the following. 5.~.1' Propo8ltlon. II the Banach space E has a norm C" away Irom its origin. k ~ I. then E has bounded C"-carriers.
Proof. By 4.2.13 there exists cp: R ..... R.C"" with compact support and equal to one in a neighborhood of the origin. If 11'11: E\{O} ..... R is C". k ;a. I, then cp °11'11: E\{O} ..... R is a nonzero map which is C". has bounded support 11'11- 1 (suppcp), and can be extended in a C~ manner toE. • Theorem 5.5.12 now follows from 5.5.19,5.5.18. and 5.5.16. The situation with regard to Banach subspaces and submaru.foJds is clarified in the following proposition, whose proof is an immediate consequence of 5.5.18 and 5.5.16.
320
TENSORS
5.5.20 ProposlUon. (i) If E i.f a Banach space admitting C A partitions of unity then so docs any closed suhspace. (ii) If a manifold admits C k partitions of unity subordinate to any open covering, then so does any submanifold. We shall not develop this discussion of partitions of unity on Banach manifolds any further, but we shall end by quoting a few theorems that show how intimately connected partitions of unity are with the topology of the model space. By 5.5.18 and 5.5.19. for separable Banach spaces one is interested whether the norm is C k away from the origin. Restrepo [1964] has shown that a separable' Banach space has a C I norm away from the origin if and only if the dual is separable. Bonie and Reis [1966] and Sundaresan [1967] have shown that if the norms on E and E· are differentiable on E\{O} and E·\{O}. respectively, then E is reflexive. for E a real Banach space (not necessarily separable). Moreover. E is a Hilbert space if and only if the norms on E and E· are twice differentiable away from the origin. This result has been strengthened by Leonard and Sundaresan [1973]. who show that a real Banach space is isometric to a Hilbert space if and only if the norm is C 2 away from the origin and the second derivative of e ..... UeU 2/2 is bounded by I on the unit sphere; see Rao [1972] for a related result. Palais [1965b] has shown that any paracompact Banach manifold admits Lipschitz partitions of unity. Because of the importance of the differentiability class of the norm in Banach spaces there has been considerahle work in the direction of determining the exact differentiabilit\ . l:J~s of concrete function spaces. Thus Bonic and Frampton [1966] haH: shown that the canonical norms on the spaces LP(R).IP(R). p ~ I. p < 00 are COO away from the origin if p is even. C p - I with DOl-liP-I) Lipschitz, if p is odd. and C[pi with Dlp)(II'UP) HOld"r continuous of order p -[pl. if p is not an integer. The space Co of sequences of real numbers convergent to zero has an equivalent norm that is Coo away from the' origin. a result due to Kuiper. Using this result, Frampton and Tromba [1972] show that the A-spaces (closures of Coo in the HOlder spaces) admit a Coo norm away from the origin. The standard norm on the Banach space of continuous real valued functions on [0. I] is nowhere differentiable. Moreover. since CO([O.I).R) is separable with nonseparable dual. it is impossible to find an equi\ .• lent norm that is differentiable away from the origin. To our knowledge it is still an open problem whether CO([O.I).R) admits C k partitions of unity for k;;d. Finally. the only results known to us for nonseparable Hilbert spaces are those of Wells [1971]. [1973]. who has proved that nonsep-
PARTITIONS OF UNITY
321
arable Hilbert space admits ("2 partitions cif unity. The techniques used in the proof. however. do not seem to indicate a general way to approach this problem.
Exercl... 5.5A
5.5B
Show that any closed set Fin R" is the inverse image of 0 hy II C' real-valued positive function on Rn. Generalize this to any n-manifold. (Hint: proof of 5.5.11 for R"\F.) In a paracompact topological space. an open suhset need not he paracompact. Show that (i) If every open subset of a paracompact space is paracompact. then any subspace is paracompact. (ii) Every open submanifold of a paracompact manifold is parac:ompICt. (HiIU: Uk chart domains to coru:lude metIUabilily.)
Let ..: £ - M be a vector bundle. £' c £ a 5ubbundle and assume III . admits C· partitions of unity subontiDate to any open coverin&. Show that £' splits in E, i.e.. there exists • subbundle £11 such that E - E'eE". (H;III: The result is trivial for local bundles. Construc:t for Cvay element of a locally fmite covering (ll;) a vector 'bundIe map /, wbose kernel is tt, .. complement of E', u,. for u" II» a C· partitioa of unity, put; <...;&;/, and show that E - E'eter/.) S.SD Let ..: E - III be a vector bundle OWl" the base M that admits C· partitions of unity and with the fibers 01 E modeled OIl a Hilbert space. Show that E admi~ a C· bundle metric, i.e., a C· map ,: M -Tz°( E) that is symmetric stronaJy aondeaenerate and positive definite at every point n • . ;. N. S.SC
«
Let E - III be a line bundle OWl" the manifold III admittina· C· partitions of unity subordinate to any open coveriDa- Sbow that Ex E is trivia) (HIIII: Ex E - L(E·, E) and coastnact.1ocal bale that am be extended.) S.SF Assume M admits C· partitioos 01 unity. Show that any submanifold . of III diffeomorphic to S' is the intepaJ CUIW of a C· wctor Ileld OIl M. 5.50 Let /II be a connected paracompact manifold. Show that tIMn exists a C· proper mappinl/: M - R·. (Hlltt: /II is secoad countable, beina Riemannian. Show the statement for Ie -1. where 1and is a countable partition of unity.) 5.5H Let /II be a connected paracompact ,...manifoJd and X e ~(/II). Show that there exists II E ~(M).II > 0 such that Y -AX ii ~ S.SE
<.,)
t::..i.,;.
322
TENSORS
plete. (Hint: With! as in 5.50 put h = e C~'lfl" so that I YUIi ~ 1. Hence (f 0 c)(Ja. h[) is bounded for any integral curve (' of Y and Ja. b[ in the domain of c.) 5.51 Show that every compact n-rnanifold embeds in some Rk for k big enouah in the following way. II {(lJ,.cp/)}/.' •... N is a finite atlas with cp/(lJ,) the ball of radius 2 in R", let X e COG(R"). X -Ion the ball of radius 1 and X - 0 outside the ball of radius 2. Put /, - (X • cp/). cp/: M - R", where /, - 0 outside lJ,. Show that /, is COG and that "': M _R N" XR N, "'(m)"" (f,(m) .....!N(m). x(cp,(m»..... X(cpN(m))) is an embedding. 5.SJ Let , be a Riemannian metric on M. (i) Show that if N is a submanifold. its ,-normal bundle "n( N) = {oe T"Mln eN. v.l T"N} is a subbundle of TM. (ii) Show that TMIN = "g(N)eTN (iii) If h is another Riemannian metric on M. show that ".II( N) is a vector bundle isomorphic to "4 ( N ).
CHAPTER
6
Differential Forms
Differential k-forms are tensor fields of type (~) that are completely antisymmetric. Such tensor fields arise in many applications in physics. engineering; and mathematics. A hint at why this is so is the fact tbat the classical operations of grad. div. and curl and the theorems of Green. Gauss. and Stokes,can all be expressed concisely in terms of differential forms. However. the examples of Hamiltonian mechanics and Maxwell's equations (see Chapter 8) show that their applicability goes well beyond this. The goal of the chapter is to develop a special calculus of diHerential forms. dye largely to E. Cartan [1945). The exteri(lr derivative (Iperator d plays a central role. The properties of d and the e'lIHc<.sion of the Lie derivative in terms of it will be developed.
'.1
EXTERIOR ALGEBRA
We begin with the exterior algebra of a vector space and extend this fiberwise to a vector bundle. As with tensor fields. the most important case is the'tangent bundle of a manifold. which is considered in the next section. We shall recall a few facts about the permutation group on k elements. Proofs of the results that we cite are obtainable from virtually any elementary algebra book. The permutation group on k elements. denoted S". consists of all bijections 0: {I •...• k} ..... {1 •...• k} usually given in the form of a table
323
324
DIFFERENTIAL FORMS
together with the structure of a group under composition. Clearly. Sic has order k!. Letting (R, X) denote R\{O) with the multiplicative group structure, there is a basic homomorphism sign: Sic -+ (R, X); that is, for 0, -rES", sign(oo-r)-(signo)(sign-r). The image of sign is the subgroup {-I, I}.
A permutation 0 is called even when sign 0 =- + I and is odd when sign 0 = -I. A transposition is a permutation that swaps two elements of {I,.: .. k). leaving the remainder fixed. An even (odd) permutation can be written as the product of an even (odd) number of transpositions. The expression of 0 as a product of transpositions is not unique. but the number of transpositions is always even or odd according as 0 is even or odd. If E and F are Banach spaces. an element t of T~)(E. F) - LA(E: F); i.e.• a k-multilinear continuous mapping of Ex· .. x E -+ F is called skew symmetric when
for all el .... ;e" E E and 0 Eo SA' This is equivalent to saying that t(t', ..... ed changes sign when any two of e l ... .• eA are swapped. The subspal'e of skew symmetric elements of L"(E; F) is denoted L!(E; F) (the suhscript a stands for "alternating"). Some additional shorthand will be useful. Namely, let AO(E. F) = F. N(E. F) = L(E. F) and in general. A"(E, F) = L!(E; F), the vector space of skew symmetric F-valued multilinear maps or exterior F-valued k-forms on E. If F-R. we write AO(E)-R, N(E)-E* and A"(E) = L!(E; R); elements of A"(E) are called exterior k-forms. t To form elements of N(E. F) from elements of T"o(E; F), we can skew-symmetrize the latter. For example. if t E T2o(E). the two tensor At defined by
is skew symmetric and if t is already skew. At coincides with t. More generally. we make the following definition. 8.1.1 Deftnltlon. The IIhentllt;OII mapping A: TAO ( E. F) -+ T"o( E. F) (for notational simplicity we do not index the A with E. For k) is defined by
At( e l ••• .. e,,) = kl,
L
(sign 0 )t( eo(l) .... • eO(Ie))'
. oES"
where the sum is over all k! elements of Sk' fSome authors write 11."(£-) where we write 11.'(£).
EXTERIOR ALGEBRA
••1.2 Proposition. A is a linear mapping onto /\4 ( E. the identity. and A 0 A = A. Proof.
Linearity of A follows at once. If t At(el ... ·.ek ) =
i,
L
(signo)t(e a ("
. "e S.
=
i,' L
E Ak(E. F).
. "es.
.....
i,
L
n. A IN( E. n
is
then e n (4»)
-
t(el ... ·.ek )
since (signa)2 = 1 and Sk has order k!. Second. for t
·At(e, ..... ed=
325
E
71."( E. F) we.have
(signa)t(e,,(I) ..... enfA))
. "es.
=
~, 1:
(signaT)t(e".(I, ..... enT(k,)
. "es.
since a ...... aT is a bijection (its inverse is p ...... a - Ip) and the map sign is a homomorphism. This proves the fiTst two assertions. and the last follows from them. _ From A = A 0 A. it follows that IIAII $IIAII2, and so. as A'" O.IIAII ~ I. Since A = Id on Ak(E. F).IIAII $1. Thus IIAII = I, so A is continuous. We may define the wedge. or exterior product as follows .
••1.3 DeflnlUon. If aE Tt<E) and p E 1j1l(E). define their wedge prodflCt CIA
PE
Ak+t(E) by ClAP=
(k + I)! k!/! A(CI@P).
For F-valued forms. we can also define A. where @ is taken with given bilinear form BE L(F1• F2 ; F3 ). Since A and @ are respect to continuous. so is A. There are several possible conventions for defining the wedge product A. The one here conforms to Spivak (1979). and Bourbaki (1971) but not to
a
326
DIFFERENTIAL FORMS
Kobayashi and Nomizu [1963] or Guillemin and Pollack [1974]. See Exercise 6.1 G for the possible conventions. Our definition of u/\ P is the one that eliminates the largest number of constants encountered later. The reader should prove that for exterior forms.
(I) where E' denotes the sum over all (k./) shufflest ; that is. permutations a of
{1,2, ..• ,k + I} such that a(l) < ... < a(k) and a(k + I) < ... < a(k + I). Formula (I) is a convenient way to compute wedge products. as we see in
the following examples. 8.1.4 Examples A. If u is a two-form and
P is a one-form, then
Indeed the only (2, 1)-shuffles in S) are
2 2
2 3
2 3
of which only the second one has sign - I. B. If u and I) are one-forms. then
since S2 consists of two (I. 1) shuffles.
..
Basic properties of the operation /\ are given in the following.
8.1.5 Pr0p08It1on. For
uE
Tko(E).
PE T,°(E). and '( E
T,2(E). we have
(i) u/\ P - Au /\ I) = u/\ AI); (ii) /\ is bilinear (iii) (iv)
W'ts /\
u/\ P - (u; u/\(I) /\ 'Y)" (u/\ ts)/\ 'Y.
t1be reason for this name is that this is the kind of permutation made when a deck + I cards is shufflc:d. with k cards in one hand and I in the other.
of k
EXTERIOR ALGEBRA
327
Proof. For (i). first note that if aESk and we define at(el ... ·,ek)t(, ..O),. ••• e..(k)' then A( at) - (signa)At. for
A (at)( '1'"
:! L
,,'d =
(sign p )t( eplJ(I).···
pES,
=
:1
L
.
.epOI~)
(signa )(sign pa )t( epa(l).··· "po(k)
. pES,
=
(signa )At( e l , ... • ek )
since p ..... pa is a bijection. Therefore,
A(Acx@lI)(el.···,ek'···,eH/)=A(Aa('I.···,eA )1I(ek • I'···.'A "»
L
-A(k\
(sign'T)cx(eT(I) •...• eTlk»II(ek., ..... eH/») .
'rE~
A
=
(:!
= :,
L
(SiSn'T)('Tcx@lI)(el •...• ek ..... e'kt/))
rES,
L
(sign'T)A('Ta@lI)(el,· ... e'H/)
(linearity of A)
. rES,
=ki L .
(sign'T')A'T'(cx@lI)(e, ..... ek + / )
. rES,
where 1" E SHI is defined by
'T'(I ..... k .... ,k + I) = (1'( 1) ..... T( k), k + 1, .... k + I) so sign l' = sign 1" and 'TG@II = A(Aa@lI) becomes
k\ •
L
TES,
1"( a@/3).
Thus the preceding expression for
(sign'T')(sign'T')A(a@/3)(e l ,···.ek+/) .
328
DIFFERENTIAL FORMS
Thus A(Aa®p) = A(a®p); that is. (Aa)A P = a A p. The other equality in (i) is similar. Now (ii) is clear since ® is bilinear and A is linear. For (iii). let 0 0 E SIc+' be given by oo(\ •...• k + I) = (k + I •.. .• k + I. I•...• k). Then a®p(el •...• elc+,)=p®a(e"o(l) •...• e"o(k+,). Hence. by the proof of (i). A(a®p)= (signoo)A(p®a). But sign 0 0 = (-I)k'. Finally. (iv) foIlows from (i). • Conclusions (i)-(iii) hold (with identical proofs) for F-valued forms when the wedge product is taken with respect to a given bilinear mapping B. Associativity can also be generali7.ed under suitable assumptions on the bilinear mappings. such as requiring F to be an associative algebra under B. Because of associativity in 6.1.5. a A PAy can be written with no ambiguity.
6.1.6 Examples A. If a ' • ; ... I •...• k are one-forms. then we will show that (a l A ... A a k )( e l •... •ek ) =
L (sign 0 )a l (e (Il)' •• a k (en! ~) O
" (2) This equality holds for k = 1.2. as the examples in 6.1.4 show. Inductively. let us assume the relation is true for k - I. For each (k - I. I )-shufOe T which moves I to the last spot. 1-1.1 1-1.1+ I
k-l.k) k .1'
denote by S; _I the subgroup of Sk (isomorphic to Sk· elements which leave I rixed; these are of the form
ijE{I •... ,I-I,I+I •...• k}.
0=
can be written as a product P=
(~lJ
consisting of
1.1 + I I. i, + I
UI
wlSre
I)
k)
(JI 0
=
JIe POT.
1·1.
J,
whereo(k) =jlc
=1.
where
/-1.1.1+1
J,
J=I •...• k.J.,.
k
JIe
I) Es; I
. EXTERIOR ALGEBRA
and. as above. T
=
(!
1-1.1 1-1.1+1
329
k-I.") k. I .
Thus denoting by E'. the sum over all (k -I.I)-shurnes in Sk' and by the sum over all elements in S: _ I we have by Eq. (I ).
_ ~'(s'gn )~ITI(' ) I( ) A I( -~ 1 T~ slgnp. e IlTIII ••·• •
~
T
E .-s..
EIT)
) A( ) ',,,,4_1,. eTI~1
"
(sipG).'( ••("),, .•• ( .... )
since p(T(k»- T(k). If ., •...••• and .'•... ,•• are dual bases. observe that as a special case,
(.' /\ ... /\ .'X." ... ,.,)-l.
B. If at least one of. or, is of even depee. then 6.I.5(iU) says that ./\ , - , /\ .. If both are of odd depee, then ./\ , - - , /\ .. Thus. if. is a one-form, then • /\ • - •. But if • is a two-form, then in geaeraI./\ • - •. For exampte. if • -.' /\ ,2 + " /\ ,4 e A2(R4)' where ,',,2,,',,4 is the stanclanl dual basis of R 4, then • A • - lei I\. ,2 /\ . ' /\ .4 _ •• e. The properties listed in 6.1.5 make the computations of wed. products similar to polynon-. . . "'iplicatioD, caR beiDa taken with c0mmutativity. For example. if " 't!'e ODe forms on R 5, • - 2.' /\ . ' + .2/\ - ' _ 3-' /\ .4 e A2(RS, ........ ~ - _.1 /\ ..~ " .' + 2.1 /\ .' /\.4 e A'(R'). then the wedp product ./\ , is computed usina the bilinearity and commutation properties of /\: . / \ ' __ 2( .1" .')/\ (.1 /\ .2 /\
.')-(.z /\ .')/\ (.' /\ .2;" .S)
+3(.' /\ .4)/\(.'/\ .2" .')+4(.'/\ .')/\(.1/\ -' /\ .4)
_ 3.3 "
.4 ".' /\ .2 /\ .'
330
DIFFERENTIAL FORMS
To express the wedge construction in coordinate notation, suppose E is finite dimensional with a basis e ... ,ell • The components of t E T,o(E) are " the real numbers (3)
For tEAic(E).t",. is anti symmetric in its indices il ..... i A • For example. t E A2( E) yields til' a skew symmetric n x n matrix. From the definition 6.1.1 of the alternation mapping and (3) we have (At);, . .;o =
~, E • 0
(signa)to(", .. O(;";
ES.
i.e.• At antisymmetrizes the components of t. For example. if t then (At)ij = Ht;j - tji )·
E
T2o(E).
If aE TIc°(E) and IJ E T,°(E), then (I) and (3) yield
where E' is the sum over all the (k. I }-shuffles in Sic + /. 1.1.7 DetinlUon. The direct sum of the spaces AA(E) (k =0.1.2 ... ) together with its structure as a real vector space and multiplication induced by ". is called the exterior IIIgebru of E. or the Grussmlllln IIIgebra of E. It is denoted by J\( E).
Thus A( E) is a graded associative algehra. i.e.. an algebra in which every element has a degree. in our case the degree of the form (a k-form has degree k). and the degree map is additive on products (hy 6.1.2 and 6.1.3). Elements of A( E) may be written as finite sums of increasing degree exactly as one writes a polynomial as a sum of monomials. Thus if II. h. c E R. aEAI(E) and IJEA2(E) then a+ba+cp makes sense in A(E). The one-form a can be understood as an element of N(ln and also of A(E). where a is identified with 0 + a + 0 + 0 + .... We can now find a system of generators of A'(E) and if E is finite dimensional, a basis of Aic(E). 8.1.8 Propoaltlon. (i) The set
n. N( E) = (O). while for 0 < k s n. Al( E) has dimension The exterior
(n
EXTERIOR ALGEBRA
331
algebra over E has dimension 2ft. Indeed, if 1- (e l .... ,eft) is an ordered basis of E and 1*" (e l .... ,e ft ) its dual basis, a basis of Ak(E) is
Proof. (i) This follows directly from the fact that the set of p"@ ... @p" spans Tt( E) (see 5.1.2 and 6.1.2). (ii) First we show that the indicated wedge products span AA(E). If exE Ak(E). then from 5.1.2.
where the summation convention indicates that this should he summed over all choices of i l , .... i k between 1 and n. If the linear operator A is applied to this sum. we have ex= Aex= ex(e" ..... e,. )A(e"@ ... ®e'·).
so that
·./d = ex( ei,..... e;J :1 L
ex( /1'"
(signa)( e"@ ... @e" )( /.,(1) .... ./o(k))
. "eS.
by formula (2). Therefore. Ii,
ex=ex ( ei" .. ·.e,. ) k!~' /\ ... /\ e '. The sum still runs over all choices of the i I .... ' i k and we want only distinct. ordered ones. However. since ex is 'skew symmetric. the coefficient ex( e ..... ei , ) is 0 if il ..... ;k are not distinct. If they are distinct and 11 E SA' th~~
ex( •
". ).i,,, ... "e i, -- ex(.
S;fl,.···~i,
~
~o(;,).···t
ecrt;,) )e"(i,) /\
...
"e,,(i,)' ,
since both tlI and the wedge product.change by a factor of signa. Since there are k! of these rearrangements. we are left with tlI=
L ;1<'"
ex(e" ..... e,,)ei',,"·/\e'·.
332
DIFFERENTIAL FORMS
Secondly. we show that
are linearly independent. Suppose that
E
(1:;, ... ;,
e l , 1\
•.. 1\
el •
=
O.
;, < ... < i.
j.
For fixed +I
i; ..... i". let j. + I"
< ... <
K Then r
i-I
a 'I" .
'1.
•• •j~
denote the complementary set of indices.
e l , 1\ .•• 1\ e" 1\ e'" , 1\ •.• 1\ ej~ =
0
;1 < ... <;,
However. this reduces to (l:li"'I,
el
1\ •.• 1\." -
O.
The proposition now follows. •
8.1.9 Corollary. If dimE=n. then dimA"(E)-1. If (al •...• a") is a basis for E*. then a l 1\ ••• 1\ an spans A"(E). Proof.
This follows from 6.1.8 and the fact that
8.1.10 Corollary. Let al •...• a le dent iff a l 1\ • • . 1\ a k - O. Proof.
E
(~) =
1. •
E*. Then al •...• a k are linearly depen-
If a l•...• a k are linearly dependen t. then
al
=
E Cjaj j.j
for some i. Then, since a 1\ a = O. for a a one-form. we see a l 1\ ••. 1\ a k = O. Conversely. if a l 1\ ••. 1\ a k .. O. then for any vectors el ••••• ek in E. 0 = (a l 1\ •.. 1\ ale)(el'" .• ek ) -= det[a'(ej ») by Example 6.1.6A. We can assume a l .,., O. Choose. by the Hahn-Banach theorem and a Gram-Schmidt argument. a vector.1 such that a l(., ) ~~ I and a 2 ( el) = ... = a k ( "I ) ~.' Il Then
EXTERIOR ALGEBRA
333
set _) = e, for j -l, ... ,n and expand det(a;(ej )] = 0 along the first row to get a' written as a linear combination of az •... ,ak • •
1.1.11 Exam.,... A. Let E=R2. (e"ez) be the standard basis of R2 and (el.e z ) the dual basis. Then any element.., of N (R 2) can be written uniquely as
and any element.., of N(R2) can be written uniquely as
B. Let E =R3. (e,.e Z.e3) be the standard basis. and (e l.e 2.e 3) the dual basis. Then any element.., E N(R 3) can be written uniquely as
Similarly. any elements "Ie AZ(R3) and
t e A3(R 3) can be uniquely written
I'S
and
t - r l23e l " e 2 " e 3. Since R 3, N(R 3), and A2(R 3) all have the same dimension; they are isomorphic. An isomorphism R3 &I N(R 3) - (R 3)* is the standard one associated to a given basis given by demanding e l .... e l , ; -1,2,3. An isomorphism of N(R') with AZ(R3) is given by demanding that
This isomorphism is usually denoted by *: A'(R') ..... A2(R]): we shall study this map in general in the next section under the name Hodge star operator. C. Note that the standard isomorphism of R] with AI(Rl) = (R 1 )* is given by the index lowering action II of the standard metric on Rl; i.e., bee;) - e;. Then R3 -+ A2(R3) has the foUowing property:
.0":
(4) for aU ." w E R 3, where X denotes the usual cross-product of vectors; i.e.
334
DIFFERENTIAL FORMS
This relation follows from the dermitions and the fact that if ex = al~1 + a)~) and II'" fJl~1 + fJ2~2 + fJ3~3. then ex 1\ P = (a 2 fJ) - a)P2 )~2 1\ ~3 +(a"~2
+ (a)PI - alP) )~)
+ a2~2
1\ ~I
- a2PI)~11\ ~2.
Exercl...
6.IA
Compute ex 1\ a, a 1\ P. P 1\ p, and P 1\ a 1\ P where a "" 2~1 1\ ~! ~21\ ~l E A2(Rl) and = - ~I + ~2 -2~J for (e l .e 2 .e-1 ) a basis of
p
(R l
6.18
) •.
Ir k! is omitted in the definition of A (6.1.1). show that 1\ fails to be associative.
Let "I" ..• v" be linearly dependent vectors. Show that a( VI" ..• v" ) = 0 for all aE A"(E). 6.10 Show that A"(E·) is isomorphic to (A"(E»·. (Hint: Define cp: (A"(EW ..... A"(E·) by cp(o)(exl •...• a") = o(exll\ .. , 1\ a") and construct its inverse using the system of generators in 6.I.8(i).) 6.1 C
6.lE
6.1 F
Let (~, •...• ~") be a basis of E with dual basis {~I •...• ~"} and (f1, ...•I"') be a basis of F. Show the following: (i) Every P E A"(E. F) can be uniquely written as P = E:"-IP;l for P; E A"(E). where (y/)(v, •... ''',,) == y( Vi .... •v,,)/ E F for ", •...• "" E E.I E F. and y E A"(E), (ii) {(~;'I\ "'I\~;')~I;I<'" <;,,) is a basis of Ak(E.F) and thus dim(A"(E. F»= (k)m, (iii) dim(A(E.F»=m2". (iv) If BE L(R, F; F). where B(I, J) ... II and 1\ is the wedge product 1\: A'(E)xA"(E,F) ..... Ak+'(E.F) defined by B. show that al\p=E;'!.I(al\p;)/,. If E=Rl, F=R2 and a= e l 1\ ~2 -2~1 1\ ~3 and P = (~' 1\ ~))/I + 2(~2 1\ e 1 )/2 (~I 1\ ~2)/3' compute a 1\ p. Let (el •...• ~.) and (II, ...•h) be linearly independent sets of vectors. Show that they span the same k-dimensional suhspace iff II 1\ '" 1\ I. - -11\ ... 1\ e•• where Q .... O. (Oive a definition of 1,1\ ... 1\ I" as part of your answer.) Show that in fact Q - detcp where cp: span{~I .... ,~,,) ..... span{ II, ....I,,} is determined by cp(~;)'" in 3.1.80.
l. ; -I •.. .• k.
Use this to relate
Ak
with G k
.
6.IG (P. Chernoff. J. Robbin). Let 1\' be another wedge product on forms that is associative and satisfies a I\'p ... c(k./)exl\ II. where ex is a k-form and II is an I-form. c(k. /) is a scalar. and that forms of degree zero act as scalars.
DETERMINANTS. VOLUMES. AND THE STAR OPERATOR
(i)
335
Prove the "cocycle identity"
c( k .I)c(k + I. m) = c{k.1 + m )e(l. m). (~i)
Deduce that c( k
(iii)
+ J. m) = c( k. m + l)c( I. m)jc( I. k).
Define 1/-( I) inductively by 1/-(0) = 1/-( I) = I and I/-(l+I)=c(I.I)I/-(l).
Show that c(k.1) = I/-(k + /)/I/-(k )1/-(/). Deduce that c(k. /) = c(l. k); i.e.• A' satisfies a A'P = (-I) k1 P A'o. automatically. (iv) Show that e given by (iii) yields an associative wedge product. (I/-(k)-l/k! converts our wedge product convention to that of Kobayashi and Nomizu [1963]).
6.2 DETERMINANTS, VOLUMES, AND THE HODGE STAR ~EAATOR
According to linear algebra. the determinant of an n X n matrix is a skew symmetric function of its rows or columns. Thus. if Xl •... •X" E R". then c.l(XI •...• X,,) = det[xl ..... x~]. where [xl ..... x,,1 denotes the n x n matrix whose columns are XI" ..• x". is an element of A"(R"). Also recall from linear algebra that det[x, •...• x,,] is the oriented volume of the parallelipiped spanned by XI ..... X" (Fig. 6.2.1). Finally. if Xi has components x', the
~------------~P
x, Volume( PI - dct(x,. x,. x,) - dctlx,. x,. x,)
Figure 6.2.1
336
DIFFERENTIAL FORMS
determinant is given by det[xl •. ··.x,,] ==
E
(signa )x l,,(I)'
•• :
·x",,<,,) .
"es. In this section determinants and volumes are approached from the point of view of the exterior algebra of the previous section. Throughout this section E is assumed to be a finite-dimensional vector space with dim E = n. Recall from 6.1.9 that A"(E) is one-dimensional. If..,: R 3 .... R 3 is a linear transformation. we recall from linear algebra that det.., is the oriented volume of the image of the unit cube under (see Fig. 6.2.2). In fact det.., is a measure of how .., changes volumes. In advanced calculus. this fact is the basis for introducing the Jacobian determinant in the change of variables formula for multiple integrals. The same thing is done in the next chapter. This background will lead to the development of the Jacobian determinant of a mapping of manifolds. First. however a review of a few facts about the pull-back is needed. p
Figure 8.2.2
Recall that the pull-back ..,*a of aE Tko(F) by .., E L(E. F) is the element of TkO(E) defined by (..,*a)(e l •.... ek ) = a(..,(e l )•. ... ..,(ek If.., E G1.(E. F). then .., .... (..,-1)* denotes the push-forward. The following proposition is a consequence of the definitions and 5.1.9. (The same results hold for Banach space valued forms.)
».
6.2.1 (i)
Propoaltlon. Let.., E
I.( E. F). and", E I. \ ! . (i). Then
..,*: Tko(F)-+Tt(E) is linear. and..,*(AA(F»C Ak(E);
(ii) <",olp)*=..,*o",*; (iii) 1/.., is the identity • .w is lp*;
DETERMINANTS, VOLUMES, AND THE STAR OPERATOR
337
(iv) if"eGL(E.F).then"oEGL(Tt(F). T"o(E».(,,·) I=(cp I). and ,,·(A"(F»- A"(E); (v) If "eGL(E.F). then cp.EGL(T"o(E). Tko(F». ("-1)0=,, •• and (".)-1 = (cp-I).; if'" E GL(F. G). then cp)o = cp.; (vi) If ClE A"(F). and PE N(F). then cpO(CI" IJ) = cp.CI" "op.
(+ 0
"'.0
o,r'" ... " r'
For example, if P = Pa, ... e /\"(F) (the sum over a l < ... < a k ) and" E L( E, F) is given by the matrix (AU, )-i.e .• relative to ordered bases i of E and of F, one has cp( ~j) = A O, fa - then
i
That is,
"'0, ...0, Aa,. ···AU'·a
*4) . . =k!Q ( ... ,.. t" lI··J,
11
,,'
I
< ...
(I)
Now we are ready to consider the determinant. Recall that ,,: /\"(E) ~ A"(E) is a linear mapping and that A"(E) is one-dimensional. Thus. if Ill" is a basis and w - cWo, then cp·w = ccp°Wo = hw for some constant h. clearly unique.
••2.2 DeftnlUon. Let dim(E) - nand cp E L(E. E). The unique constant det cp, such that cpo: A"( E) -+ A"(E) satisfies
for all w e A"( E) is called the detlrm;"", of ".
The definition shows that the determinant does not depend on the choice of basis of E. nor does it depend on any norm or metric structure on E. To compute det" one can use just one nonzero element of N( E). Thus, to find det cpo let us choose a basis e l ••• •• ~" of E with dual basis ~I ••••• e". Let cpEL(E,E) have the matrix (Al,); i.e., cp(ej ) = Lj_IA',er
33IJ
DIFFERENTIAL FORMS
Then by Example 6.1.6A.
",*(e i "
•• ,
"e")(el •...• e,,) = (e l " ... "e")( "'(el) •...• "'(e"» = det[ e i ( '" (e i »] = det[ Aii ]
Thus. since (e l " ... "e")(t'I ..... t'")=I.det"'=det[Ai,). the classical expression of the determinant of a matrix with x, •.... x" as columns. where x, has components AI,. Thus the definition of the determinant (6.i2) coincides with the classical one. From properties of pull-back. we deduce corresponding properties of the determinant. all of which are well known from linear algebra. 8.2.3 PropoaHion. Let ",. '" E L (E, E). Tht'n
dele '" (det '" )(det "'); if'" is the identity. del", - 1; (iii) '" is an isomorphism iff del", * 0, and in this case det( '" - ') = (det "') - ,. (i) (ii)
0 "') -
For (i), ('" 0 ' " )*w = det( '" 0 ' " )w; but ('" 0 ' " )*w = ("'* 0 ",*)w. Hence, ('" 0 ' " )*w'" "'*(det ",)w = (det '" )(det ",)w and (i) follows. (ii) follows at once from the definition. For (iii), suppose '" is an isomorphism with inverse", _. I. Then, by (i) and (iii), 1= det( '" 0 ' " - ' ) == (det q? )(det '" - '). and, in particular, det",,. O. Conversely. if '" is not an isomorphism there is an e, * 0 so ",(e,) = O. Extend to a basis e l, e 2 ..... t'". Then for all n-forms w, (",*w )(e l•.. . ,e,,) "" w(O, ",(e 2 ), •••• ",(e,,» = O. Hence, det", == O. • Proof,
Recall from Chapter 2 that there is a unique vector space topology on L( E, E) since it is finite dimensional. One convenient norm giving this topology is the following operator norm:
where lIell is a norm on E. (See Section 2.2). Hence, for any t' E E.
11",( t')11 ~ 1I",lIlIell·
8.2.4 Propoaltlon. del: L( E. E) -+ R iscontinuous.
DETERMINANTS, VOLUMES, AND THE STAR OPERATOR
339
Proof. This is clear from the component formula for det. but let us also prove it abstractly. Note that
111.0)11 = sup{l 1.0) ( e ••... •ell )1 1lie. II = ... = lIe,,1I =
=
I}
sup{l 1.0) ( e ...... e" )l/lIe.ll· .. lie" II 1e ...... e l "" O}
is a norm on N( E) and I1.0)( e ...... e.)1
S
1l1.o)1I1Ie.ll· .. lie. II· Then. for cpo '" E
L(E. E).
Idet cp - det '" 1111.0)11 = IIcp*1.o) - ",*1.0)11 =
sup{ 11.0) ( cp(" ) ..... cp(,,,))
c.>( "'('.l ..... "'(e,,»IIII'dl
= .... =
11',,11 = I}
~ sup{11.o) ( cp (e. ) - '" (e. ). cp ( e 2)'" .. cp ( e. »1 + ...
+ 11.0) ("'(e.)."'(e2) ... ·.cp(e.)-"'(e.»llIledl = ... = lIe,,1I =
I}
s 1lI.o)lI11cp - "'1I{llcpll·-' +lIcpll,,-211"'1I+ ... +II"'II·-I} ~ 1lI.o)II11cp - "'" (II cp II + "",")" . I Consequently, Idetcp-det"'I~lcp-"'I(lIcpll+II"'")"·· and the result fol~ lows. • In Chapter 2 we saw that the isomorphisms form an open subset of L(E, F). Using the determinant, we can give an alternative proof in the
finite-dimensional case.
•
6.2.5 . Proposition. Suppose E and Fare finite-diml'nsional and let GL(E, F) denote those cp E L(E, F) thai are isomorphisms. Then GL(E. F) is an open subset of L(E, F). Proof. If GL(E.F)=0, the conclusion is true. Otherwise. there is an isomorphism", E GL(E, F). A map cp in L( E. F) is an isomorphism if and only if'" - 'cp is also. This happens precisely when det( '" Icp) "" O. Therefore. GL(E, F) is the inverse image of R\{O) under the map taking cp to det(",-'cp). Since this is continuous and R\{O) is open. QUE. F) is also open. •
The basis elements of A"( E) enable us to define orientation or .. handedness" of a vector space.
6.2.6 DefinlUon. The nonzero elements of the one-dimensional space A"( E) are called voIIIme ekmellts. If 1.0). and 1.0)2 are volume elements, we say 1.0). and 1.0)2 are equivaktlt iff there is a c> 0 such that 1.0). = C1.o)2' An equivalence class
340
DIFFERENTIAL FORMS
(w) 01 ooIl11M ~kments on E is called an oriMI""itNt on E. An 0";"'. wctor IJItIee (E,(wn is a vector space E together with an orientation [w) on E; [- w) is called the rn.woriMtIItion. A basis {~I, ... ,e,,} olthe oriented vector space (E,(w» is called JI08itiwly (resp. Mg"'iwly) orinI,ed, il w(e l , . .. ,e,,) > 0 (resp. < 0). Note that the last statement is independent of the representative of the orientation [w). for if w' e [w]. then w' = cw for some c> 0, and thus w'(el •...• e,,) and w(el, ... ,e,,) have the same sign. A vector space E has always exactly two orientations: one given by selecting an arbitrary dual basis el, ... ,e" and taking [el " ... "e"); the other is its reverse orientation. The definition of orientation, given previously, is related to the concept of orientation from calculus as follows. In R), a right-handed coordinate system like the one in Fig. 6.2.1 is by convention positively oriented, as are all other right-handed systems. On the other hand. any left-handed coordinate system, obtained for example from the one in Fig. 6.2.1 by interchanging XI and Xl' is by convention negatively oriented. Thus one would call a positive orientation in R' the set of all right-handed coordinate systems. The key to the abstraction of this construction for any vector space lies in the 9bservation that the determinant of the change of ordered bases of two right-handed systems in R3 is always strictly positive. ThUs, if E is an /I-dimensional vector space. define an equivalence relation on the set of ordered bases in the following way: 1- {el'" .• e,,} and , - {e, •.•. ,e~} are equivalent iff del. > O. where. e OL(E) is given by fP(e/) - e;. i ... 1••••• n. We can relate n forms to the bases by associating to a basis el, ... ,e" and its dual basis el •...• e" the n-form w - e l " ... "e". The following proposition shows that this association gives an identification of the corresponding equivalence classes. 8.2.7 Propoeltlon. An orientation i/l a vector space;s uniquely determined by an equivalence class 01 ordered bases. Proof. If [wI is an orientation of E there exists a basis {el' ...• e,,} such tlt(lt w(e l... .• e,,)'" 0 since w'" 0 in A"(E). Changing the sign of e l if necessary. we can find a basis that is positively oriented. Let {e;, ... ,e~} be an equivalent basis and.e OL(E), defined by cp(ej)=e:, ; = 1, ... ,/1, be the change of basis isomorphism. Then
w( e" ...• e~) ... w( .(el ) •...• cp(e,,» ... (cp·w)(el.···,e,,) .. (detcp)w(el, ...• e,,) > 0 That is, [w) uniquely determines the equivalence class of {el, ... ,e,,}.
DETERMINANTS. VOLUMES. AND THE STAR OPERATOR
341
Conversely. let {el ••.. •e,,} be a basis of E and let Co) - e l " ..• " e". wbere {el ••••• e"} is the dual basis. As before. Co)'(e; ••• .• e;) > 0 for any Co)' E [Co)] and {e; ••..• e~} equivalent to {e l •.•. ,e,,}; tbus. the equivalence class of tbe ordered basis {e l ••••• ell } uniquely determines the orientation (Co)). • Next we shall discuss volume elements in inner product spaces. An important point is that to get a particular volume element on E requires additional structure. although tbe determinant does not. The idea is based on the fact that in R) the volume of tbe parellelipiped P == P(x i • X2' x) spanned by, three positively oriented vectors XI' X2' and X) can be expressed independent of any basis as
wbere (x/.xj ») denotes the symmetric 3x3 matrix whose entries are (x/. Xj). If XI' x 2 • and X) are negatively oriented. det(x,. x)J < 0 and so the formula has to be modified to Vol(P) == ( Idet(x,. x)1 )
1/2
.
( I)
This indicates that besides the volumes. there are quantilles involving absolute values of volume elements that are also important. This leads to the notion of densities.
'.2.8 Deftnltlon. Let a be a real number. A continuow mopping II: Ex· .. X E ..... R (n factors of E for E an n-dimensional vector space) is called an a· . . .ity if 11(.,( 1)•...• .,( q,» -Idel .,1-11(01 •... •q,).for all "I' ...• q, E E and all., E L(E. E). LeI IAI-(E) denote the a·densities on E. With a -=1. l-densities on E are simply called dsuitin and IAII(E) is'denoted by IAI(E) ..
°
The determinant of ., in this definition is, taken with respect 'to any volume element of E. As we saw in 6.2.2. this is independent of the choice of the volume element. Note that IAlfl(E) is one-dimensional. Indeed. if d l and 112 E IAla(E). III" O. and e l ••••• ell is a basis of E. then 112 (e l ••• •• ell ) == adl(e l ••..• ell ). for' some constant a E R. If "I •...• "" E E. le~ "/ = .,(e,). defining., E L(E. E). Then
i.e.• 112 = ad l •
342
DIFFERENTIAL FORMS
Alpha-densities can be constructed from volume elements as follows. If ,.,eA"(E). define l"'I"eIAI"(E) by 1,.,1 ..(el ..... en)=I,.,(el ..... enW where e l •... • e~ e E. This association defines an isomorphism of N( E) with IAI"(E). Thus one often uses the notation 1,.,1" for a-densities. We shall construct canonical volume elements (and hence a-densities) for vector spaces carrying a bilinear symmetric nondegenerate covariant two-tensor. and in particular for inner product spaces. First we need to recall a fact from linear algebra.
6.2.9 Propoaltlon. Let E be an n-dimensional vector space and g = (.) e T2°(E) by symmetric of rank r; i.e., the map e e E ...... g(e.·) e E* has r-dimensional range. Then there exists an ordered basis {e l .... • e n } of E with dual basis el ..... e" such that
,
g
L
=
c;e;®e;.
;-1
where c, = ± I and r
s
n, or equivalently. the matrix of g is
o c,
o
o
o
This basis {el ..... e"} is called a g-orthonormal basu. Moreover, the number of basis vectors for which g(e,.e;)=1 (resp. g(e;.e;)= -I) is unique and equals the maximal dimension of any subspace on which g is po.vitive (resp. negative) definite. The number s = the number of + Is minus the number of -Is is called the s;gtUltun of g. The number of - Is is called the index of g and is denoted Ind(g). If g is an inner product (i.e .. is positive definite) this proposition (and the proof that follows) is the Gram-Schmidt argument proving the existence of orthonormal bases.
Proof.
Since g is symmetric. the following polarization identity holds:
Thus if, .... O. there is an e. e E such that '(e •• el) .... O. Rescaling. we can
DETERMINANTS, VOLUMES, AND THE STAR OPERATOR
343
',e,..•• )-
assume (' •• ± 1. Let E. be the span of '1 and E 2 " {, e EI,(e •• ,)'" O}. Clearly E. n E2 = {O}. Also, if % e E. Ihen ;: - c.g(%. e. lei e E2 so that E'"," E. + E2 and thus E = E.(f)E2. Now if g '* 0 on E2• thert~ is an e 2 e E2 such that ,(e2• e2) = ('2 = ± 1. Continue indl'ctively 10 complete the proof. For the second part. let E. = span{',lg(e,. e,) = I}. E2 = span{e,lg(e,. e,)'" - I} and kerg =' (elg(e. e') = 0 for all " e E}. Nole Ihat kerg = span{eilg( ei • e,) = O} and thus E = E.(f)E2(f)kerg
Let F be any subspace of E on which g is positive definite. Then clearly F nkerg = {O}. We also have E2 n F= {O} since any ve E2 n F mu~t simultaneously satisfy ,(v, v) > 0 and g(v. v) < O. Thus F n( F2 (f)kerg)'" {O} and consequently dim F s dim E •. A similar argument shows that dim E2 is the maximal dimension of any subspace of E on which g is negative definite. • Note that the number of ones in the diagonal representation of , is (r
+ s)/2 and the number of minus-ones is Ind(,) ... (r - ,f)/2. Nondegen-
eracy of , means thaI r = n. In this case any e e E may be written' e = 1:7_.[g(e. ei)/c;]e;. where c, = gee,. e,) = ± I and {e,} is a ,",orthonormal basis. For g a positive definite inner product. r'" nand Ind(,) '"' 0; for, a . Lorentz inner product r = nand Ind(,) ... 1.
8.2.10 Proposition. Let E be an n-dimensional vector space and T20( E) be nondegenerale and symmetric. (i)
,,=[w) If
is an orientation of E there exists a unique volume element ,,(g) E [wI. called the ,-~. such that ,,( ..........,,) - I lor all positively oriented g-orthonormal bases {" .... ",,} 01 E. In lacl. if {, •• , .. •e"} is the dual basis. then" ....' 1\ ••• 1\ .". MOI'e gennrJlly. il {f....../,,} is a positively orienled basis wilh dual basis (It·..... It .. ). then " ... Idet(,( /;.
(ii)
,E
·/2
ti )) 1 It' 1\ ••. 1\ It"·
There exists a unique a-density 1"1". called the g-a-Malty. such that I lor all g-orthonormal btue.f (., ........) 0/ E. II {.' ...... "}. is th. dual basis. then hal"· 1.'/\ ... II: ."f". MOI'e If!IIUo ally. il "._ .... "" E E. then 1111"( v, •.... ",,) -Idet,( ",. ,~./2.
1"1·('.......,,) -
Proof. First a relation must be established between the determiDaDtI of the foilowiDa three matrices: j)-cfiaa(CI..... C") (see 6.2.9). -'.~) for
-<./.•
344
DIFFERENTIAL FORMS
an arbitrary basis {fl •... •In}. and the matrix representation of 4f E GL(E) where .< ~I) - /, - Aji~j" By 6.2.9 we have
g(/,.~) .. (
i:. Cp~P(i!;~P)(Aki~Ic.A~e/)
p-I
(sum onp) Thus (2) By 6.2.9. Idet(g(ei.e,»1 = I. (i) Clearly if {e l•... •e.} is positively oriented and g-orthonormal. then I'(~I' ... '~n) = I uniquely determines I' E (10)) by multilinearity. Let {II.··· .f,,} be another positively oriented g-orthonormal basis. If cp E GL( E) where .<_;) == /,. ; -I •.. .• n. then by (2) and 6.2.9 it follows that Idet cpl = I. But O
(ii)
follows from (i) and the remarks following 6.2.8. •
A symmetric nondegenerate two-tensor g on E induces one on Ak(E) for every k = O•...• n in the following way. Let
a=ae i , / \ ···/\e " 'I .. 1 I.
~
and
P=P
e " /\ ···/\ei'EN(E) '
, I" .. of It.
I
and let
be the components of the associated contravariant k-tensor, where (gkj) denotes the inverse of the matrix with entries gij = g( ei • ej ). Then put g(k)( a, p) =
1: ;1<'"
a i ,...i, pi, .. .;,
(3)
<;,
If there is no danger of confusion, we will write (a. P> = g(k)(a. Pl. We now show that this definition does not depend on the basis. If i is another ordered basis of E • let a- if"I"'a,I"' /\ ... /\ la, and ,., II -= Ii i a, /\ ... /\ PD""Il, ,"'. Define the isomorphism. E GL(E) with matrix representation CP(~i) =
DETERMINANTS, VOL UMES, AND THE STAR OPERA TOR
345
So defined, g(k) is clearly bilinear. It is also symmetric since
= I)), " ···1)."0 ) , ' , .I,
PI,""
where (gij) - ( gll)-I. and gil - g('i' e1 ). glk) is also nondegenerate since if g(k)(Cl.P)=O for all peAk(E). choosing for P all elements of a basis. shows that Oi, ...i, ... O. i.e .• that Cl = O. The following has thus been proved .
•.2.11
ProposIUon. A nondegenerate symmetric covariant two-tensor g = (.) on the finite-dimensional vector space E induces a similar tensor on Ak(E) for all k = I." ·.n. Moreover. if {el'" "'k} is a g-orthonormal basis of E. in which n
g=
L
c1ei®e'.
ci=±I.
i-I
then {e" and
/I ... /I
e"li l < ... < id is orthonormal with respect 'to glk)= (.).
Introduced next. with the aid of the g-volume f& on E and g = (.). is the Hodge star operator.
•.2.12 Proposition. Let E be an oriented n-dimensional vector space and g=(.)eT2o(E) a given symmetri, ,",,' fwndegenerate tensor. Let" be the corresponding volume element of E (.~ee 6.2.10). Then there exists a unique isomorphism.: A"(E) -0 A"-"(E) satisfying (5)
346
DIFFERENTIAL FORMS
If {e l ,' •. ,e,,} is a positively oriented g-orthonormal basis of E and {e l , ...• e"} is its dual basis, then
wh"r".,(1)<··· <.,(k)and.,(k+I)<··· <.,(n).
First uniqueness is proved. Let - satisfy Eq. (5) and let P = eo(l) /\ ... /\ eO(.) and a be one of the g-orthonormal basis vectors e i , /\ ••• /\ ei'of Ak(E), i l < ... < i •. Then by (5). a/\ - P= 0 unless (il'" ·,i.) = (.,(1)", ".,(k». Thus - P= aeo(k+ I) /\ ••• /\ eo(n) for a constant a. But then p /\ - P - asign(a)1L and by (4), (P. P) "" co (I)'" cOCk)" Hence a = Co(l) Proof.
... c,,(k)sign(a) and so - must satisfy (6). Thus - is unique. Define. by means of (6), recalling that eo(l) /\ ••• /\ eo(k) for a(l) < ... < a(k) forms a g-orthonormal basis of Ak(E). As before, Eq. (5) is then verified using this basis. Clearly - defined by (6) is an isomorphism, as it maps theg-orthonormal basis of Ak(E) to that of An-k(E). • The isomorphism -: Ak ( E) --- An - k (E) of this proposi tion is called the Hodge star operator. Some basic properlies are summarized in the following.
6.2.13 Proposition. Let E be an oriented n-dimensional vector space, g'" (,) E T20(E) symmetric and nondegenerate of signature s, and IL the associated g-volume of E. The Hodge star operator satisfies the following, for a. P E Ak(E).
a/\ - P =
P /\ - a =
(7)
(a. P)IL.
_1- (IL. -IL)'" (_I)'nd(.). _. a= ( - I) 'nd(.)( -
I)k(n-k) a.
(8) (9)
( 10) (Recall that Ind(x) is the number of minus-ones in the canonical form of (.
».
Equation (7) follows from (5) (note the symmetry of (<<. P». Equation (8) follows directly from (6), with n = O. n. respectively. and .,.., identity (note that c i . . . CII = (-I)'nd(·). It suffices to verify Eq. (9) for a = 1'0(1) /\ ••• /\ eo(k). By Eq. (5). Proof.
for a constant b. To find b use Eq. (5) with a =
P=
e,,(k+ I) /\ . . . 1\ 1'''(11)
DETER~/NANTS,
VOLUMES, AND THE STAR OPERATOR
347
to give
Hence b -
e",A
• • ( a"(I) 1\
I
II' ••
.•• 1\
C"'HI( -1 )., .. ~ A)8igno. Thull by Eq. (6) •
a"''')) ... C,,(I) . . . C"'AI sign 0
.
= c,,(1)'" c"lk)C"lk+ I ) ' " C",") ( slgno
•
(a"O , I, 1\
. . . 1\
)2( - I)AIII . - k) a ,,(I) 1\
an( n , )
••• 1\
a "IA)
Finally for Eq. (10). we use (7) and (9) to give (. a •• P)I' = • al\ •• p = (_I)lndI.)( -I)k'" - A). CIA = ( _ I )lndW
p 1\ •
P
a = ( - I )lndI.1 (a. P)1'. •
'.2.14 Examplee A. The Hodge operator on N(Rl) where Rl has the standard lfletric
,.1 .• ,.2
,.3.
and dual basis is given from Eq. (6) by .,.' = ,.2 /\ = - ,.' /\ and, ,.2. (This is the isomorphism considered in 6.1.118). B. USi,ng (5). we can compute • in an arbitrary oriented basis. 'Write
.,.1 _,.' /\
• (,./, /\ ... I\"/')=c/''''''. II. ,"·lA • ., ,."" /\ ...
1\
e ' ·'·
(sum overi.+1 < ... < i.+H) and apply Eq. (5) with II =,." /\ ... /\ ,./, and a = ,.j, /\ ... I\,.h where i,.' .. JA is a complementary' set of indices. One gets
Hence
• (,.1, 1\
... /\
,.1,)
-ldetgI1f2Esign(
~
(II)
1,
where the sum is over all (k. n - k) shuffles
(JI
l).
348
DIFFERENTIAL FORMS
C.
In particular, if k = I. Eq, (II) yields
.~I-'det(g),1/2
n
E (-I)/~lg'j~I"
.. ,
"?,, ... ,,~"
(12)
j-I
since sign(jl' i2'" ',in) = (-1)1- 1• for 12 < ... < in. il = i. and where? means that ~j is deleted, . D. From 8 we can compute the components of • II, where liE Ak(E). relative to any oriented basis: Write a = a i,...il' " ... " ~i. (sum over i I < ... < i k) and apply Eq. (II) to give
. (I
(. a) ""',detg,I/2Eslgn .
JI
Hence
~)
.III
(13)
(summed over complementary indicesil < ... < id. E. Consider R4 with the Lorentz inner product. which in the standard basis '1"2'~3'~4 has the matrix
(~
o I
o o
o t o o
gj. -I
Let ~I,~2.~3,~4 be the dual basis. The Hodge operator on N(R4) is given by
If R4 had been endowed with the usual Euclidean inner product. the ,4) V. "have formulas for • ~4 • • (~I" ~4)•• (,2" ,4). and
.(,3"
DETERMINANTS. VOLUMES. AND THE STAR OPERATOR
349
opposite signs. The Hodge. operator on A3(R 4 ) follows from the formulas on N(R4) and the fact that for It E A3(R 4 ), •• It = It (from formula (9». Thus we obtain
.(~I" ~2"
,4) = ,1, _(,1",2"
F. If It is a one form and "I'~'" normal basis. then
'.""
,~) =
,4.
is a positively oriented ortho-
(- It)(~ •...• q,) = It( "I)' This foUows directly from (5) taking fI- J. the first element in the dual basis.
Exercl. . 6.2A
Let ~1.,2. ~3 be the standard dual basis of R 1 and fI:z,1 " ,2 1,2" ,3 E A2(R 3 ). It "" 3,1 - ~2 + 2~1 E N(Rl) and.., E,I.(R 2,R 1 ) have the matrix
[~
0] . .
-\
I
Compute ..,-fl. With the aid of the standard metrics in R 2 and R 3 compute. fI, • It•• (..,-fl). and .(..,-p). Do you get any equalities? Explain. 6.28 A map. e L(E. F). where (E. w).(F. Ii) are oriented vector spaces with chosen volume elements is called volume p"seroing if ..,-" = Cal. Show that if E and F have the same (finite) dimension. then.., is an isoll¥>l'Pbism. 6.2C A map • e L(E, F), where (E.l,,» and (F.lw» are oriented vector . spaces is called orientation preserving if.·" E I w]. If dim E - dim F. and • is orientation preserving, show that. is an isomorphism. Give an example for F - E - R 3 of an orientation-preserving but not volume-preserving map. • 6.20 Let E and F be n-dimensional real vector spaces with nondegenerate . symmetric two-tensors" E T2o(E) and" E T2o( F). Then.., E L(E. F) is called an ;som,try if ,,(.(,)• ..,(~'»=,( •• e'). for all ,.e'E E. (i) . Show that an isometry is an isomorphism. and ,,(II). (ii) Consider on E and F the,. and "-volumes Show that if .., is an orientation-preserving isometry. then cp-
lie,)
350
DIFFERENTIAL FORMS
commutes with the Hodge star operator; i.e.• the diagram Ak(F)----· A,,-k(F)
~·l
1~.
Ak(E)
6.2E
• A,,-k(E)
commutes. If ~ is orientation reversing. show that • (~·a) == - ~.(. a) for aE Ak(F). Let g be an inner product on R3. Use 6.1.I1C and 6.2.14C to show that if {el.e2.e3} is a positively oriented basis then ei X ej = (detg)'/2g k1el
where ( 6.2F
.1·~·k3) E S3 has sign + I.
I. j.
Let E be an n-dimensional oriented vector space and g E T2o(E) be symmetric nondegenerate of signature s. Using the g volume. define the Hodge star operator .: Ak(E. F) .... A,,-k(E. F). where F is another finite-dimensional vector space by •a
= (•
a' ) /, .
where a i E A*(E). (fl'" ·./m} is a basis of F and a= aJ,. Show the . following: (i) The definition is independent of the basis of F. (ii) •• - (_1)(,,-·)/2+k(,,-*) on N(E; F)
If" E Tzo(F) and if it( /.a)= (. ai)h(f. /,). then • it( /. a) = h(/•• a). (iv) If 1\ is the wedge product in A(E; F) with respect to a given bilinear form on F. then for Cl, II E Ak(E. F).
(iii)
( • a) 1\ II =
(•
II) 1\ a
and a
1\ ( •
II) = II 1\ ( • a).
(v) Show how g and h induce a symmetric nondegenerate covariant two-tensor on A*(E. F) and find formulas analogous to (7)-(10). 6.2G Prove the following identities in R 1 using the Hodge star operator:'
II" X vll2 = 1I,,1I 211v1l 2- (". V)2 . and
• X(vX.,) - (.,.,)O-(M· v).,.
DIFFERENTIAL FORMS
6.2H
351
(i) Prove the following identity for the Hodge star operator:
(.ex.P)'" (ex A P.,,). where exE A"(E) and PE A"-"C E). (ii) Prove the basic properties of • using (i) as the definition. Let E be an oriented vector space and S c T20(E) be the set of nondegenerate symmetric two-tensors of a fixed signature s. (i) Show that S is open. (ii) Show that the map Vol: g ..... /-I(g) assigning to each g E Sits g-volume element is differentiable and has derivative. at g given by" ..... !
6.21
6.3
DIFFERENTIAL FORMS
The exterior algebra can now be extended from vector spaces to vector bundles and in particular to the tangent bundle. First of all. we need to consider the action of local bundle maps. As in Chapter 3. V x F denotes a local vector bundle. where V is open in a Banach space E and F is a Banach space. From V x F we construct the local vect~ bundle V X A"(F). Now we want to piece these local objects together into a global one. '.3.1 Definition. Let cp: V x F -> V' x F' he a locul ,'('ctor bUildIe map that is an isomorphism on each fiber. Theil define '1'.: I.' X A"(F) .... V'x A"(F') by (u,w) ..... (cp(u).CPu.w). where IJ'" is the second factor of cp (an isomorphism for each u). 6.3.2 Proposition. If cp: V x F -> v' x F' is a IlIt'ol ('ector bundle map that is an isomorphism on each fiber. then so i,J CP •. Moreover, if cP is a local vector bundle isomorphism. so is CP •. Pf'fJO/. 6.3.3
This is a special case of 5.2.4.
Definition.
•
Suppose 'IT: E .... B is a vector bundle. Define
A"( E)IA = U A"(E,,). heA
where A is a subset of Band E" = 'IT - I( b) is the fiber over b E B. Let N(E)lB= A"(E) and define A"('IT): A"(E) .... ·B by A"('IT)(I)-h if tE N(E,,). 6.3.4 Theorem. Suppose {E-I U;. cp;} is a vector bundle atlas of 'IT. where CPt: EIU; .... U;'x F(. Then (A"(E)IU;.cp;.};s a vector bundle atlas of A"('IT):
352
DIFFERENTIAL FORMS
A"(E) - B. wMn ~.; A"(E)Ilf; -If;' X N'(F') is defined by cp;.IEb (cp;IEb ).
Proof. We must verify (VBI) and (VB2) of 3.3.4: (VBI) is clear; for (VB2) let CP" CPj be two charts for "'. so that cP, 0 CPj- I is a local vector bundle isomorphism. (We may assume If; = ~.) But then. CP;.o CPj. -I = (cp; 0cpj-I)•• which is a local vector bundle isomorphism by 6.3.2. • Because of this theorem. the vector bundle structure of .,,: E .... B induces naturally a vector bundle structure on A"( .,,): AIe ( E) .... B. Let us now specialize to the important case when .,,: E .... B is tangent bundie. If TM : TM - M is the tangent bundle of a manifold M, let A"(M)==A"(TM). and A'/.,=A"(TM)' so A'/.,: A"(M)-M is the vector bundle of exterior k forms on the tangent spaces of M. Also, let (1o(M)= iff(M), gl(M)'" 6j,o(M), and (1"(M)'" rOO(A"M)' k = 2,3"" .
"3.5 PropoaItIon. Regarding 6j"o(M) as an iff(M) m~e, O"(M) is an iff(M) submoduJe. That is, O"(M) is a real subspace of 6j"o(M) and if f E Iif(M) and aE O"(M). thenfaEO"(M).
If '1,'2 E O"(M) and f E Iif(M). then we must show ft l +'2 E This follows from the fact that for each mE M, the exterior algebra on T,.,M is a vector space. • Proof.
Olc( M).
1.3.1 PropoalUon. If a" p:
M -
aE
(11c(M) and
pE
(1'(M), k,l- 0, I,···, defin~ Then a" p E
AH/(M) by (a" p)(m) = a(m)" p(m).
OIc+'(M), and" is bilinear and associative. Proof. First," is bilinear and associative since it is true pointwise. To show P is of class Coo, consider the local representative of P in natural charts. This is a map of the form (a" P) .. - B o (a.. X p.. ). with a .. , p.. , Coo and B - ", which is bilinear and continuous. Thus (a" P) .. is COO by the Leibniz rule. •
a"
a"
1.3.7 Deftnltlon. Let O(M) denote the direct sum ofO"(M), k == 0,1", ',. together with its structlll'e as an (infinite-dimensional) real vector space and with the multiplication" extended componentwise to O( M). (1/ dim M = n < 00, the direct sum need only be taken for k = 0,1,' . ·,n). We call OeM) the algebra of ex,erior di/leNII,iIII/omu on M. Elements of (l1c(M) are called k-/omu. In particular, elements of IX,.( M) are called OM-/OIYIM.
Note that we generally regard O( M) as a real vector space rather than an Iif(M) module [as with 6j(M»). The reason is that Iif(M)=Oo(M) is included in the direct sum, andf "a= fflJa= fa.
DIFFERENTIAL FORMS
353
8.3.8 Remalb and Exampl.. A. A one-form • on a manifold M assigns to each m E M a linear functional on Tift M. B. A two-form Col on a manifold assigns to each m E M a skew symmetric bilinear map Col,": T",M x T"'M ..... R. C. For an n-manifold M, recall that a tensor field 1 E '.'f:(M) has the local expression I(U)=I",,,j, .. II
where
U
E U, (U, qJ)
t j I ... j ,
(U)~®'" ®~®dxll®
I.
ax"
ax"
... «>dxl·(u)
is a local chart on M, and u =1
. ()
J,'''J.
(d'x
I
•••
d'x' -axil' a- ... • -axiJ a )(
..,
U
)..
Corroborated with Section 6.1. this gives the local expression for Col E 0· (M), 'namely
where
D. In O( M), the addition of forms of different degree' is .. purely formal" as in the case M = E. Thus, for example, if M is a two-manifold (a surface) and (x, y) are local coordinates on U eM, a typical element of OeM) has the local expression / + gdx + h dy + k dx 1\ dy, for /, g, h, k E ~(U).
E. As in Section 6.1, we have an isomorphism of trivial vector hundles .: N(R 3 ) .... A2(R 3 ) given by
On the other hand, the index lowering action given by the standard Riemannian metric on Rl defines a vector bundle isomorphism II: T(R l ) .... T*(R 3 ) - N(R l ). These two isomorphisms applied pointwise define maps
Then 6. .1.98 implies
354
DIFFERENTIAL FORMS
for any X. Y e 'X(R l ), where X x Y denotes the usual cross-product of vector fields on R 1 from calculus. That is, X x Y- (X2yl -
X3y2)~ +(Xlyl- Xly3)~
ax 2
axl
+(Xly2_
X2yl)~ ax J
i=I,2,3. F. The wedge product is taken in O(M) in the same way as in the algebraic case. For example, if M=RJ.cx""dx l -x l dx 2 eO I (M)
and
P=x 2 dx l /l.dx J-dx2/1.dx J,
then
••3... Definition. Suppose F: M - N is a Coo mapping of manifnldf. For '" e OIt(N), define p",: M - A"(M) by P",(m) = (TmF)·o '" F(m); i.e .• 0
(P", )"'( "I" .. ,.,,) = "'F(",)(T",F- "I" . " Tm F· ",,), where "I"" ,"k e T",M. We say p", is the puI'-1Mdc of'" by F. Especially. note that for g e OO(N). F*g = g 0 F. See Fig. 6.3.1. Some basic properties of pull-back are given in the next proposition .
••3.10 Proposition. Let F: M - Nand G: N - W be Coo mappings of manifolds. Then (i) P: Ok(N)_O"(M); (ii) (Go F)· = poG·; (iii) if H: M - M is the identity; then H·: Ok (M) - Ok( M) is the identity; (iv) if F is a diffeomorphism, then P is a vector bundle isomorphism and (p)-I - ( r I).;
(v)
P( cx /I.
tt) -
Pcx /I.
F*P for cx e
0"( M) and tt e 0'( M).
DIFFERENTIAL FORMS
,
355
'F"
.,
.. ---- .... .... -
Figure 6.3.1
Proof. Choose charts (U, q», (V, ~) of M and N so that F( U) c V. Then the local representative F"I/- = ~ 0 F 0 q> - 1 is of class Coo, as is c.>1/- = (T1/I)* 0 c.> 0 1/1- I. The local representative of Pc.> is
which is of class Coo by the composite mapping theorem. For (ii), we merely note that it holds for the local representatives; (iii) follows at once from the definition; (iv) follows in the usual way from (ii) and (iii); and (v) follows from the corresponding pointwise result. • We close this section with a few optional remarks about vector-bundlevalued forms. As before. the idea is to globalize vector-valued exterior forms. 6.3.11
Definition.
Let 77: E
-+
B. p: F
-+
B be vector bundles Oller the
same base. Define
the vector bundle with base B of vector bundle homomorphisms over the identity from N( E) to F. If E == TB. Ak(TB: F) is denoted by N( B; F) and
356
DIFFERENTIAL FORMS
is called the vector bundle 0/ F-va/wd k-forms 011 M. 1/ F = B x F, we denote it by A"(B, F) and call its elements J¥ctol'-f)a/uedk-fomu ill M. The spaces 0/ sections 0/ these bundles are denoted respectively by O"(E; F). O"(B; F) and OIr(B; F). Finally,O(E; F) (resp. O(B; F), O(B, F» denotes the direct sum o/O"(E; F), k -1,2", ',n, ... together with its structure as an infinitedimensional real vector space and <J ( B)-module.
Thus, IIEO"(E; F) is a smooth assignment to the points h of B of skew symmetric k-linear maps IIh: Eh X ... X Eh .... Fh • In particular. if all manifolds and bundles are finite dimensional, then liE DIr( M,R r) may be uniquely written in the form 11= Ef_llljej• where ul.···.UPEOA(M). and (e l•·· ·,ep } is the standard basis of Rp. Thus uEOIr(E.RP) is written in local coordinates as l . dx j , ( a ',".'A-
"
•••
"dXi • ... aP
'
"1
0
. dx" " ... " dx i.) ",.
for i l < ...
Exercises 6.3A
Show that for a vector bundle 7r: E ..... B. N(E) is a (smooth) subbundle of T"o( E). Generalize to vector-bundle-valued tensors and forms. 6.38 Let,,: R3 -+R2 be given by ,,(x, y, z)= (x 2 , yz). For II'" v 2 du + dv E 01(R2) and p - uvdu" dv E 02(R2), compute II" p, ,,·11, ,,~ and ".( a " P). 6.3C (Cartan's lemma). Let M be an n-manifold and a l, .. ·,a" E Ol( M), k 5. n, be pointwise linearly independent. Show that pi, ... ,p". E 1l1(M) satisfy E~_lai " pi = 0 iff there exist C"" functionsa i i E <J(M) satisfying a i i = aj such that pi = aijai • (Hint: work in a local chart and show first that a i can be chosen to be dxi; the symmetry of the matrix (ail) follows from antisymmetry of " and the given condition.)
6.4 THE EXTERIOR DERIVATIVE, INTERIOR PRODUCT, AND LIE DERIVATIVE We have already defined the operator d: Oo( M) -+ Ol( M); I ..... dl that gives the differential of functions. Now we extend this to a map d:
EXTERIOR DERIVA TlVE, INTERIOR PRODUCT, LIE DERIVATIVE
·357
O·(M) .... OHI(M) for any k. This operator turns out to have marvelous algebraic properties. After developing these we shall show how d is related to the basic operations of div. grad. and curl on RJ. Then we develop . formulas for the Ue derivative. We first develop the exterior derivative d for finite-dimensional manifolds. Later in the section the infinite-dimensional case is discussed.
'.4.1 Theorem. Let M be a finite-dimensional manifold. Then there is a unique lamily 01 mappings dk(U): Ok(U) .... 0' >1(U) (k = O.I.2..... n. and U is open in M). which we merely denote lIy d. called thl' exterior derit""iw on M. such that (i) d is a A-IIIItitkriwltiott. That is. dis R linear and for caE Ok(U) and II E O'(U).
d(ca A II) - dca A II + (-I)kca A df) (product rule) (ii) II f E iff(U). then dl is as defined in 4.2.5; (iii) d 2 = dod = 0 (that is. d H I(U)o d"(U) = 0); (iv) dis IIIIhllYII wi", rnpect to rntrictioru; that is. if U eVe M are open and ca E 0' ( V). then d( cal U) = (dca) IU. or the lollowing diagf.'am commutes:
IU O"(V)
d
• OIc(U)
1d
J
OHI(V)_ O"+'(U)
IU .' As usual. condition (io) means that d is a I«IIl """,'or.
Proof. We first establish uniqueness. Let (U.CP). where cp(u)= (x' ..... x .. )•.
, ,
i".
be a chart and ca- IIi ... 1 dx l , A ••• A.. dx i , E O"(U). i, < . " < If k = O. the local formula dll- (alii axl) dx' applied to the coordinate functions Xl. i - ....... n shows that the differential of Xl is the one-form dx'. From (iii). d( dx l ) - O. which corroborated with (i) shows that d( dx i , 1\ .•• A dx i ,) == Thus. again by (i).
o.
(I) and so d is uniquely determined on U by properties (i)-(iii), and by (iv) on any open subSet of M. "
358
DIFFERENTIAL FORMS
For existence, define on every 'chart (U,.,,) the operator d by Eq. (I). Then (ii) is trivially verified as is R-linearity. If p.,., Pio" ..},dx il /I. •.• /I. dx h E g'(U). then
..
.... p." ".
d(a /I. A) = d(a." I -
(101""10 ( _ _ _ II
(Ia,···i '_I dx' (Ix i
= __
+( - I)
.
~io""l'
(Ixi
It.
a·
/I.
dXi,
...
+0
1""/0
dx i, /I.
/I. '"
/I.
dxio
(I~""h).. - - dx' /I. dx" (Ix I
••• /I.
dx i,
.
/I.
.
p.
" ..."
/I.
dx io
/I. ••• /I.
/I. ••• /I.
dx il
dx h )
.. dx'o /I. dXl'
/I. ••• /I.
/I. ••• /I.
. dx"
dx h
apj, ..." . .
.
. dx" /I. ••• /I. dx" /I. - - d x ' /I. dx 1' /I. ••• /I. dx"
ax'
',""0
=-da/l. P +( -I)"a/l. tIP. and (i) is verified. For (iii), symmetry of the second partial derivatives shows that
Thus. in every chart (U• .,,). Eq. (I) defines an operator d satisfying (i)-(iii). It remains to be shown that these local ds define an operator d on any open set and (iv) holds. To do so, it suffices to show that this definition is chart independent. Let d' be the operator given by Eq. (I) on a chart (U'• .,,'), where U' () U .. ". Since d' also satisfies (i)-(iii), and local uniqueness has already been proved, d'a = da. on U () U'. The theorem thus follows. •
6.4.2 Corollary. Let.., E g"(U), where U c E is open. Then d..,( u)(~ •...• v.) ...
Ie
E (-I); D..,( u )·v,( ll(, •••• • e
,-n
i .....
v,,)
where v, denotes that v, ;.~ deleted. Also. we den()fe element.v (u. v) 0/ TU merely by v lor brevity. [Note that D..,(u)·VE L!(E.R) since ..,: U ..... L!(E.R).)
"-f.
It suffICeS to show that d dermed this way coincides with Eq. (I). For functions and one-forms this veriracation is trivial. The rest follows by induc:tion on the degree of..,. •
EXTERIOR DERIVATIVE, INTERIOR PRODUCT, 'LIE DERIVA TlVE
359
8.4.3 Exampl..
A.' On R2.let a= f(x. y)dx + g(x. y)dy. Then da= df A dx
+ fd(dx)+dg
A
dy + gd(dy)
, by linearity and the product rule. Since d 2 = O. da-df
Adx +dg Ady -
ag + ( axdx
(¥XdX
+ if;dy)A dx
ag ) v I'dy. + ayd.
Since dx A dx = 0 and dy A d.y = O. this becomes da= af dyAdx+ a8dXAdV=(ag - af)dxAdy.
ay
ax'
ax
ai'
B. On R 3; letf(x. y. z) he given. Then af af af df = -dx + -dr + -dz, ax ay' ilz
so the components of df are those of grad f. That is, (grad.f )' = df, where t> is the index lowering operator defined by the standard metric of R3 (see Section 5.1). C. On R3. let
Computing as in Example A yields • (aF aF,- aF, ) dF"= -2- aF, - ) dxAdv- ( ax ay 'az ax' dXAdz aF3 - aF 2) +(dVAdz. ay az .
Thus associated to each vector field G = Gli + Gd + G3 i on R J is the one-form Gb and to this the two-form .(Gb) by
360
DIFFERENTIAL FORMS
where. is the Hodle operator (see Section 6.2); it is clear that 4Fb_(curIF)b. D. The divergence is obtained from 4 by 4- Fb - (divF) dx 1\ dy 1\ dz;
i.e.,
_4_,'0- divF. Thus, by associating to a vector field F on R J the one-form F" and the two-form - F,4 gives rise to the operators curl F and div F. From Db - • (curl F)b it is apparent that
tl4F b - 0 - 4-(curl F)'o - (div curl F) dx 1\ dy 1\ dz·. That is, 4 2 == 0 gives the well-known vector identity div curl F .... O. Likewise, MI-O becomes 4(gradf)b-O; i.e., -(curl gradf)"=O. So here 4 2 -0
becomes the identity curl grad 1- O.
•
The effect of mappings on 4 can now be considered. Recall that O(M) is the direct sum of all the O*(M). Let F: M - N be a C' map. As F*: O*(N)-D*(M) is R-Iinear, it induces a mapping on the direct sums, F*: O(N)-O(M). -+ N be of class C'. Then F*: D(N) -O(M) is a homomorphism of differential algebras; that is,
8.4.4 Theorem. Let F: M
(i) F*(+ 1\ Col) ... F*+ 1\ F*Col, and (ii) 4 is IItIIIIrrII willi rap«I 10 mtIpp;"p; that is, F*(4Col) - 4(F*Col), or the following diagram commutes:
Proof. Part (i) was already established in 6.3.11. For (ii) we shall show in fact that if m E M, then Ihl're is a neighborhood U of m EM such that 4(F*ColI U) "" (F*4Col)1 is sufficient, as F* and 4 are both natural with respect to restriction. Let (V, cp) be a local chart at F( m) and U a neighborhood of m E M with F( U) c V. Then for Col E O*(y), we can write
u. " '.
EXTERIOR DE'RIVATIVE, INTERIOR PRODUCT, LIE DERIVATIVE
where
and so
361
a a =-iJx;. '0
and by (i)
If ~ E gO(N), then d(F*~) = F*d~ by the composite mapping theorem, so
d(F*IiJIU)-F*(dliJ
~y
".
'... . )/\F*dx " /\ ... /\
F*(h;"
(i). •
•.4.5 Corollary. The operator d is natural with respect to push-forward by diffeomorphisms. That is, if F: M ..... N is a diffeomorphism, then F .dliJ = dF ,j,IiJ, or the following diagram commutes:
F.
gA(M)--_ glc(N)
14 .
1
d gAil(M)
• gk+ I(N)
F.
Proof. Since F. = ( r I)., the result follows from 6.4.4(ii). • The next few propositions give some important relations between the Lie derivative and the exterior derivative.
1.4.1 Propo8ItIon. Let X E CX(M). Then d is natural with respect 10 Lx. TIuJt is, for CIt E; OIc(M) we heme LxliJ E Olc( M) and
or the following diagram commutes: Lx Ok(M) ---gk(M)
d1
14
gk+l(M)
glc+l(M) Lx
362
DIFFERENTIAL FORMS
Proof. Let F, be the (local) now of X. Then we know that Lx..,(m) = ad (F,*..,)(m)1 I
1-0
.
Since F,*.., e nk( M). it follows that LxG) e nk( M). Now we have F,* d.., = d(F,*"'), Then. since d is R linear. it commutes with dldt and so dL x '" = Lx d..,. • In Chapter 5 contractions of general tensor fields were studied. For differential forms. contractions playa special role. 1.4.7 DetlnHlon. Let M be a manifold. X e ~(M). and.., e g"+ I(M). Then define i x '" e 6j"o(M) by
If.., e gO(M). we put i x '" = O. We call i x '" the intnior product or CfIfItNd_ of X and..,. (Sometimes X J.., is written for i x "") 1.4.8 TheoNm. We have ix: gk(M)_gk-I(M).k=I ..... n. and if II e M).II e M). and f e go( M). then
n"(
(i)
n'(
ix is a lI-IIIItitkrivtltiOll. That is. ix;s R-linear and i X (lIl1fl)= (ixG) II II + (- I)kll II (ixll);
(ii) ilxll= fixll; (iii) ixdf - Lxf; (iv) LxG- ixdG+ dixll; (v) LIXIl" /LxII + df II ixll.
Proof. That ixlle nk-I( M) follows at once from the definitions. For (i). R-linearity is clear. For the second pan of (i). write ix( II II 11)( X2 • X] •...• Xk+,) - (II II II)(X. X 2 .···.Xk+,)
and
+(
_I)d k + I-I)! A(II®i.\fl). k!(/-I)!
Noor." _"rite out the dd"mitioD 0( A in terms 0( permutations (rom defmition
'EXTERIOR DERIVATIVE, INTERIOR PRODUCT, LIE DERIVA TlVE
363
6.1.1. The sum over all permutations in the last term can be replaced by tlie sum over aao. where ao is the permutation (2,3 •..•• k + 1.I.k +2•...• k + I) ..... (1.2.3 •...• k + I) whose sign is (_1)k. Hence (i) follows. For (ii). we note that CI is ~(M)-muhilinear. and (iii) is just the definition of L"I. Fo~ (iv) we proceed by induction on k. First note that for k = o. (iv) reduces to (iii). Now assume that (iv) holds for k. Then a k + I form may be written as Dlh " "'i' where ..,/ is a k form. in some neighborhood of m E M. _But Lx(tll"..,) = Lxtll " .., + til " Lx'" since Lx is a tensor derivation and commutes with A. (or from the definition of Lx in terms of nows). and so
ixtl(tll" ..,)+ tlix (til" ..,) =
-
= -
ix (til" tI..,)+ tI(ixtil " .., - til" i x "') ixtll " tI.., + til" ixtl.., + tlixtll "
Ii)
+ ixtll " tI.., + til " tlix'" =
til" Lx'" + tlLxl " ..,
by the inductive assumption and (iii). Since tiL "I =: 1."tll. the result fol-lows. Finally for (v) we have
L,x'" = i,xtl.., + di,x'" = lixtl.., + tI( li x "') =
lixtl.., + til "ix'" + Itlix'"
=
/Lx'" + til" i x""
•
Note that (i) and (ii) are valid without change in Banach manifolds. Formula (iv)
(a "magic" formula of Cartan) is particularly useful. It can be used in the foUowing way. 6.4.9 Exampl.. A. rr CI is a k-form such that tlCI = 0 and X is a vector field such that tli XCi = O. then P,CI = CI. where F, is the now of X. ~ndeed. :
P,CI
=
p,LxCl= p'(ixtlCl + dixCl) = O.
so FiCi is constant in t. Since Fo - identity. FiCl- Cl for aU t.
364
DIFFERENTIAL FORMS
B. Let M .. R], suppose div X = 0 and let u == dx /\ dy /\ dz. Thus tlu-O. Also
So tlixu-tl. Xb "" (divX)u== O. Thus by Example A, P'(dx /\ dy /\ dz) == dx /\ dy /\ dz. As we shall see in the next section in a more general context, this means that the flow of X is volume preseroing. Of course this can be proved directly as well (see for example Chorln and Marsden [1979]). For related applications to fluid mechanics, see Section 8.2. • The behavior of contractions under mappings is given by the following proposition. (The statement and proof also hold for Banach manifolds.) 6.4.10 Proposition. Let M and N be manifolds and f: M ~ N a C' mapping. If Co) E nk(N), X E 'X (N). Y E 'X (M). and Y is J-related to X. then
In particulttr, iff is a diffeomorphism, then
That is, interior products are natural with respect to diffeomorphisms and the following diagram commutes: Ok(N)
r
nk(M)
1irx
ix 1 nk-I(N)
Similarly for Y
E ~(M)
r
' nk-I(M)
we have the following commutative diagram:
nk (M) ----'f'-._ nk ( N)
iyj nk - I (M)
li,.y f.
nk
I(
N)
EXTERIOR DERIVA TlVE, INTERIOR PRODUCT, LIE DERIVATIVE
Proof. Let .,' ..... .,k-' e Tm(M) and
PI =
365
f(m). Then
iyr..,(mH .,' .... '.,k-I) = r..,(m)· (Y( m),.", ... ,v~ I) = ..,(n)·«TI 0 Y)(m). TI( v, ) ..... Tf( Vk-I»
... ..,(n)«X 0 f)(m), TI( v, ) ....• Tf( v"
_,»
.,,,_,»
=
ix..,(n )·(Tf( Vi) ... ·• Tf(
=
rix..,(m)·(." , .. ·,1'''_1) •
The next proposition expresses d in terms of the Lie derivative (cf. . Palais [1954]). 8.4.11 PropoalUon. we have
Let Xi e 'X(M).;,. O..... k. and.., e D"(M).Then
(i) (Lx •..,)(XI.· .. ,X,.) = Ld..,(XI.···,X,,»-
L" ..,(X" ...• LX.X, ... ~.Xk)
1-'
and Ie
(ii) d..,(Xo• XI •...• XIc )
...
L (-I)iLx,(..,(Xo ..... X, ..... XIc)) 1-0
E
+
(-I)i+j..,(Lx.(Xj ). X{) ..... Xi .... ~XJ .... 'Xk)
O,,;<j""
where
X, denotes that
X, is deleted.
Proof. Part (i) is exactly condition (001) in 5.3.1. For (ii) we proceed by induction. For k "'" O. it is merely d..,(Xo) = Lx."" Assume the formula for k - I. Then if .., e DIe ( M). we have, by 6.4.8(iv)
d..,(Xo• XI'· ..• X Ic )
""
(ix.d..,)(XI ... ·.XIc )
=
(Lx.'" )(X, ..... XIc ) - (d(ix."'))( X, .... ,XIc )
=
LX.(..,(XI.···.Xh »
- E" ..,(XI, .. ·,Lx.X, .... 'Xh ) I-I
-(dixo"')(XI .... ,Xh )
(by (;».
But ixo..,eD"-I(M) and we may apply the induction assumption. This
366
DIFFERENTIAL FORMS
gives. after a simple permutation k
(d(;XoCo)))(X, •...• X k ) =
L
'-I
(-I)'-'Lx/(Co)(Xo • X, •...• X, ..... X A »)
Substituting this into the foregoing yields the result.
•
Note that (i) and the next corollary hold as well for infinite-dimensional manifolds.
6.4.12
Corollary.
LetX.YE~-X(M).
[Lx.iy]=i[x.y[ and
Then [Lx.Ly]=L[x.y[
In particular. ix 0 Lx = Lx 0 i x ' Proof. It is sufficient to check the first formula on any k-form Co) E nk(U) and any X, •...• X k _, E ~ (U) for any open set U of M. We have by 6.4.II(i)
(iyLxCo)( X, ..... X k
_,)
=
(LxCo)( Y. X, ..... Xk _,)
=Lx(Co)(Y.X, •...• X k
_,»-
k - I
L
Co)(Y.X, ..... [X.X,] ..... X k _,)
I-I
k - ,
=
Lx «iyCo)( X, ..... Xk
,»- L (iyCo)(X, ..... (X. X,] ..... X '-I
A _ ,)
- (i,x.y,)( XI" ",Xk _,) =
(LXiyCo)( X, ..... X k-,)- (i,x.YlCo) }(X, ... "Xk _,)
one similarly proves (Lx. Ly] = L[x.y)' •
6.4A THE EXTERIOR DERIVATIVE ON INFINITE-DIMENSIONAL MANIFOLDS Now we are ready to discuss the eltterior derivative on infinite-dimanifolds. Tht\~rem 6.4.1 i~ rather awkward. primarily . ~'au!'(' we cann(\t. with,")ut a I(\t l,f techni~·alitie~. palOS fr(lm. for
men5-i\~nal
EXTERIOR QERIVA TlVE. INTERIOR PRODUCT. LIE DERIVA TIVE
367
example. one-forms to two-forms by linear comhinations of decomposable two-forms. i.e.• two-forms of the type I I " p. However. there is a simpler alternative available. , Adopt formula 6.4.ll(ii) as the definition of d nn any open subset of M. Note that at first it is defined as a multilinear function on vector fields and note that Lx is already defined. 2. In charts. 6.4.II(ii) reduCCI to the local formula 6.4.2. This or a direct computation shows tbat d: D"(M)-D"+'(M) is well defined. depending only on the point values of the vector fields. (cr. 5.2.19). 3. One checks the basic properties of d. This can be done in two ways: directly. using the local formula. or using the definition and the following lemma. easily deducible from the Hahn- Banach theorem: II a k-Iorm II) is zero on any set X, •...• X" E IX(U) lor all open sets U in M. then II) == O. This second method is slightly faster if one first proves the key formula I.
(2)
Proof 0/ Form,,1II (2). Let
II be a k-form and X, •...• X" a set ot k vector fields defined on some open subset of M. Writing Xo = X.
k
- I: (-I)'Lx,(II(Xo •...• X, ..... X,,))
'-0
~ L..
+
(
. . ) -I )' + j II ( Lx,X,. XO ..... X,., ..• X, ....• X4
o<;<}<" 'k
+
I: (-1)'- 'Lx,(II(Xo. X, ..... X, .... ;X4 )
'-I.
I:
(-1);+ JII(Lx,Xj • XO' X, ..... X; ..... Xj ..... X,.)
I <;<j<" k
... Lxo(II(XI ..... X,,»+
I: (-I)JII(LxoXj' X, ..... Xj ..... X,,) i-I
~
(Lxll)(X, ..... X,,)
by 6.4.II(i).
368
DIFFERENTIAL FORMS
This and 6.4.12 will allow us to give a proof of the infinite-dimensional version of 6.4.6:
Lxod-=daL x · F:or functions / this formula is proved as follows. By 6.4.II(i).
(Lxdf)(Y) - Lx(df( Y»-df([X, Y)) - X[Y[J]]-[X, Y][fJ - Y[X[J]] -d(X[J))(Y) = (dLxf)(Y). Inductively, assume the formula holds for (k -I)-forms. Then for any k-form a and any vector field Y defined on an open subset of M, dLxil'G - LXdiyG. Thus by 6.4.12,
Henced 0 Lx = Lx ad. Next. the remaining properties of d are checked in the following way. R-linearity and 6.4.I(iv) are immediate consequences of the definition. For 6.4.l(ii), note that df(X) = ixdf = Lx / - dix / = Lx / - XI J] = df(X). To show that d 2 = 0, first observe that
so that for any k-form
G
and any vector fields XI'" .• X~ n. we have
=ix
'~2
ddix A + I (a(XI.···,X,»=O. 1\
The antiderivation property of d is proved along the same lines using induction. Finally, the formula 1* a d = d a 1* for a diffeomorphism f
, EXTERIOR DERIVA TlVE. INTERIOR PRODUCT. LIE DERIVA TlVE
369
follows by definition and the properties !*(L.\·fP) = l.rx!*fP and !*[X, Y)= (f*X,!*Y). Thus, with the preceding procedure. d is derined on Banach manifolds and satisfies all the key properties that it does in the finite-dim~sional case. These key properties are summarized in Table 6.4.1 below. Next we discuss the important concepts of closed and exact forms and , the Poincare lemma. This lemma is a generalization and unification of two well-known facts in vector calculus: l. If curl F= O. then locally F= vf. 2.. If div F = 0, then locally F = curl G.
8.4.13 Definition. We call Co) E (lA( M) closed if dCo) = O. and exact if there is an a E '(lk '( M) such that Co) = da. . 8.4.14 Theorem. (i) Every exact form is closed. (ii) (Poincan lemma) If Co) is closed. then for each neighborhood U of m for ...;' ... 'I U E (l4(U) is exact.
mE
M. there is a
Proof. Part (i) is clear since dod = O. Using a local chart it is sufficient to consider the case Co) E (lk( U), where U c E is a disk about 0 E E, to prove (ii). On U we construct an R-linear mapping H: (l~ (U ) -, (lk - '( U) such that doH + Hod is the identity on (llc( U). This will give the result, for dCo) .. 0 implies d( HCo) = Co). For e, .... ,ek E E, define
HCo)(u)(e" ... ,ele_') = 1't le -'Co)(tu)(u,e" ... ,ek ,) dl. o
Then, by 6.4.2
dHCo)(u)·(e, ... ,ek) =
E" (-Ir + 'DHCo)(u)·e,(e"
.... e,. .... ek)
;-1
E (_I)i+' f tk-'Co)(tu)(e;.e, ..... e., .... ek)dt. k
=
I
;-1
k
,
+E ( -Ir + '1't k DCo)( tu)·e,( u. e, ..... ii"" ,e.) dt. ,_I
0
370
DIFFERENTIAL FORMS
(The interchange of D and / is permissible. as w is smooth and bounded as a function of I e (0.1).) However. we also have. by 6.4.2 Hdw( u)· (e l , .. . ,e,,) -
foi,,, dW(/u)( u,e l , ... ,e,,) dl
=
foi,,, Dw( lu )·u( e l , .. .• e,,) dl It
+
L
(-I)i1'/" Dw(/u)·ei(u.el.· ..• ii' ... ,ed dt.
i-I
0
Hence
=
fol ~ [tkw(tu).(el •.... e,,)] dt
=w(u)·(el .. ··,ed. which proves the assertion. • There is another proof of the Poincare lemma that is useful to understand; It will help the reader master the proof of Darboux' theorem in Section 8.1 and is similar in spirit to the proof of Frobenius' tbeorem. 4.4.3. Ahel'Jllltive Proof of 1M PO;"C4n umlftll. Again let U be a ball about 0 in E. Let, for t > 0, F,(u) -Iu. Thus F, is a diffeomorphism and, starting at t -I, is generated by the time-dependent vector field X,(u) = ult;
that is. FI(u)- u and dF,(u)/dt = X,(F,(u». Therefore. since w is closed,
=
F*(di , x, w)
EXTERIOR DERIVATIVE, INTERIOR PRODUCT, LIE DERIVA TlVE
For 0 < to
oe;;
371
I, we get CAl- P,'CAl =
"
Letting to -- 0, we get
CAl =
d
.. f 'P,"xCAldt. 10
tIP. where
Explicitly,
(Note that
thi~
formula for
Pagrees with that in the previous proof.) •
The proofs of (ii) show in fact the following: if CAl E nA( U) is closed and U is contractible. then CAl is exact. See Exercise 6.41 for a relative Poincare lemma. It is not true that closed forms are always exact (fpr example. on a sphere). In fact, the quotient groups of closed forms by exact forms (called the deRham cohomology [{roups of M) are important algebraic objects attached to a manifold. A discussion may be found in Section 7.5. Table 6.4.1 summarizes some of the important algehraic identities involving vector fields and differential forms that have been Ilbtained. Table 6.4.1 I.
Vector fields on M with the bracket IX. VI form a Lie a1,ebra; that is. IX. VI is n:a1 bilinear. skew symmetric. and Jacobi'~ identity holds: [[X. Y).ZI +[[Z.X). Yl+ [(V.z).Xl- O.
Locally. [X.Yl- DY·X - DX·Y
2. 3. 4.
and on runctions. IX. YHJI = XI YI/II- YIXI/II· For dirreomorphisms ",.,y. ",.1 x. Y 1- I '" ~X. CPo VI and (cp 0,y). X = ",.1Jt. x. The forms on a manifold are a real associative algebra with 1\ as ~ultiplicalion. Furlhermore. u 1\ p - ( - I)"P 1\ U for k and '·forms u and p. respectively. For maps CP.,y. cpO( U 1\ p) - cp'u 1\ ",·P.(", 0,y )·u - .,....",.111.
372
DIFFERENTIAL FORMS
Table 6.4.1 Continued S.
tI ia a real linear map on forma and tI tI. - O. tI(. 1\
IS) -
tlo 1\ IS + ( - I)' a
1\
dII
for II a Ie·form
6 . . For II a Ie·form and Xo." .• X. vector fields: k
dll(Xo."·.X.)-
E (-I)'X/(II(Xo..... X.. " .. X.)) /-0
+
E (-I)" 'II([X,.X,).Xo." .• X, ..... X, .. ".X.) ;<j
Locally. k
tI", ( x )( 1\) ..... "k )
7. II.
9.
-
E (- I) / Dw ( x ). II, ( 1\) ..... ti,. " .• Ok)'
.-0
,,0
For a map ". da - d"oll. (Poincare lemma) U dll - O. then a is locally exact; that is. there is a neighborhood U about each point on which a - dIS. iXIl is real bilinear in X. II and ror h: M -- R. iua- hixll- ixha. Also ixixa = O. a~d
··60. For a diffeomorphism ". ,,°;1/(11- ;._1/(,,°11. II. Lxa-dix«+;xtl•. 12. Lx. is real bilinear in X. II and LX(II II IS> - LXII II II + II II LxlS. 13. For a diffeomorphism "."oLxa- L.-X"oll. 14. (LXIl)(X, •...• X,) - X(II(X, •... •X. »-t~_,I1( X, .... •IX. X/I.... •X.). Locally. k
(LXIl).·( 0, ..... Ok) -
Da.· X( II(). (""" ""k)+ Ea.· (""'''' DX.· 0/ ..... 0.) .. i-I
I S.
The following identities hold:
~XTERIOR
DERIVATIVE. INTERIOR PRODUCT. LIE DERIVATIVE
373
For vec:tor-valuJ k-forms one proceeds in a similar manner, adopting Palais' formula, 6.4.1 I(ii), as the definition on an open subset of M. Note again that this definition uses the fact that Lx is defined for vector-valued tenson, and apin one hal to prove that the local formula 6.4.2 holds. Then all properties in Table 6.4.1 are verified in the same manner as previously. For vector-valued forms we have however an additional formula on Ok(M; F)
·for any A E L(F, F). If F is finite dimensional, the definition and properties of d become quite obvious; one notices that if n
"''''' I: "'j"EOk(M;F),
where "'jEOk(M).
j-I
then, d", is given by n
d", =
I: d"'j"
j-I
and this formula can be taken as the definition of d in this case.
• OX I... DIFFERENTIAL IDEALS AND PFAFF/AN SYSmlttS
This box discusses a reformulation of the Frobenius theorem in terms of differential ideals in the spirit of E. Cartan. Recall that a subbundle E c TM is called involutive if for all pairs (X, Y) of local sections of E defined on some open subset of M, the brackef[X, Y] is also a local section of E. The subbundle E is called integrable if at every point m E M there is a local submanifold N of M such that T",N ... E",. Frobenius' theorem states that E is integrable iff it is involutive (see Section 4.4). J)efore starting the general theory, which is fairly heavy on algebraic formalism. let us illustrate by a simple example how forms and involutive subbundles are interconnected. Let (oj E 02( M) and assume that E.. = {vE TMli,,"'''' O} is a subbundle of TM. H X and, Y tlbe method described does not work for vector-bundle-valued forms. Additional '>cable to lift Lx. structure on the bundle is require'!
374
DIFFERENTIAL FORMS
are twO'sections of E... then
Thus, E.. is involutive iff ixiydll) = O. In particular, if II) is closed, then E.. is involutive. In this box we shall want to express conditions such as ixiydll) - 0 in terms of one forms for subbundles E of TM that are not necessarily defined by exterior forms. For any subbundle E, the k-annihilator of E defined by
is a subbundle of the bundle N( M) of k-forms. Denote by f( V. E) the cae sections of E over the open set V of M and notice that . oc
g(E)~
E9 f(M.
EO(k»)
k-O
is an ideal of OeM); i.e., if 11)1.11)2 Eg(E) and pEO(M), then + 11)2 E geE) and p 1\ WI E geE).
WI
6.4.15 Propoaltlon. The subbundle E of TM is involutive iff for all
n
n
open subsets V of M and 01111) E V, EO( I», we have dw E V, EO(2». If E is involutive, then II) E f(V. EO(k» implies dw E feU, EO(k + I). Proof.
For any
0 E
nU. E°(l»
do( X. Y) = X( o( Y
and X. Y E f( u. E). 6.4.11(ii) yields
»- Y( o( X»- 0(1 x. Y)
Thus E is involutive iff do( X. Y) = 0; i.e .• do E
= -
0(1 x. Y).
n U. E()(2».
•
The Frobenius theorem in terms of differential forms takes the following form.
6.4.16 Corollary. The subbundle E c TM is integrable iff for all ope" subsets U of M.w E
nu. E°(l» implies dWE feU. EO(2».
EXTERIOR DERIVATIVE, INTERIOR PRODUCT, LIE DERIVATIVE
375
The following considerations are strictly finite dimensional. They can be generalized to infinite-dimensional manifolds under suitable splitting assumptions. We restrict ourselves to the finite-dimensional situation due to their importance in applications and for simplicity of presentation.
6.4.17 DeflnlUon. Let M be an n-manifold and '\ C 0.( M) be an ideal. We say that 9 is locoIIy generuted by II - k ilukpelldellt one-forms. if every point of M has a neighborhood V and n - k pointwise llnearly independent one-form.r I&)I •...• I&),,_~ E gl(V) such that: n-k
(i)
E
If I&) e ~. then 1&)1 V ,
8/1\ 1&), lor
some 8, e 0.( M);
;-1
(ii) If I&) e 0.( M) and M is covered hy open sets V,fuch that for each V in this cover. n-k I&)IV=
L
,-1
8;1\1&);
for some 8; Eg(M).
then I&) E 'l. The ideal
9C
0.( M) is called a differelltial ideal if d'i c ,\.
Finitely generated ideals of 0.( M) are charal'lcrizcd hy heing of the form 1( E). More precisely. we have the following. Let!t be an ideal ofg( M) and let n = dime M). The ideal 9 is locally generated by n - k linearly independent one-forms iff there exists a subbundle E c TM with k-dimensional fiber such that g - geE). Moreover. the bundle E is uniquely determined by g.
6.4.18 Proposition.
If E has k-dimensional fiber. let X,,-H I ..... X" be a local basis of f( V. E). Complete this to a basis of ex (V) and let 1&); E 0 1(U) be the, dual basis. Then clearly I&) I'" .• 1&)" _ k are linearly independent and locally generate 1( E). Conversely. let 1&)1 ..... 1&)". k generate
. Proof.
Differential ideals are characterized among finitely generated ones by the following.
376
DIFFERENTIAL FORMS
'.4.11 Propo8ItIon. Let g be an ideal 01 O( M) locally generated by n - k linearly independent lorms WI" •• ,10',,_ I< e Ol( V), n - dim( M), and let WI" • " " 10',,_1< - 10' e O,,-I«V). Then the loIlowing are equivalent: (i) g is a differential ideal. (ii) 1110', =- Ej:fwlj " wjlor some wI} e OI(V) andlor every Vas in the hypothesis. (iii) 1110';" 10' - 0 lor all open sets V, as in the hypothesis. (iv) There exists 8 e Ol( V) such that 1110' = 8 " 10' lor all open .fet.' V, as in the hypothesis.
That conditions (i) and (ii) are equivalent and (i) implies (iv) follows from the definitions. Condition (iv) means that
Proof.
,,-I<
E (-1)'1110'," WI" ... "WI" ... " 10',,_1< =8" WI" ••• "10',,-1<,
I-I
so that mUltiplying by WI we get (iii). Finally, we show that (iii) implies (ii). Let 10'1'''''10'" e OI
1110'1 =-
E alj,wj "
w,, where aljl e lff( V).
j<1
But
0-1110'1 " 10' -
E
,,-I«j<1
alJlw} "
w, " WI "
•••
"w,,-I<
and thus aljl - 0 for n - k < j < I. Hence
1110'1-
"tl< (- l-j+1 t aliIW/) "
wi'
•
j-I
Assembling the preceding results, we get the following version of the Frobenius theorem. 6.4.20 Theorem. Let M be an n-manilold and E c I'M be a .mhlmndie with k-dimensionalliber. and ~( E) the assoc:iatecl i,J,,(/I. The fol/(}Wing are equivalent. (i)
E is integrable.
(ii) E is invaluti~. (iii) !t( E) is a differential ideal locally generated by n - k linearly independent one-Iorms 10'1, .•• ,10',,_1< e OI(U).
EXTERIOR DERIVATIVE, INTERIOR PRODUCT, l.IE DERIVATIVE
For every point 0/ M there exists an open set V and CA)I" ••• CA)".
377 k E
DI(V) generating ~(E) such that II-Ie
(iv)
dCA);'"
1: CA);j /\ CA)j
for some CA)/j E D'( V).
j-I
(v)
(vi)
dCA);/\CA)I/\ ···/\CA),,_k=O.
There exists 8 E D'(V) such that dCA) = CA)I/\ ••• /\ CA)"-k'
8/\ CA). where, CA) =
6.4.21 Examples A. In classical texts (such as Cartan [1945] and Flanders (1963». a system of equations CA)I =O ..... CA),,_k =0.
where CA);EDI(V)
and
VcR"
is called a Plat/ian system. A solution of this system is a k-dimensional submanifold N of V given by x' = x ' ( u l ... .• u k ) such that if one plugs in these values of x' in the system. the result is identically zero. Geometrically. this means that CA)1 ..... CA)"_k annihilate TN. Thus. finding solutions of the Pfaffian system reverts to finding integral manifolds of the subbundle E=(vETVICA)/(v)-O.i=I •...• n-k} for which Frobenius' theorem b
CA); =
1:
a;jdbj"
j-I
To see this. recall that by the Frobenius theorem thae are local coordinates b l ... .• b" on V such that the integral manifolds of E are given by b, == constant .... •b,,-Ic - constant. so that db;. i == I ... .• n - k annihilate 'the tangent spaces to these submanifolds. Thus the ideal g generated by CA)I ..... CA)"_ Ie is also generated by db l •·· .• db". k; i.e.. CA), == E7':."a;jdbj for some smooth functions a;j on U. B. Let us analyze in detail, the case of one Pfaffian equation in R2. Let CA)=P(x.y)dx+Q(x.y)dyED 1(R 2 ) usint-! standard (x.y) coordinates. We seek a solution to CA) = O. This is e4uivalent to dy/dx = - P(x. y)!Q(x. y). so existence and uniqueness of solutions for. ordinary differential equations assures the local existence of a function /(x.y) such that/(x,y)-constant give the integral curvesy(x). In other words /(x. y)'" constant is an integral manifold of CA) - O. The
378
DIFFERENTIAL FORMS
same result could have been obtained by means of the Frobenius theorem. Since dll) A II) E 03(R 2). we get dll) A II) ... O. so integral mani- , folds exist and are unique. In texts on differential equations, this problem is often solved with the aid of integrating factors. More precisely, if II) is not (locally) exact, can a function f and a function g, called an integrating factor. be found, such that II) = g df? The answer is yes, and g is found by solving the partial differential equation,
a(gP)
a(gQ)
ay=~
This always has a solution and the connection between g and f is given by I af I af g=--=-pax Q ay'
f(x, y) - constant being the solution of 11)'" O. C. Let us analyze a PCaman equation II) - 0 in Alt. As before. we would like to be able to write II) == gdfwith df., 0 on U c A". for then f(x' ..... x") - constant gives the (n - I)-dimensional integral manifolds; i.e., the bundle defined by II) is integrable. Conversely. if the bundle defined by II) is integrable. then by A, II) = g df. Now integrability is (by the Frobenius theorem) equivalent to dll) A II) = 0, which. as we have seen in B, is always verified for n = 2. For n ;;. 3, however. this isa genuine condition. For n = 3. let II) = P(x. y. z) dx + Q(x. y. z) ~y + R(x,y,z)dz. Then
dIl)AII)=[p(aR _ aQ)+Q(ap _ aR)+R(a Q _ ap)]dXAdYAdZ; ay az az ax ax ay so II) -= 0 is integrable iff the term in the square brackets vanishes. D. The Frobenius theorem is often used in overdetermined systems of partial differential equations to answer the question of existence and uniqueness of solutions. Consider for instance the following system of A. Mayer [1872] in Rp+q = {(Xl •.. .• xp. yl ... .• yq)}:
dy"
-d' =Ao,(xl •. ".Xp.yl., ..• yq).
i=l ..... p.
a=I. .... q .
.~
We ask whether there is a solution y = f(x. c) for any choice of initial conditions c such that f(O. c) - c. The system is equivalent to the
EXTERIOR DERIVATIVE, INTERIOR PRODUCT, LIE DERIVA TlVE
379
following Pfaffian system:
SinCe the existence of a solution is equivalent to the existence of p-
where C
a
" )
k
a
oA"j aA k aA j II aA"k II =-----. +--A k - - - A . axk ax) a yll ayll)
Since dx' •...• dxp."" •...• "'q are a basis of D'(RP'q). we see that d"," - El_,,,,a ll /\ ",fJ for some· one-forms ",all iff C"" = O. Thus the Mayer system is integrable iff calk = O. ... In Section 8.4 we shall give some applications of Frohenius' theorem to problems in thermodynamics. constraints and control theory. Many of these applications may alternatively he understood in terms of Pfaffian systems; see for example. Hermann r1977. Ch. 18).
Exercl188 6.4A
Compute the exterior derivative of tlte following differential forms on Rl:
p= 6.4B
3e JC dx /\ dy + 8cos(xy) dx /\ dy /\ dz.
Using 6.4.3 and the properties of d and •• prove the following formulas in Rl: (i) grad(fg) = (gradf)g + f(grad g) (ii) curl(fF) = (grad f)x F + f(curl F) . (iii) div(fF) = grad(fF) + fdivF (iv) div(F x G) = G·curl F - f·curl G (v) LTG = (F'V)G -(G'V)F= FdivG - GdivF-curl(F x G) , wheref. g: Rl-+R and F.G E 'X, (R 3 ).
380
DIFFERENTIAL FORMS
Let ,,: S' xR+ .... R2 be defined by ,,(0, r)'" (rcosO, rsinll). Compute "·(dx,, dy) from the definitions and verify that it equals d("·x),, d(,,·y). 6.40 On S' find a closed one-form CI that is not exact. (Hint: On R 2 \{0) consider CI- (ydx + xdy)/(x 2 + y2)'/2.) 6.4E Show that the following properties uniquely characterize ; x on finite-dimensional manifolds (i) ix: O~(M) - O~ - '( M) is a " antiderivation. (ii) i x ! = O;! E Cff( M); (iii) i x ., - .,(X) for., E 0'( M); (iv) ix is natural with respect to restrictions; Use these properties to show i,x.y,-Lxiy-iyL x ' Finally. show ix oix =0. 6.4F Show that a derivation mapping Ok ( M) to 0" + '( M) for all k ~ 0 is zero (note that d and ix are antiderivations). 6.4 Let $: T2M - T2M be the canonical involution of the second tangent bundle (see exercise 3.40). . (i) If X is a vector field on M, show that s 0 TX is a vector field on TM. (ii) If F, is the now of X. prove that TF, is a now on TM generated bysoTX. (iii) If" is a one-form on M, ji.: TM - R is the corresponding function, and we T2M. then show that 6.4C
»+ dP,( w).
dP,(sw) - d,,( 'rTM( w), T'rM( w
6.4H Prove the following relative Poincare lemma: Let ., be a closed k-form on a manifold M and let N c M be a closed submanifold. Assume that the pull-back of w to N is zero. Then there is a (k -I)-form CI on a neighborhood of N such that dCl= W Ilnd CI vanishes on N. If w vanishes on N. then CI can be chosen so that all its first partial derivatives vanish on N. (Hint: Let ", be a homotopy of a neighborhood of N to N and construct an H operator as in the Poincare lemma using ",.) 6.41 (Angular variables). Let SI denote the circle. SI "" R/(2'IT) "" {z E Cllzl"'I}. Let y: R -SI: x ...... eix, be the exponential map. Sho~ that y induces an isomorphism TS ' "" SI XR. Let M be a manifold and let w be an "angular variable." thaI i~ a smooth map w: M - SI. Define dw, a one-form on M by taking the $; •ljection of Tw. Show that (i) if w: M - Sit then d 2w .... 0; and (il' ,; I: M - N is smooth. thenr(dfA»-d(rfA». whererfA) = fA) 0 f.
ORIENTATION. VOLUME ELEMENTS. AND CODIFFERENTIAL
6.4J
381
Prove the identity
LXiy - LyiX - ilx .y)'" [d, ix 0 iy 1· 6.4K
(i)
x
Let X - (Xl. 2, 0) be a vector field defined on the plane S - {(x. y,O)lx. y E R) in R3. Show that there exists Y E <x'(R l ) such that X=curlY on S. (Hint: Let Y(x.y.z)= (zX 2(x. y). - ZXI(X. y).O).)
Let S be a closed surface in R J and X E 'X(S). Show that there exists Y E (R 3) such that X == curl Y on S. (Hint: By 5.5.9 extend X to XE 'X(R 3 ) and put Co) = • Xb. Locally find 01 such that dOl- Co) hy (i). Use a partition of unity (CJ>;) to write Co) = ECJ>;Co) and let dOl; = CJ>;Co), 01 = EOl;.) (iii) Generalize this to forms on a closed submanifold of a manifold admitting C·-partitions of unity. (i) Let M be a manifold and let i,: M .... [0.1 JX M he the mapping i,(m)-(t,m). DefineH: (}k+I([O.I]XM) .... UA(M) by (ii)
6.4L
ex
(HOl)(XI.· .. 'Xk ) =
(ii)
fol(i~;i1la'CJ)( XI'···'Xk) dl.
Show that H is wen-defined and that doH + Hod = it - i~. (Hint: Use formula 6 in Table 6.4.1.) Two smooth mappings /. g: M .... N are called C-homolopic, r ~ 0, if there exists'a C mapping F: I X M .... N such that F(O, .) "" / and F(1,') == g. Let OlE (}k(N) he closed. Show that if / and g are C'-homotopic. then g·Ol- f·Ol E Ok( M) is exact. (Hint: Show that G - HoP with H as defined in (i) satisfies doG -G od- g.-
r.)
(iii)
Oeduce from (ii) that if M is contractible (see 1.6.13)· then
every closed form is exact. (Hint: use (ii) and the fact that the identity mapping is homotopic to the map sending all of M to a point.) , (iv) Use (iii) to give another proof of the Poincare lemma. the ideal generated by Co)I = x 2 dx I + Xl dx 4 , Co)2 = 6.4M Show that on x 3 dx 2 + x 2 dx l is a dirferential ideal. Find its integral manifolds.
R".
1.5 ORIENTATION, VOLUME ELEMENTS, AND THE CODIFFERENTIAL This section globatizes the setting of Section 6.2 from linear spaces to manifolds. All manifolds in this section are finite dimensional. t t For infUlite-dimensional analogues of orientability, see for instance, E1worthy and Tromba [197Obj.
382
DIFFERENTIAL FORMS
1.5.1 DefInition. A voIIuIw form on an n-manifold M ;s an n-form 1& E 0"( M) such that 1&( m) .. 0 for all m E M; M is called if there exists some volume form on M. The pair (M.I&) is called a vohune lIIIIIIi/o/iJ.
tJIV",,,,,
Thus. 1& assigns an orientation. as defined in 6.2.5. to each fiber of TM. For example. R' has the standard volume form 1& =- dx " dy" dz. 1.5.2 (i)
(ii)
P~ltIon.
Let M be a connected n-manifold. Then
Mis orientabie iff there is an element 1& E O"(M) such that every other 1&' E 0"( M) may be written 1&' == fl&for f E ~(M). M is orientable iff M has an atlas {( If;. 'Pi)}' where 'Pi: If; ..... If;' C R". such that the Jacobian determinant of the overlap maps is positive (the Jacobian determinant being the determinant of the derivative. a linear map from R" into R").
Proof. For (i) assume first that M is orientable. with a volume form 1&. Let 1&' be any other element of O"(M). Now each fiber of A"(M) is one-dimen-
sional. so we may define a map f: M ..... R by
We must show thatf E '!f(M). In local representation. we can write
and
I&(m)"'"(m)dx i ,,,
. . . ,,
dx i •
But ,,(m) .. 0 for all me M. Hence f(m)'" ,,'(m)/,,(m) is of dass Coco Conversely. if 0"( M) is generated by 1&. then 1'( m) .. 0 for all m E M since each fiber is one-dimensional. To prove (ii), let {( If;. 'P,)} be an atlas with 'Pi ( If;) -If;' C R". Also. we may assume that all If;' are connected by taking restrictions if necessary.. Now 'Pi. 1& = /; dx I " ••• " dx" = /;1&0' where 1'0 is the standard volume form on RIO. By means of a reflection if necessary. we may assume that /;( u') > 0 (/; .. 0 since I' is a volume form). However. a continuous real-val- ' ued function on a connected space that is not zero is always > 0 or always < O. Hence. for overlap maps we have
/;.
----".!...,-~I dx
h
0
Ifj 0 Ifj-
I
" ... " dx".
ORIENTA TlON, VOLUME ELEMENTS, AND CODIFFERENTIAL
383
But. 1/J*(u)(ul
"
...
"u") = D1/J(u)*'u l
"
D1/J(u)*'u 2 "
•••
"D1/J(u)*·u" •
. where D1/J(u)··u'(e)-u l (D1/J(u)·e). Hence. by definition of determinant we have
We leave as /in exercise for the reader the fact that the canonical isomorphism L(E; E) '"' L(E*; E*). used before does not affect determinants. For the converse of (ii). let {( V".1/J a)} be an atlas with the given property. and {(lI,. (JJj' gj)} a subordinate partition of unity. Let
and let _(
lA, m
)={g;(m)IA,(m) 0
Since supp g/' c lI,. we have
if m E l~ if m!ll~
ii; E 0"( M). Then let
. Since this sum is finite in some neighborhood of each point. it is clear from local representatives that Il E 0"( M). Finally. as the overlap maps have positive Jacobian determinant. then on U, () ~.Ilj * 0 and so
Since Ljg/ = 1. it follows that lA(m)* 0 for each mE M . • Thus. if (M.IA) is a volume manifold we get a map from 0"( M) to namely. for each 1" E 0"( M). there is a unique IE '!f(M) such that
~t( M);
lA' ,.. Ill·
.
6.5.3 DeftnlUon. Let M be an orientable manilold. Two tlo/ume lorms IAI and 1'2 on M are called equioalent il there is an I e ':t ( M) with I( m) > 0 lor
3tU
DIFFERENTIAL FORMS
1111 me M such that ". - f"2' (Thu is clearly an equivalence relation.) An __tIItitM of M is an equivalence class [Ii] of volume forms on M. An oriatal _i/oltl. (M.["D. is an orientable manifold M together with an orientation [,,] on M. If [,,] is an orientation of M. then [- ,,]. (which is clearly another orientation) is clllied the rewru oriallltitM.
The next proposition tells us when [,,] and [- ,,) are the only two orientations.
1.5.4 PropoalUon. Let M be an orientable manifold. Then M is connected iff M has exactly two orientations.
Pro9/. Suppose M is connected. and
Ii. Ii' are two volume forms with ,,' ... f". Since M is connected. and f(m) ... 0 for all m E M'/(m) > 0 for all m or elsef(m) < 0 for all m. Thus Ii' is equivalent to" or - Ii. Conversely, if M is not connected. let V ... '" or M be a subset that is both open and closed. If" is a volume form on M. define Ii' by
'(m) ... { "
,,(m) - ,,(m)
mEV mEV.
Obviously. Ii' is a volume form on M, and Ii' E [,,)U[ - ,,]. • A simple example of a nonorientable manifold is the MObius band (see Fig. 6.5.1 and Example 3.3.8C.
Flgu....5.1
ORIENTATION, VOLUME ELEMENTS, AND CODIFFERENTIAL
385
'.5.5 PropoeItIon. The equivalence relation in 6.5.3 is natural with respect to mappings and dif!eomorphisms. That is, ill: M -0 N is 01 class Coo. ,. Nand "N equivalent volume lorms on N. and IV ) is a volume form on ' M. thenr("N) is an equivalent volume form. III is a dil/eomorphism and "'M and,.:., are equivalent volume lorms on M. then 1.(",,,,) and 1.( ... :.,) are equivalent volume lorms on N.
r(,.
are.
Proof. This follows from the fac, that r(gw)== (gof)rw. which implies
1.(gw)=(gorl)/.w
when I is a diffeomorphism. •
8.5.' D......ltIon. Let M be an orientable manilold with orientation ItA). A chart (U.",) with ",(U) - U' e R" is called positiwly orimled if ",.( ... 1U) i.f equivalent to the standard volume lorm
From 6.5.5 we see that this defmition does not depend on the choice' of the representative from [tA). . If M is orientable. we can find an atlas in which every chart has positive orientation by choosing an atlas of conneCted charts and. if a chart has negative orientation. by composing it with a reDection. Thus. in 6.S.2(ii), the atlas consists of positively oriented charts. If M is not orientable. there is an orientable manifold M and a 2-to-l COD suljective local diffeomorphism w: M- M. The manifold if is called the orientable double covering and is useful for reducing certain facts to the orientable case. The double covering is constructed as follows. Let M = . {(m.I,.",)lm e M,l,.",] an orientation of T",M). Define a chart a~ (m,l,.",» in the following way. If,,: U-U'eR" is a chart of M at m, then 1".(,.",)] is an orientation of R". so that setting
D- {(u,[,..))lueU,[",.(,..)] -
c;: D.... u'.
[",.(,.",)]}.
C;(u.[,..]) = ",(u).
we get a chart (D. c;) of M. It is straightforward to check that the family
{( D, C;)}
constructed in this way forms an atlas. thus' making if into a differentiable n-manifold. Define w: if .... M by w( m, I ...... ]) ... m. I n local
386
DIFFERENTIAL FORMS
charts fT is the identity mapping. so that fT is a surjective local diffeomorphism. Moreover fT-'(m)={(m.["",J),(m. -["",))). so that fT is a twofold covering of M. Finally. M is orientable. a natural orientation on M is given on the tangent space 1( .... (.. ~J)M by I(T...fT)-"",].
'.5.7 Propoamon. Let M be a connected n-manifold. Then M i.' connected iff M is nonorientable. In fact. M is orientable iff M consists of two disjoint copies of M. one with the given orientalion. the other with the reverse orientation. Proof. The if part of the second statement is a reformulation of 6.5.4 and it also proves that if M is connected, then Mis nonorientable. Conversely if M is a connected manifold and if M is disconnected, let e be a connected component of M. Then since fT is a local diffeomorphism, fT(e) is open in M. We shall prove that fT(C) is closed. Indeed, if mE c1(fT(C».let "',,"'2 E M be such that fT(!",) = fT("'2) = "!: If there exist neighborhooc;!s V" VI' of "'I' "'2 such that V, () C - 0 and V2 () C = 0. then shrinking VI and V 2 if necessary. the open neighborhoods fT( V,) and fT( V2 ) of m have empty intersection with fT( C). contradicting the fact that mE c1( fT( Thus at least one of m ,. m 2 is in c1( C) = e; i.e.• m E fT( C) and hence fT( C) is closed. Since M is connected. fT(C) = M. But fT is a double covering of M so that M can have at most two components. each of them being diffeomorphic to M. Hence M is orientable. the orientation being induced from one of the connected components via ft. •
e».
Another criterion for orientability is the following.
'.5.'
Propoamon. Suppose M is an orientable n-manifold and V is a suhmanifold of codimension k with trivial normal bundle. ThaI is, there are Coo maps N;: V-TM, i-I •...• k such that N;(v)eTv(M), and N;(v) span a subspace w" such that Tv M = T., V $ Wv for all v e V. Then V is orientable. Let" be a volume form on M. Consider the restrict,ion "I V: . III V is a smooth mapping of manifolds. This follows by using charts with the submanifold property. where the local representative is a restriction to a subspace. Now define 110: V -+ A" A( V) as follows. For Proof.
V -+ h"( M). Let us first note that
0
X"oo.,X,,_1t e 'X(V), put
ORIENTATION, VOLUME ELEMENTS, AND CODIFFERENTIAL
387
It is clear that ILo( v) "" 0 for all v. It remains only to show that ILo is smooth, but this follows from the fact that ILl V is smooth. •
, If • is a Riemannian metric. then ,b: TM - T- M denotes the index,lowering operator and we write (,b)- I. For I E ~f( M), grad I - r(tl/) is called the gradient of I. Thus, grad I E ~X (M). In local coordinates, if gij=,(a/ax i; a/axil and gil is the inverse matrix, then
r-
( gradl ) i =
..
al
g'J_..
ax J
(I) ,
6.5.9 Coronary. Suppose M is an orientable manilold. H E ~(M) and c E R is a regular value of H. Then V = H- I( c) is an orientable submanifold of M of codimension one, if it is nonempty. Proof. Suppose c is a regular value of H and H - 1( c) = V .. 0. Then V is a submanifold of codimension one. Let g be a Riemannian metric of M and N = grad( H)IV. Then N( v) E T"V for v E V. because T..V is the kernel of dH( v), and dH( v)[ N( v») = ,( N. N)( lJ) > 0 as dH( I') .. 0 by hypothesis. Then 6.5.8 applies, and so V is orientable. •
Thus if we interpret V as the "energy surface:' we ~ce that it is an oriented submanifold for "almost all" energy values (hy SaHl's theorem; see Appendix E). Let us now examine the effect of volume forms under maps more closely. •
6.5.10 Definition. Let M and N be two orientable n-manifolds with volume , forms I'M and ILN' respectively. Then we call a Coo map I: M - N vobune prese"'i1Ig (with respect to IL M and I' N ) il f*IL N = IL M' and we call f orimtatiOll pNse",in, if f*( I' N ) E [I'M)' and orieNtation reversin, if f*( IL N ) E [ - I' At). From 6.5.5, (f*ILN) depends only on IILN)' Thus the first Pllrt of the definition explicitly depends on ILM and ILN' while the last two parts dClpend only on the orientations IILM J and IILN). Furthermore. we see from 6.5.5 that if f is volume preserving with respect to I'M and ILN' then I is volume preserving with respect to hIL", and IDAN iU h = g 0 I. It is also clear that if f is volume preserving with respect to IL", and ILN' then f is orientation preserving with respect to (I'M) and (IL N).
6.5.11
Proposition. Let M and N be n-manilolds with !'olume lorms IL M and ILN' respectively. Suppose f: M -+ N is 01 class Co<, Then (i) /*( ILN) is a volume form iff f is a local diffeomorphism; that is. for each III E M. there is a
388
DIFFERENTIAL FORMS
neighborhood V of m such that II V: V -> f( V) is a diffeomorphism. (ii) If Mis connected. then f is a local diffeomorphism iff f is orientation preserving or orientation reversing.
r("
Proof. If f is a local diffeomorphism. then clearly N)( m) ., 0, by 6.4.4(ii). Conversely, if r("N) is a volume form, then the determinant of the derivative of the local representative is not zero, and hence the derivative is an isomorphism. The result then follows by the inverse function theorem. (ii) follows at once from (i) and 6.5.4. • Next we consider the global analog of the determinant. 8.5.12 DeIInlUon. Suppose M and N are n-manifolds with volume forms "M and "N' respectively. Iff: M -> N is of class COO. the unique Coo function J(,. ... ,.,."f e 'S(M) such that r"N'" (J(,. ... ,.,.,'/)"M is called the JlICObilm line",.. . ., of f (with respect to and "N)' If f: M -> M, we write J,..J'" )f. .
"M
'<,. ... ,...
The basic properties of determinants that were developed in Section 6.2 also hold in the global case. as follows. First, we have the following consequences of 6.5.11. 8.5.13 PropoalUon. f is a local diffeomorphism iff J(,. ... ,.,.,)f(m)., 0 for allme M.
. Second, we have consequences of the definition and properties of pull-back. 8.5.14 PropoalUon. (i)
Iff: M
->
Let (M. ,,) be a volume manifold.
M and g: M -> M are of class COO, then J,.(/og) = [(J,.f)og][J"g).
(ii) If h: M -> M is the identity. then J"h = I. Iff: M -> M is a diffeomorphism. then
(iii)
J,,(/-I) Proof.
= \/[ (J"f)
0
f
1].
For (i). J,,(/og),,= (/og)*"=g*r,, = g*(/)&fhl. = ((J"f) 0 g )g*" "" ((J,./) 0
g)( I)&g)"
ORIENTA TlON. VOLUME ELEMENTS, AND CODIFFERENTIAL
389
Part (ii) follows since h· is the identity. For (ill) we have
8.5.15 PropoIIUon. Let (M.[I'M)) and (N,[I'N)) he oriented manifolds and f: M -+ N be of class COO. Then I is orientation presen.Jing ill J(,..,.,.",,!( m) > 0 for all me M, and orientation reversing iff -,<,.." .. ",l!( m) < 0 for all me M. Also,/is volume preserving with respect to I'M and I'N if! -'<""'''Nl/'= I. This proposition follows from the definitions. Note that the first two assertions depend only on the orientations [I'MI and ["NI since
which the reader can easily check. Here g E (fj (N), h E ~f (M), g( n) .. 0, and h(m) .. 0 for alI n E N, m E M. We have seen that in R3 the divergence of a vector field is expressible in terms of the standard volume element I' == dx A fly A dz by the use of the metric in R3 (see Example 6.4.3D). There is, however, a second characterization of the divergence that does not require a metric but only a volume form 1', namely
as a simple computation shows. This can now be generalized,
8.5.18 DeflnlUon. Let (M, 1') be a volume manifold; i.e., M is an orientable manifold with a volume lorm 1'. Let X be a vector field on M. The unique function div,.X E '!X.,(M), such that LxI' = (div,.X)" is called the divergenu of X. We say X i~ incompressible (with respect to 1') if div,. X = o. '
8.5.17 Propoaltlon. Let (M, ,,) be a volume manifold and X a vector field onM. . (i)
Iff E (fj(M) andf(m). o for all m EM, then ' .. X = d'IV.. X + Lxf d IV, f'
(ii)
Proof.
For g E (fj(M),div.. gX = gdiv,.X + LxK'
Since 'Lx is a derivation, we have
390
DIFFERENTIAL FORMS
As fp. is a volume form, (div/.. X)(fp.) = (Lx/)p. + f(div"X)I" Then (i) follows. For (ii), we have Lgxp. = gLxp. + dg 1\ ixI'. Now from the antiderivation property of ix, dg 1\ ixp. == - ix(dg 1\ 1')+ ixdg 1\ 1'. But dg 1\ I' E 0"+ '(M), and hence dg 1\ I' = O. Also, ixdg = Lxg and so LgxI' = gLxl' + (Lxg)p.. The result follows at once from this. •
1.5.18 PropoelUon. Let (M, 1') be a volume manifold and X a vector field on M. Then X is incompressible (with respect to 1') iff the flow of X is volume preserving: that is, the local diffeomorphism F,: U .... V is volume preserving with respect to "I U and 1'1 v. Proof. If X is incompressible, LxI''' O. then I' is constant along the flow of X; p.( m) - (F,.,,)( m). Therefore F, is volume preserving. Conversely, if (F,·,,)(m)" ,,(m), then Lxp. == O. •
1.5.19 Corollary. Let (M. 1') be a volume manifold and X a vector field with flow F, on M. Then X is incompressible iff J.. F, = I for all t E R. The considerations regarding the Jacobian and the divergence can also be carried out for one-densities. If II'",I.II'NI are one-densities and f: M .... N is Coo, we write f*1I'NI = 1.I(1 .. ",I.I"1V11flll',,,I, where the pull hack is defined as for forms. Then 6.4.13 and 6.5.14 go though for one-densities. The Lie derivative of a one-density is defined by Lxl,,1 = ; l,_oP,II'1 and one defines the divergence of X with respect to 1,,1 as in 6.5.16. Then it is easy to check that 6.5.17-6.5.19 have analogues for one-densities. We shall now globalize the concepts pertaining to Riemannian volume forms and densities, as well as the Hodge star operator discussed in Section 6.2. Let (M. R) he a pseudo-Riemannian mani/old 0/, Signature s; i.e .• R( m) has .figllature s for all mE M. (i) If M is orientable. ,hen there exists a unique Imlume eleme", ,,- ,,(g) on M, called tire g-volume (or pseudo-Riemannian volume 0/ g), such that I' equals one on all positively oriented orthonormal bases on the tangent spaces to M. If Xl .... •X" is such a basis in an open set U of M and if (I ..... (" is the dUllI basis. then" ... (I 1\ .•• 1\ (". More generally, if ", ..... "" E T"M are positiwly oriented, then ,,(x)( "I .... ,c,.) -I(det g(x) ("i' ",)1 1/ 2• (ii) For ewry a E R there exists a unique a-density lILia, called tile g-a-deruity (or"" "...-RiemtlMitur a-tlsuity of II. such that IILI" equals
1.5.20 Propoaltlon.
ORIENTATION, VOLUME ELEMENTS, AND CODIFFERENTIAL
391
I on all orthonormal bases of the tangent spaces to M. If X" ...• X n is such a basis in an open set V of M with dual basis (', ... ,(". then Il1l a = 1(' " ... " .(nla. More generally. if V, •••• , VII e Tx M, then
The proor is straightrorward rrom 6.2.10 and the ract that 11 and'l"l a are smooth. As in the,vector space situation, g induces a pseudo-Riemannian metric on Ak(M) by'
where ex,peAk(M)x and pi''''i. are the components of the associated contravariant tensor via g( x) or p. As in 6.2.11. ir X, ....• X" is an orthonormal basis in V eM with dual basis (' ..... E". then the denwnts Ei, 1\ ... 1\ (". where i, < ... <; k rorm an orthonormal basis of ,\~ (I ' ) The Hodge operator is derined pointwise on an onentable pseudoRiemannian manirold with pseudo-Riemannian volume form 11 by .: Ok( M) .... 0" -k( M).{. a)(x) = • a(x). The propertie~ in Propositions 6.2.12 and 6.2.13 carry over since they hold pointwise. The exterior derivative and the Hodge star operator enable us to . introduce the following linear operator. (The reason ror the strange-looking factor (- I) in the definition is so a later integration by parts formula , proved in 7.2.13 will come out simple.) 6.5.21
Definition.
The codi/lerwrtiol8: Ok( M) .... Ok-'( M) is defined by
8(Oo( M» = 0 and on k-forms a by
8a =
(-
I )",k t
')'
I 'Ind(.).
do. a.
Since d 2 = O. and •• is a scalar, we have 8 2 = O. The previous rormulas give the general local expression of 8a. For example. let (xl, ...• x") be a positively oriented chart on M such that a/ax· .... ,a/ ax" is orthonormal and a be a (k + I)-form. Then in this 'chart.
" 8«"" (-I ) k+I+Ind(.) - a ( g'la.
axi
',"".
)dx"" . ... "dx". i < ... ,
<;k (3)
392
DIFFERENTIAL FORMS
The fonnula for
lea in general coordinates if ea is a k fonn is as follows:
(lea)I",,;, _ -
I g .•. g ~(gJo/lgh/l ... gh-,/,-,gml'a1,...1, 1."ldet ..D I1/2) Idetgll/2 lai, 1,-ai'-'8xm
(3') where ;1 < ... < '''_I and II < ... < 1,,_1 < I". (Without these conventions facton of (k -I)! and k! must be inserted.) This ronnula is messy to prove directly. However it follows fairly readily from integration by parts in local coordinates and the fact that I is the adjoint of fI, a fact that will be proved in Chapter 7 (see 7.2.13 and Exercise 7.S0).t We now shall express the divergence of a vector field X e ~(M) in terms of I. We define the divergence div,(X) of X with respect to a pseudo-Riemannian metric g to be the divergence of X with respect to the pseudo-Riemannian volume I' - 1'(,) of g; i.e. LxI''' div.(X)I" To compute the divergence, we prepare a lemma.
1.5.22 Lemma. ixl' "" • Xb
Proof.
At points where X vanishes this relation is trivial. So let X(x) '* «» and choose "I ..... 'l.-I e T,.M such that {X(x)jg(X. X)(x). "2 ..... "n} is a positively oriented orthononnal basis of T,. M. Then (ixl')(x)( "2 ..... 'l.) = 1'( X(x). "2, .. ','l.) = g( X, X)(x) ... Xb(X)(x)
.. ( • Xb)( x )( "2 •... , 'l.) by 6.2.14F.
•
tThere is another useful formula for' that involves the covariant derivative from Riemannian geometry. which we mention for completeness. Let liE 04( M) and define the covtUiImt derivative of II by ila ,l ...j , (Vill)jo-··J.--"-i-oX
It
E aJI"" •. IIJ•• ,..."r,',•. Ir-I
where
are the Christoffel symbols (summation on repeated indices is understood). Then -I
(3.)""".,- (k_I)!sJ'(v,a)"", ,.
ORIENTATION, VOLUME ELEMENTS. AND CODIFFERENTIAL
393
This may also be seen in coordinates using formula (12) of Section 6.2. 8.5.23 Propo8lUon.
Let If be a pseudo-Riemannian metric on the orienta-
ble n-manilold M. Then
(4) In (positively oriented) Irxal coordinates
(5)
f'roDf. Lx" - dl x " - d- X b by the lemma. But then (div X),,- - - axb by the lemma, the definition of a and the formula for - •. ~ince • I ... 1', we get (4). To prove (5) write" -Idet 8ijl'/2 the' " ... "the" and compute Lx" -= t6 x !" in these coordinates. We have
16Xr" =
(~ldet8' .1l/2 X h ) the'" axh '1
... "the"
.
8.5.24 DeflnlUon. The lAp,," - <rtutri OJIelYltor on lunctions on an orienlable pseudo-Riemannian manilold is defined by V 2 = div 0 grad.
Thus in a positively oriented chart, equation (5) gives
(6)
ExerclMa
I: R" .... R" be a diffeomorphism with positive Jacobian and 1(0) = O. Prove that there is a continuous curve!, of diffeomorphisms joining I to the identity. [Hint: First join / to Dj
6.5A Let
1(lx)!I.) 6.5B If t is a tensor density of M, that is, t = t' ®I'. where IL is a volume form, show that
394
DIFFERENTIAL FORMS
6.5C (T. Hughes). A map A: E -+ E is said to be derived from a variational principle if there is a function L: E -+ R such that dL{x)·v= (A{x). v).
where (.) is an inner product on E. Prove Vainberg's theorem: A comes Irom a variational principle il and only il DA(x) is a symmetric linear operator. Do this by applying the Poincare lemma to the one-form a(x)·v = (A(x). v). 6.5D Show in three different ways that the sphere S" is orientable by using 6.5.2 and the two charts given in 3.1.2. by constructing an explicit n-form. and by using 6.5.9. 6.5E Use formula (6) to show that in polar coordinates (r. 9) in R 2.
and that in spherical coordinates (P. 9. q» in R .1,
6.5F
where p. = cosq>. Let (M. ,,) be a volume manifold. Prove the identity div,,[X. Y] = X [div"Y] - Y [div"X].
6.5G Let I: M -+ N be a diffeomorphism of connected oriented manifolds with boundary. If Tml: TmM -+ ~(m)N is orientation preserving for some m E Int( M). show that J( f) > 0 on M; i.e.• 1 is orientation preserving.
CHAPTER
7
Integration on Manifolds
The integral of an n-form on an n-manifold i~ ddll1l'd in terms (If integrals over sets in An by means of a partition of unitv ~l1hordinale to an atlas. The change-of-variables theorem guarantees thai the integral is well defined. independent of the choice of atlas and partition of unity. The basic theorems of integral calculus-namely. the change-of-variables theorem and . Stokes' theorem-are discussed in detail. along with some applications. 7.1
THE DEFINITION OF THE INTEGRAL
The aim of this section is to define the integral of an n-form on an oriented n-manifold M. We begin with a summary of the basic results in An. Suppose f: An ..... A is continuous and has compact support. Then If d:(1 ... dx" is defined to be the Riemann intt:gral over any rectangle containing the support of f. 7.1.1 Definition. Let U c An be open and support. If. relative to the standard basis of An.
WE
O"(U) have compact
I (). . I w(x)= n!"';, ... ;" x dx"/\ ···/\dx'·="'I .. ,,(x)dx /\ ... /\d.'(". where the components of w are defined by
"';, .. ;.<x) =
"'( x)( e;,."
.. eJ. 395 .
396
INTEGRA TION ON MANIFOLDS
then we define jll)= j"'I .. ,,(x)dx l
••
·dx".
The change-of-variables rule takes the following form.
7.1.2 Propoeltlon. Let U and V be open subsets of H" and suppose f: U .... V is an orientation-preseroing diffeomorphism. Then if II) E (1"( V) has compact support, /*11) E O"(U) has compact support as well and jf*1I) = jll), that is, the following diagram commutes: (1"(U)
r
(1"(V)
i\'/f H
Proof. If II) = "'I ... " dx l " ••• "dx", then f*1I) = ("'I." f)(1I1./)(1(;. where (10 = dx l " ••• "dx" is the standard volume form on Hn. As was discussed in Section 6.5, }gJ > 0 is the Jacobian determinant of f. Since f is a diffeomorphism, the support of f*1I) is I(SUpp 11). which is compact. Then 0
r
by the usual change-or-variables formula, namely,
we get jf*1I) =
III). •
Suppose that (U, cp) is a chart on a manifold M. and II) E (1"( M). Then if supp II) C U, we may form II) I U, which has the same ·support. Then CP.( II) I U) has compact support. and we may state the following.
7.1.3 Definition. Let M be ann-manifold with orientation [(1). Suppose (1"(M) has compact support C C U. where (U.CP) chart. Then we define 1(.)'" = !cp*( 11)1 U).
'" E
7.1.4 Propoeltlon. Suppose
I.~
a positively oriented
II) E 0"( M) has compact support C C U () V. where (U. cp). (V, '" ) are two posilit)e(v oriented charts on the oriented manifold
THE DEFINITION OF THE INTEGRAL
397
M. Then
f w-f (,,)
w
(oS-)
By 7..1.2. f'l'.(wl V} = f( ¥- 0 'I'~ I }.'I'.( wi V). Hence f'l'.(wl V) = [Recall that for diffeomorphlsms f. =
N.( wIV).
0
Thus we may define fw <= f('f)w. where (V. '1') is any positively oriented chart containing the compact support of w (if one exists). More generally. 'we can define fw where w has compact ~upport as follows.
7.1.5 Definition. Let M be an oriented manifold and it an atlas of positively oriented charts. Let P = {( Va' '1'41' ga)} be a partition of unity subordinate to it. Define wa ., gaw (so w" has compact support in some U,). Then define
7.1.6 Propolltlon. (i) The sum shown contains only a finite number of nonzero terms. and hence fpla) E R. (ii) For any other atlas of positively oriented charts and subordinate partition of unity Q we have f pia) s: f~. The common value;s denoted /w, the ;"tegnll of wE U"( M). Proof. For any me M, there is a neighborhood V such that only a finite number of ga are nonzero on V. By compactness of supp w. a finite numher of such neighborhoods cover supp w. Hence only a finite number of g .. are. nonzero on the union of these V. For (ii), let P = {( V... '1'". g .. )} and Q = {( VII' ¥-II' hll )} be two partitions of unity with positively oriented charts. Then the functions {gahll} satisfy gahll(m) == 0 except for a finite number of indices \ II. jJ). and EaElIgahll(m) = 1. for all me M. Since Ellhll = 1. we get
=
r. r. J a
fl
gahll w ~
1w. Q
•
The globalization of the change-or-variables formula is as follows.
398
INTEGRA TION ON MANIFOLDS
7.1.7 Theontm. Suppose M and N are oriented n-manifoldsandf: M --. N is an orientation-preseroing diffeomorphism. If w E on( N) has compact support, then f*fI) has compact support and
r
Proof. First, supp f*w -
I (supp w), which is compact. For the second part, let {( U,), 'PI)} be an atlas of positively oriented charts of M and let P-{g,} be a subordinate partition of unity. Then {(f(u,),'Plor l )} is an atlas of positively oriented charts of N and Q - {g, I} is a partition of unity subordinate to the covering U(U,)}. Then by 7.1.6, 0
f f*fI)
=
r
l:, f gJ·w = l:, f lJ'i.(gJ·W)
-l: f 'P,. (rl).(gi or I)W /
-l: f( 'Pi rl).(g, rl)w 0
0
;
=
fw. •
As in 7.1.2, we have the following commutative diagram:
We also can integrate functions of compact support as follows.
7.1.8 Definition. Let (M, ,,) be a volume manifold. Suppose f E'S(M) and f has compact support. Then we call If" the bttlgrrll off with respect to ". The reader can easily check that since the Riemann integral is R-Iinear, so is the integral just defined. The next theorem will show that the foregoing integral can be obtained in a unique way from a measure on M. (The reader unfamiliar with measure
THE DEFINITION OF THE INTEGRAL
399
theory can find the pecessary background in Royden (1968). However, this will not be essential for future sections.) The integral we have described can clearly be extended to all continuous functions with compact support. Then we have the following. . 7.1.9 Rlesz Representation Theorem. Let (M, 11) be a volume manifold. Let
The space LP(M, JIo). pER. consists of all measurable functionsf such IftP is integrable. For p~l. the norm IIfllp'-<JlfIPdp.)J/P makes L'(M,Il) into a Banach space (functions that differ only on a set of measure zero are identified). The use of these spaces in studying objects on M itself is discussed in Section 7.4. The next propositions give an indication of some of the ideas. If F: M -+ N is a measurable mapping and p. M is a measure on M. then F -I'M is a measure on N defined by F .p.M(A) = P.M(r J(A». If F is bijective we set P(P.N) = (r 1).p.N' If f: M -+ R is an integrable function, thenfp.M is the measure on M defined by that
for every measurable set A in M. 7.1.10 ProposlUon. Suppose M and N are orientahle n-manifolds with volume forms 11M and IlN and corresponding meavures I'M and P-N' Let F be a C I diffeomorphism of M onto N. Then
PI'N = (J(P.M.P.N)F )P.M' 'Proof.
Let
f be any Coo function with compact support on
M. Then by
.7.1.1. /. P{fJloN) = /. {foF)(~P.M.P.N)F)flM /.Nfdp.N=/.fflN= N M M
400
INTEGRA TION ON MANIFOLDS
As in the proof of 7.1.9. this relation holds for f the characteristic function of F( A). That is,
7.1.11 Propoeltlon. Let M be an orientable manifold with volume form I' and corresponding measure ". Let X be a (possibly time-dependent) C l vector field on M with flow F,. The following are equivalent (if the flow of X is not complete. the statements involving it are understood 10 be where defined): (i) (ii) (iii) (iv) (v)
div" X - O. J,.F, -I for all 1 E R. F,." -" for all t E R. r,I' ... I' for aliI E R. Ifd" = l(f 0 F,) d" for all f E LI(M. 1') and all t E R.
Proof. Statement (i) is equivalent to (ii) by 6.5.19. Statement (ii) is equivalent to (iii) by 7.1.10 and to (iv) by definition. We shall prove that (ii) is equivalent to (v). If J,.F, = 1 for all 1 E R and f is continuous with compact support, then
l(f F,) d" for all integrable
Hence. by uniqueness in 7.1.9. we have If d" =
0
f. and so (ii) implies (v). Conversely, if
then taking f to be continuous wi th compact support. we see that
Thus. for every integrablef. f(f 0 F,) dp. and so (v) implies (ii). •
=
f( f F,)( J"F,) dp.. 0
HenceJ"F, = I.
The following result is central to continuum mechanics (see below and Section 8.2 for applications).
THE DEFINITION OF THE INTEGRAL
401
'7.1.12 Trarwpott Theorem. Let (M.~) be a volume IIltinifold and X a vt!ctor field on M with flow F,. For f E ~'i' ( M x R) and lettin/{ /, ( m) = f( m. t). we h.'lvt! '
d / -d 1
/,dp. - /
F,(UI
F,(UI
(!!1at + dlV... ( /,X).) dp.
for an open set U in M. Proof. · have
By the flow characterization of Lie derivatives and 6.5.17 (ii). we
=
a f -. F,*( a,~)+ F,*[( Lx/')~ + /,(dlv~X)~]
=
F,* [(
~~ + div~ ( /, X) ) ~ ] .
· Thus by the change-of-variables formula.
.
-/F,*[( aaft +diV~(/,X»)~] =/ I'
1-,1 U I
(i/}/I hli\~(/,X»)~ . • I
7.1.13 Example. Let p(x. t) be the density of an ideal fluid moving in a compact region with smooth boundary of R 3. One of the basic assumptions of fluid dynamics is conservation of mass: the mass of the fluid in the open · set U remains unchanged during the motion described by a flow F,. This , means that
:t f
F,(UI
p(x,t)dx=O
for all open sets U. By the transport theorem, this is equivalent t<;l the equation of continuity
~
+div(p,,) = 0,
402
INTEGRA TlON ON MANIFOLDS
where" represents the velocity of the fluid particles. We shall return to this example in Section 8.2. As another application of 7.1.11. we prove the following.
7.1.14 Polnee... Recurrence Theorem. Let ( M. 1') be a compa('t volume manifold and X a time-independent divergence-free vector field with flow F,. Then for each open set U in M and T ~ O. there exists S ~ T such that UnFs (U)-0. Proof. Since M is compact. po( M) = 1",1' < 00. Since X is divergence-free. F, is measure preserving by 7.1.11. Thus. U. FT(U). FlT(U)..... all have the same measure. Since po( M) < 00. they cannot all be disjoint. so fi. T( U)n F,T( U) .. 0 for k > I. Since Fo ' - F: (as X is time independent). we get 1;, _/IT(U)nU -0. • The Poincare recurrence theorem is one of the forerunners of ergodic theory. a topic that will be discussed briefly in Section 7.4. So far only integration on orientable manifolds has been discussed. A similar procedure can be carried out in order to define the integral of a one-d,ensity (see Section 6.5) on any manifold. orientable or not. The only changes needed in the foregoing definitions and propositions are to replace the Jacobians with their absolute values and the redefinition of divergence with respect to a given density as discussed in Section 6.5. All definitions and propositions go through with these modifications. F-valued one-forms and one-densities can also be integrated in the following way. If W - E~.lw;/;, where /1 ••.••/ .. is an ordered basis of F. then we set I",w = E:.I(/",w;) /; E F. It is easy to see that this definition is independent of the chosen basis of F and that all the basic properties of the integral remain unchanged. On the other hand, the integral of vector-bundlevalued n-forms on M is not defined unless additional special structures (such as triviality of the bundle) are used.
Exercl... 7.1A
Let M he an n-manifold and I' a volume form on M. If X is a vector field on M with flow F, show that
; (J.. ( F,») =
J..( F,)(div.. X 0
F,).
(Hillt: Compute (d/dI)P,,. using the Lie deri~'ative formula.'
STOKES' THEOREM
7.1 B
403
Prove the following generalization of the transport theorem
-dtd
f.
"',=
F,( V)
f. (-a",at + Lx"', )• F,(V)
where "', is a time-dependent k-form on M and V a k-di~ensional submanifold of M. 7.IC (i) Let cp: SI .... SI be the map defined by cp(e;')=e 2 ;'. where 9 E (0,2,,). Let, by abuse of notation, d9 denote the standard' volume form of Sl. Show that fs,cp·Cd9)- 21... ,d9. (ii) Let cp: M .... N be a smooth surjective map. Then cp called a k -/old covering map if every n E N has an open neighborhood V such that cp-I(V) - VI U ... U V•• where Vi .... ' V. are disjoint open sets each of which is diffeomorphic by cp to V. Generalize (i) in the following way. If ", E nne N) is a volume form. show that iMcp·", = kIN"" 7.1 D Let cp: M .... N be a smooth orientation-preserving map where M and N are orientable manifolds. n = dime N). and ex E U'C M). Define CP.ex: O.. -k(N) .... R as a linear function by (cp.ex)(P)=
f. ex M
1\
cp*p.
.
for all p E O"-k(N); i.e.• CP.ex is a distributional k-/orm on N. If there is aye Dk(N) with (cp.GI)(P)'" fy 1\ p, identify CP.ex and y. Prove the following: . (i) If cp is a diffeomorphism. then CP.ex is the usual push-forward. (ii) If ex is a volume form, this dermition corresponds to that for the push-forward of measures. (iii) If cp is a surjective submersion. prove CP. ex is defined by an element of O"(N). For surjective submersions. CP.ex is said to be obtained by integration over the fibers 0/ cpo Explain this statement. (iv) If cp is a surjective submersion. X E 'X.( M). ex E Ok ( M). Y E ~.(N) and X and Yare cp-related, prove CP.(ixex)=;y(CP.ex), fJI.dex = dcp.ex and cp.(LxGl) = LyfJI.GI. 7.2 STOKES' THEOREM Stokes' theorem states that if ex is an n -I-form on an orientahle n-manifold M, then the integral of dex over M equals the integral of ex over aM, the boundary of M. As we shall see in the next section. the classical
404
INTEGRA TION ON MANIFOLDS
theorems of Gauss, Green. and Stokes are special cases of this result. Before stating Stokes' theorem fonnally, we need to discuss manifolds with boundary and their orientations. 7.2.1 DellnItIon. Let R~ = {x'" (XI ..... X,,) E R"lx" ;;. O} denote the 1Ippe' ""'I-.""a 01 R" and let U C R~ be an open set (in the topology induced on R~ from R"). Call Int U = U n{x E R"lx" > O} the 01 U and au - U nCR ,,- I X (O}) the botmdary 01 u.
_rior
au.
au
We clearly have U = Int U u InW is open in U, closed in U n Int U = Ilj. The situation is shown in Fig. 7.2.1. Note is not the topological boundary of U in R", but it is the topological that boundary of U intersected with that of R~. This inconsistent use of the is temporary. notation A manifold with boundary will be obtained by piecing together sets of the type shown in the figure. To carry this out, we need a notion of local smoothness to be used for overlap maps of charts.
(not in R "). and
au
au
au
7.2.2 Definition. Let U, V be open sets in R ~ and I: U -. V; We shall say that I is SIrItItIIII illor t!tICh point x E U there exist open neighborhoods U I 01 x and VI oll(x) in R" and a smooth map II: U I -+ VI such that II U n U I III U nUl' We then define DI(x)" DII(x). We must prove that this dermition is independent of the choice of II' that is. we have to show that if cp: W -+ R" is a smooth map with Wopen in
Xo
RO, inlU
au FIgu... 7.2.1
STOKES' THEOREM
405
R" such that'l'l W nR~ = 0, then Dcp(x) = 0 for all x E W nR.~. If x E Int(WnR'!.). this fact is obvious. If xEa(WnR~). choose a sequence x"elnt(WnR'!.) such that x" .... x; hut then 0= Dcp(x.. ) .... Dcp(x) and. hence Dcp(x) = O. which proves our claim. 7.2.3 Lemina. Let V C R~ be open. '1': V .... R~ be a smooth map. and assume that for some Xo e IntV.cp(xo)e aR~. Then Dcp(xn)(R")c aR~.
Let p.. : R ...... R be the canonical projection on to the nth (last) factor and notice that the relation
Proof.
'1'( Xo + IX) = '1'( Xo ) + Dcp( x n )· Ix + fI(tX) , where lim,~oo(/x)11 = O. together with the hypothesis (1'./0 'I')(y) ~ 0 for all y e V. implies 0 :IIi; (p" 0 'I' )(xo + Ix) = () + ( P.. 0 Ocr')( Xu )'Ix + p"C o(tx)). whence for I > 0 O:lli; (P.. 0 Dcp)(xn)'x
+ p,,( ~(tl ).
Letting t .... O. we get (P.. 0 Dcp)(xo)'x ~ 0 for all x e R". Similarly. for 1< 0 and letting I .... O. we get (P.. Dcp)(xo)'x:lli; 0 for all x e R", The conclusion is 0
Intuitively. this says that if cp preserves the condition XII ~ 0 and maps an interior point to the boundary. then the derivative must be zero.in the vertical direction. The reader may also wish to prove 7.2.3 from the implicit mapping theorem. Now we carry this idea one step further. 7.2.4 Lemma. Let V. V be open sets in R~ and f: U .... Va diffeomorphism. Then f restricts 10 diffeomorphism" Intf: IntV .... Int V and
av .... av.
.
at:
Proof. Assume first that av = 121. that is. that V n(R" I X (O}) = 121. We . shall show that av == 121 and hence we take Int f - f. If av. 121. there exists x e V such that f( x) E av and hence by definition of smoothness in R~. there are open neighborhoods VI C V and VI C R". such that x E VI and f( x) e VI' and smooth maps fl: .VI .... VI' .It I: Vi .... VI such that II VI = fl' gd V n VI = II V n VI' Let XII e VI' x" .... X.,," E VI\r1V. and,),. = fCx,,). We have
r
406
INTEGRA T/ON ON MANIFOLDS
and similarly Dgl(f(x»o DI(x)
=
Id R "
so that DI(x)-1 exists and equals Dgl(f(x». But by 7.2.3. DI(x)(R")C R,,-I X{O}. which is impossible. DI(x) being an isomorphism.
r
Assume that au .. 121. If we assume av = 121. then. working with I instead of I. the above argument leads to a contradiction. Hence av ... 121 • Let x E Int U so that x has a neighborhood UI C U. UI n au = 121. and hence au, == fZl. Thus. by the preceding argumen t. aI (UI ) = fZl • and I (UI ) is open in V\ av. This shows that I(lnt U) C Int V. Similarly. working. with I. we conclude that I(lnt U) :::> Int V and hence I: Int U - Int V is a diffeomorphism. But then I( au) = av and II au: au - iJV is a diffeomorphism as well. •
r
Now we are ready to define a manifold with boundary. 7.2.5 DeflnlUon. An n-lIIQIIiJoid with boundDry is a set M together with an lit"" 0/ chlufl with IJoIuuI4ry on M: ebllrtl "';th bolllUltl", are pairs (U. q» where U C M and q>( U) C and an atills on M is a lamily 01 charts with
R:
x·'
R" ,
R" ,
Figure 7.2.2
STOKES' THEOREM
407
boundary satisfying (MAl) and (MA2) of 3.1.1, with smoothness of overlap maps rpj; understood in the sense of7.2.2. See Fig. 7.2.2. Define Int M= U urp-I(lnt(rp(U))) and aM = U 1'<1' I( iI(rp(U))) called, respectively, the interior and boIuultuy of M.
The definition of Int M and iJM makes sense in view of Lemma 7.2.4. Note that 1.
2. 3.
Int M is an n-manifold (with atlas obtained from (U, rp) by replacing rp(U)C R1 by Intrp(U)c R"). iJ M is an n - I-manifold (with atlas obtained from (U. C(') by replacing rp(U) C R~ by iJrp(U) c iJR~ = R"- I). iJM is the topological boundary of Int M in M (although Int M is not the topological interior of M).
Summarizing. we have proved the following. 7.2.6 PropoaHion. If M is an n-manifold with boundary. then its interior Int M and its boundary aM are sm()()th manifolds Il'ilhlll/l boundary of dimension n alJd n - I, respectively. Moreover. if f: M .. N /J a diffeomorphism, N being another n-manifold with boundary, thell f induces. by restri<"tion, two diffeomorphisms Int f: Int M -+ Int Nand af: ilM -+ iJN.
This set-up also works in infinite dimensions where the half-space R~ is replaced by a half space in E of the form E/ = {x E EI/( x) > O} where ., E E*, /- O. We leave the detailed formulation to the reader. Recall that before we can integrate a differential n-form over an n-manifold M, M must be oriented. If Int M is oriented, we want to choose an orientation on iJM compatible with it. In fact the reader will recall that in the classical Stokes' theorem for surfaces, it is crucial that the boundary curve be oriented in the correct sense, as in Fig. 7.2.3.
Figure 7.2.3
408
INTEGRATION ON MANIFOLDS
The rirst thing to do is to note that the earlier developments. including the tangent bundle. tensor rields. and differential forms. carry over directly to manifolds with boundary. In particular we can define T, M even if x E aM. Note that TxM is n-dimensional. even at points.v E aM. (See Fig. 7.2.4.) Thus. it makes sense to talk about volume forms and orientations on M up to and including the boundary points. .:.:: .............:......... . .'
iJM
..
",
.::,::.:. . . :::.:. . ,>::: . <:.: : :.:;.:., :;;
........: ... . :}
- -..........--.;....:..:...-..:.."-'-.:....1·........... :", ------~~~-----. .1'
T,M
T,iJM
F..,,.7.2.4 It is convenient to have in mind the following geometric interpretation of an orientation on M. An orientation on M is just a smooth choice of orientations of all the tangent spaces. "smooth" meaning that for al\ the charts or a certain atlas. called the oriented cham. the maps D( ", 0 " , - I )(x,): R" .... R" are orientation preserving. With this picture in mind. we can
Figure 7.2.5
STOKES' THEOREM
,409
define the boundary orientation of aM in the following way. At every x E aM. 1'.,( a~f) has codimension one in T,( M) so that there are""":in a chart on M intersecting aM -exactly two vectors perpendicular to x" = 0: one points inward. the other outward. Lemma 7.2.4 assures us that a change of chart does not affect the quality of a vector being outward or inward . .(See Fig. 7.2.5.)
7.2.7 DefInition. We shall say that a basis {vl ... ·.v" I} of T,(ilM) is positively o,w"ted if {- a/ax".VI ..... V" I} i.' posititlf'(r oriented in the orientation 01 M. This delines the Uuluced orientatiOll on iI M. Finally. we are ready to state and prove Stokes' theorem.
7.2.8 Stok..' Theorem. Let M be tin tJriellled .muHltil lI-maniloM with boundary and a E n" - I ( M) have compact slIpport. I.('t i: j/ M -+ M be the inc/u.fion map so that i*a E
n"- I ( aM). Then
1 i*a .. ! da iJM
M
or. for short.
1 a=/. da ;1M
M
Proof. Since integration was constructed with partitIOns of unity subordinate to an atlas and both sides of the equation to be proved are linear in a, we may assume without loss of generality that a is a form on U c R'~ with compact support. Write
a=
E" (-I);-' a ,dx ' A'"
--
Adx'A'" Ac/x".
; -I
where above a term means that it is deleted. Then da=
E" , _I
aa. -'dx ' A •.• A dx" ax'
and thus LJ /. d a= .;. U
I_I
1. R"
aa; dx " .d x i... ax'
There are two cases: au = IZJ and au"" IZJ. If au = IZJ, we have falla = O. The
410
INTEGRATION ON MANIFOLDS
integration of the ith term in the sum occurring in Juda is dx l ... -;;-... dx" 1 (1 aaidXi) ax' • • -1
(no sum)
•
and J ~:< aa; / axi) dXi = 0 since a j has compact support. Thus J"da = 0 as desired. If au then we can do the same trick for each term except the last. which is. by the fundamental theorem of calculus.
-Ill.
1 (1
00
aaa:dx") dx l ... dX,,-1
•• l O X
=
-1
a,,(xl •...• x"-I .0) dx ' ... dx"-
I•
•• - 1
since an has compact support. Thus
fuda- -1••
an(xl •...• x"-I.O)dx l .. ·dx"
1
-1
On the other hand.
1aua=l a=l iIA~
aA~
(-I)"-'a,,(x' ....• x"--'.O)dx ' /\ ···/\dx"
I.
But R,,-I - iJR~ and the usual orientation on R n 1 is not the boundary orientation. The outward unit normal is - ell = (0•...• 0. - I) and hence the boundary orientation has the sign of the ordered basis {-e"'~I.···.e" I}' which is (-I)". Thus
-
( - 1)
211-11 ••
0" (I X , ••• ,X ,,-IO)dx t
l
•••
dx ,,-1 .
1
Since (_1)2,,-1 = -I. we get the desired result. • If M has no boundary. note that JMda = O. • This fundamental theorem reduces to the usual theorems of Green. Stokes. and Gauss in R2 and R\ as we shall see in the next section. For forms with less smoothness or without compact support. the best results are somewhat subtle. See Gaffney (1954). Morrey (1966). Yau (1976). Karp (1981) and the remarks at the end of box 7.2B. Next we draw some important consequences from Stokes' theorem.
STOKES' THEOREM
411
7.2.9 Qaua' Theorem. Let M be an n-monifold with boundary and X a vector /iei(J on M with compact support. Let I' be a volume form on M. Then
1(divX)I' 1 101
Proof.
=
ixI'· aM
Recall that (divX)I' = LxI' = dixl' + ixdl' = dixl'·
The result is thus immediate from Stokes' theorem. _ If M carries a Riemannian metric. then there is a unique outwardpointing unit qormal "i/M along aM. and M and aM carry corresponding uniquely determined volume forms I'M and I'aM and associated measures I'M and p.aM' Then Gauss' theorem reads as follows.
7.2.10 Corollary
where (X. "ilM)
=
X·" aM is the inner product of X and n ,1M'
Proof. Let I'aM denote the volume element on aM inc.lul:cc.I by the Riemannian volume element I'M E 0"( M); i.e .• for any positively oriented basis v, ..... 1',,_, E 1'..( aM).
Since
,> = I'M( x >( X/ex )1'; -
(ixl'M}(X)( v, ..... 1'" _
X"(x>
= r(X)I'aM(X)(V' ..... v"
and X"
= (X. "aM)'
a!" .1', ......",- ~) ,>
the corollary follows by Gauss' theorem. _
7.2.11 Corollary. If X is divergence-free on a compact boundarylen manifold with a volume eleme"t 1'. then X as an operator is skew-symmetric: that is. for f and g E t!i( M).
f. X (f )g dp. .. - f. f X [ g] dp.. M
101
412
Proof. Thus
INTEGRA TlON ON MANIFOLDS
Since X is divergence-free, Lx(h")=(Lxh),, for any hE~'f(M).
Integration and the use of Stokes' theorem gives the result. •
7.2.12 Corollary. If M is compact without boundary, X E ~X ( M). «E O"(M) and p E O"-"(M), then
Proof. Since « " PE 0"( M). the formula follows by the use of the Stokes' theorem integrating both sides of the relation di x ( « " P) = Lx ( « " P) = Lx«" P + LxP. •
«"
7.2.13 Corollary. If M is a compact orientable, boundaryless n-dimensional pseudo-Riemannian manifold with a metric g of index Ind( g), then d and 3 are adjoints, i.e.,
Proof.
Recall from 6.5.21 that
=
liP =
( - ) )n' k + 2)
I
I + Ind(,r) •
d •• so that
d(<<" • P)
since k 2 + k is an even number for any integer k. Integrating both sides of the equation and using Stokes' theorem gives the result. • The same identity (do., P) = (a., liP) holds for noncompact manifolds, possibly with boundary-provided either « or P has compact support in Int M.
STOKES' THEOREM'
413
BOX 7.2A. STOKES' THEOREM FOR NONORIENTABLE MANIFOLDS
Let M 'be a nonorientable n-manifold with a smooth boundary --+ M. We would like to give meaning to the formula aM and inclusion map ;: iJM
1. dp M
= (
JiJM
i*( p )
in Stokes' theorem. Clearly, both sides make sense if dp and i*( p) are defined in such a way that they are densities on M and iI M, respectively. Here d should be some operator analogous to the exterior differential.. This means that p should be a section of some bundle over M analogous to A"- I( M). Call this as yet unknown bundle A~ I( M). Then we desire an operator d: A~( M) --+ AA.' I( M). k = 0.... •n, and A~(M) must be isomorphic to IAI(M). To guess what N.( M) might be, let us first discuss A~( M). The key difference between an n-form wand a density p is their transformation properties under a linear map A: Tm M --+ Tm M as follows:
for m E M and Vi .... ' V" E T", M. If l'I'" .• v" forfTI a basis then det( A') > 0 if A preserves the orientation given by VI ..... v" and det(A) < 0 if A reverses this orientation. Thus p can be thought of as an object behaving like an n-form at every m EM once an orientation of T",M is given; i.e.• p should be thought of as an n-form with values in some line bundle (a bundle with one-dimensional fibers) associated with the concept of orientation. This defj·, , '1 would then generalize to any k: AkT(M) will be line-bundle-\.. ims on M. We shall now construct this line bundle. At every point of M there are two orientations. U~ing them, we can construct the oriented double covering M--+ M. (see 6.5.7) Unfortunately, M is not a line bundle. so that some other construction is in order. At every m EM, a line is desired such that the positive half-line should correspond to one orientation of TmM and the negative'half-line to the other. The fact that must be taken into account is that multiplication by a negative number switches these two half-lines. To incorporate this idea, identify em.[",).o) with (m, - [wI, - 0) where
414
INTEGRA TION ON MANIFOLDS
mE M. a E Rand [wJ is an orientation of T",M. Thus. define the orientation line bundle a(M)={(m.[wJ.a)lmEM. aER. and [wJ is an orientation of T",M}/ - where - is the equivalence relation (m.[wJ. a) - (m.-[wJ. - a) and '11': a(M) - M. is the projection 'II'(m.[wJ. a) ... m. It can readily be checked that'll': a( M) - M is a line bundle with bundle charts given by 1/1: '11'- I( V) - cp( V) X R.IHm.[wJ.a)=(cp(m).a) where cp: V-R" is a chart for M at m. With this construction in mind we can now formulate the following.
7.2.14 DeflnlUon. A twisted k-Jorm on M is a a(M)-valued k-form on M. The bundle of twisted k-forms is denoted by A~( M) and sections of this bundle are denoted D!( M) or COC(A~( M». Locally. a section p E D! (M) can be written as p-~.
where GlE Ak(V) and ~ is an orientation of V regarded as a section of a(M) over V. The operators d:D!(M)-D!.f-'(M)
and
ix: D!( M) - D!-I( M). where X E'X (M). are defined to be the unique operators such that if p = ~ in the neighborhood V. then dp = (dGl)~ and ixp = (ixGl)~. One has /'x = i x " d + d" i x • and Lxp = (d/dt)r,pl,_o' F, being the now of X. where F,0p is defined in the usual way for bundle-valued forms. Define the map~: A~(M)-IAI(M) by ~(P)("I ...... vn)= p(vl ..... .",Xa(vl ..... where pE(A~(M»m. vl ..... V"ETmM and a(v l ..... .",) is the orientation given by Vi ..... .",. One checks that ~ is a line-bundle isomorphism identifying twisted n-forms with densities. If M is oriented by [wJ. the map p .... p[wJ is an isomorphism of A"( M) with A~( M). The proof of Stokes' theorem now can be used to prove the following version.
.",».
7.2.15 Stok..' Theorem (Nonorlentable C...).
Let M he a nonorientable n-manifold with smooth boundary iI M and p E I( M). a tM';sted (n - I )-form. Then { p. JMdp= JaM
D: .
STOKES' THEOREM
415
Of course the same statement holds for vector-valued twisted (n - I )-forms and all corollaries go through replacing everywhere (n -I)-forms with twisted (n -I)-forms. For example, we have the following.
7.2.16 Gaun' Theorem. Let M be a nonorientable Riemannian n-manifold with associated density I'M' Then for X
1. div(X)dl'M=j M
liM
E
':X ( M ).
(X·n)dl'aM
where n is the outward unit normal 01 aM. I'aM is the induced Riemannian density 01 aM and Lxl'M = (divX)I'M'
For a concrete situation in Rl involving these ideas. see Exercise 7.31.
BOX 7.2B STOKES' THEOREM ON MANIFOLDS WITH PIECEWISE SMOOTH BOUNDARY
The statement of Stokes' theorem we have given does not apply when M is, say, a cube or a cone. The trouble is that these sets do not have a smooth boundary. If the singular portion of the boundary (the four vertices and 12 edges in case of the cube. the vertex and the base circle in case of the cone). is of Lebesgue measure zero (within the boundary) it should not contribute to the boundary integral and we can hope th,t Stokes' theorem still holds. This box uisl'llsses such a version of Stokes' theorem inspired by Holmann and Rummier [19721. See Lang [1972] for an alternative approach. . First we shall give the definition of a manifold with piecewise smooth boundary. A glance at the definition of a manifold with boundary makes it clear that one could define a manifold with corners, by choosing charts that make regions near the boundary diffeomorphic to open subsets of a finite intersection of positive closed halfspaces. Unfortunately. many singular points on the boundary-such as the vertex of a cone-are not of this type. Thus, instead of trying to classify the singular points up to diffeomorphism and then make a formal intrinsic definition. it is simpler to consider manifolds alr....dy embedded in a bigger manifold. Then we can impose a condition on· the boundary to ensure the validity of Stokes' theorem.
7.2.17 Definition. Let U eRn - I be open and I: U -+ R be ~ontinu ous. A point pEr, = {(x. I(x»lx E U). the graph 01 f. is called replar
416
INTEGRATION ON MANIFOLDS
r,
il there exists an open neighborhood V 01 p such that V n is an (n - I) dimensional smooth submanilold 01 V. Let P, denote the set 01 regular points. Any point in a, = \p, is called singular. The mapping I is called p~w;. smoot" if P, is Lebesgue measurable. '11'( a, ) has measure zero in V(wherew; V xA -+ V is the projection) and/lw(a,) is locally Lipschitz; i.e .• lor each compact .ret K C w(a,) there exists a ('onstant c( K) > 0 such that I/(x)- I( y)1 ~ c( K Hlx - yll lor allx. y E K.
r,
Note that P, is open in f, and that Int(f,- )Up,. where f, X A Iy .... I(x». is a manifold with boundary Pf' Thus P, has positive orien.tion induced from the standard orientation of A". This will be called the positive orientation of f,. We are now ready to define manifolds with piecewise smooth boundary.
{(x. y) E V
7.2.18 DefInition. Let M be an n-manilold. A closed subset N 01 M is said to be a """'if0/4 wit" p~w;. smoot" bornuIIIry il lor every pEN there exists a chart (V.",) 01 Mat p.",(V) = V'x V" C A,,-I xA. and a piecewise smooth mapping I; V' -+ A such that ",(bd( N)nV)
=
ff n "'( V)
and "'( N n V) = f," n "'( V). See Fig. 7.2.6. It is readily verified that the condition on N is chart independent. using the fact that the composition of a piecewise smooth map with a diffeomorphism is still piecewise smooth. Thus. regular and singular points of bd( N) make intrinsic sense and are defined in terms of an arbitrary chart satisfying the conditions of the preceding definition. Let PN and aN denote the regular and singular plirl ;;f the boundary bd(N) of N in M.
U"
M
Figure 7.2••
417
STOKES' THEOREM
In order to formulate Stokes' theorem. we
n~
to define
IN'I. for
'I an n-form (resp.• density) on M with compact support. This is done
as usual via a partition of unity; PN and ON play no role since they have Lebesgue measure zero in every chart: PN because it is an (n -I)-manifold and ON by definition. It is not so simple to define IhdcN,t for t E U" I( M) (respectively. a density). First a lemma is needed.
t E D,,-I(U xR). ",here U i.~ op!''' in R" '. suPP(t) i.v compact and f: U ..... R is a pie('('lI'i.H' smoot" mapping. Tlren there exi.vu a smooth bounded function a: PI ..... R. ,vue" thllt
7.2.18 Lemma. Let
i*t =
a)..
where i: PI ..... U X R is the inclusion and). E D" - '( PI ) is the boundary volume form induced by the canonical volume form of U x R c R" on Int(r/ )Up"
,.rooj. The existence of the function a on P, is immediate since D" -'(PI) is a one-dimensional module with basis ).. We must prove that a is bounded. Let p E PI and v"" .. 1'" ,E T,,(P/) be an orthonormal hasis with respect to the Riemannian metric on PI induced from the standard metric of R". and denote by n the outward unit normal. Then a(p)=a(p)(dx'/\ · .. /\dx")(p)(n.v, ..... 1'" ,). =a(p».(p)(1', ..... v" ,) ... t(p)(1', ..... v" ,)
I .
~I ax I'
Let v; = vI. -a. . Smce
ax)
p
are orthonormal bases of T,,(U Hence if n
t = E £;dx ' /\
i-'
then
..... aa,,1 x
and
n.l', ..... 1'" I
I'
x R). we must have Il'/I E; I
for all i. j.
_
.,. /\ dx' /\ ... /\ dx".
la( p}1 = IU p)( 1', ..... 1',,_ ,)1
=1 i:. £;( p) E i-I
E;
nes"
(-I)' (signo)l'7 C11 •• I
"
E 1£;(p)l(n-l)! ;-,
which is bounded, since
t has compact support.
•
'l1;"",
III
418
INTEGRA TlON ON MANIFOLDS
In view of this lemma and lhe fact that can define
/ t = / tot = P,
P,
of
has measure zcro. we
/ a)... P,
Now we can define in the usual manner via a partition of unity. the integral of " e 0" - I( M) (or a twisted (n - I )-form) by
( 11=/"
'hd(N)
PN
7.2.20 Stok..' Theorem. LeI M he a paracompacl n-manifold and N a closed submanifold of M with piecewise smooth houndary. If M;s orientable and.., E n" - I( M). or (ii) M;s nonorientable and.., E n:-I(M) is a twisted (n -I)-form (see the preceding box) and supp(..,) is compact. then (i)
! d..,= ( N
..,.
'hd(N)
The proof of this theorem reduces via a partition of unity to the local case. Thus it suffices to prove that if U is open in R n I, .., E 0"· I( U X R) has compact support. and f: U ...... R is a piecewise smooth mapping, then
ir, d..,= i..,· r,
( I)
The left-hand side of (I) is to be understood as the integral ovcr the compact measurable set f,- nsupp(",). For the proof of (n we use three lemmas. 7.2.21 Lemma. ofo,in U X R.
Equation (I) holds if .., vanishes in a neighborhood
Proof. Let V be an open neighborhood of Of in U X R on which .., vanishes and let W be another open neighborhood of Of (which is closed in V) such that d(W)n(UxR)cV. The set O=(uxR)\ d( W) is open and since it is disjoint from Of' r, no is an ndimensional submanifold of 0 with hd( rf nO) = f, nO. Since
STOKES' THEOREM
supp(II..,)n r/ c r,- no and sU'pp(..,)n r, c Stokes' theorem. we have
r, n O.
419
by the usual
The purpose of the next two lemmas is to Construct approximations to II.., and .., if .., does not vanish near tIl' For this we need translates of bump functions with control on their derivatives. 7.2.22 Lemma. Let C be a box (rectangular parallelipiped) in R n of edge lengths 2/; and let D be the box with the same center as C but of edge lengths 4/,/3. There exists a Coo function cp: R n -+ [0. I J which is I on R"\C, 0 on D and Il1cp/l1x'l ~ A/I,. for a conSlalll A independent of I,. Proof. Assume we have found such a function cp: R -+ [0. I) for n = \.. Then "'(Xl •... • x") = cp(x l ) ... cp(x") is the desired function. The function cp is found in the following way. Let a = 2//3. f = 113 and ,choose an integer N such that 21 N < f. Let h: R -+ (0. Ii be a bump function that is equal to I for It I I. Then f: R -+ [0.1). defined by f(t) = 1- h(t) is a Coo function vanishing for Itl I. Let /,,{I) = f( nt) for all positive integers R and note that
If;(1)1
=
nlf'( nt)1 ~ Cn.
Define the Coo function
where the product is taken over integers z such that I;: I < 2 Na + I. Note that if Itl'< a + Ij4N and z E Z is chosen such that Ix - z/2NI < 1/2N. then fN(/- z/2N) =- 0 and Izi ~ 2NI/I +! < 2Na+ 1. so that cp(t)=O. Similarly if Itl>a+2IN and IZI<2Na+l. then It - z/2NI ~ Itl-lzl/2N > liN so that cp(/)-1. Finally. let Ito - al < 21N and let Zo E Z be such that Ito - zo/2NI < liN. All factorsfN(/ o - z/2N) are one in a neighborhood of ' 0 , unless Ito - z/2NI ~ liN. But then Ik - kol"; Iz -2Ntol + 12Nlo - zol ~ 3. Hence at most seven factors in the product are not identically I in a neighborhood of to. Hence Icp'(t,,)1
~
TCN = A/f.
.,
420
INTEGRA TION ON MANIFOLDS
a,.
7.2.23 L....m..
Let K be a compact subset of the singular set of For every e> 0 there exists a neighborhood U, of K in V X R and a C'" function ",,: V X R ..... [0. I]. which vanishes on a neighborhood of K in V" is one on the complement of V,. and is such that
f.
(i)
vol(V.>
sup 1a"" (x) I] ~ E. [ zeA" ax
i = I ..... n
_r_;_
and
(ii) vol( lI,> < I. A( V, (i p,) < I. where '" i.~ the mea.Jure on PI auocialed with the oolume form). E O,,-I(P/)' and vol(Vr ) is the Lebesgue measure 0/ lI, in R".
Proof. Partition R" - I by closed cubes
D of edge length 4//3. I ~ I. At most 2" such cubes can meet at a vertex. The set 'IT( K). where 'IT: V X R ..... V is the projection. can be covered by finitely many open cubes C of the edge length 21. each one of these cubes containing a cube D and having the same center as D. Since 'IT( K) and K have measure zero. choose I so small that for given 8 > 0,
(i) The (n - I )-dimensional volume of U :"'IC; is smaller than or equal to 8; and (ii) "('IT I(U:"'IC,)npf)~8.
Since f is locally Lipschitz and 'IT( K) is compact. there exists k > 0 such that If(x)- f( .r)1 ~ kllx -.rll for x. y E 'IT( K). We can assume k ~ I without loss of generality. In each of the sets 'IT I( C,) = C; x R. choose a box P; with base C; and height 2kl such that 'IT( K) is covered by parallelipipeds 1';' with the same center as I'; and edge lengths equal to two-thirds of the edge lengths of Pi' Let V ... U f-, Pi' Then 'IT( V) = U f_,C; and since at most 2" of the I'; intersect. vol(V) - 2k12" vol( 'IT(V»
< 2"+ Ikl8 < 2"+ 1k8
and By the previous lemma. for each I'; there exists a COD function "';: V x R ..... [0. I] that vanishes on p;,. is equal to I on the complement of PI' and sUPzER"lIa",Jax}1I < A/I. Let", == nf-,,,,,, Clearly "': V x R ..... [0. I) is C..,. vanishes in a neighborhood of K and equals one in the complement of V. But at most 2" of the P, can intersect. so that
Iax' I Ii-,_1 ax} ".,n I . -a", ....
~
-a",;
"'4.r
~"·1
.
1=
I •...• n
STOKES' THEOREM
Hence VOI(V)[ sup .. eA"
421
Iax'a'Pl] ~2"+lkI82"A/I+221'lk8A .
Now let Ii = min{E. E/2 2 ,,+ IkA}. 'P, = 'P, and V,
• .Proof of Eq. (I). Let w=E7_1""dx l l\ ... 1\ dx' 1\ ..• 1\ dx", dw - bdx l 1\ ••• 1\ dx", and i·w - a)... Then "',. band U ilre continuous and bounded on V X R and PI respectively; i.e., '''',(x)1 ~ M, Ib(x)1 ~ N for xEV X Rand la(y)l ~ N ror yE P" where M, N > 0 are constants. Let V, and 'P, be given hy the previous Iemmil applied to supp( w)U af . But then 'P,W vanishes in a neighborhood or af and lemma 7.2.21 is applicable; that is, =
V.
1r, d(~,w) =1r,'P,w
(2)
We have
1w-1r,'P,wl ~ If
1 ,r,
a( 1-
P,
'P,);\I ~ N;\( u, () PI) ~ NE
(3)
and
11r, dw - 1r, d( 'P,W)I ~ 11.I, (dw - 'P,dW)/ + 11." d'P. 1\ w/
I I
acp +E f. 1"'11 ~ ,_I r, ax II
dx l 1\ •.• 1\ d,"
II ~ N vol( V.) + ME sup [
,_1 "eA"
10!Jl(X)I] - ' - ,vol( V. ) ax
(4) From (2)-(4) we get
1r, dW-lwl~(2N+nM)E r,
1
for all E> O. which proves the equality. •
422
INTEGRA TlON ON MANIFOLDS
In analysis one also desires hypotheses on the smoothness of c.> that are as weak as possible as well as on the boundary. Our proofs show that Co) need only be C I • An effective strategy for sharper results is to approximate Co) by smooth forms Co)k so that both sides of Stokes' theorem converge as k ..... 00. A useful class of forms for which this works are those in Sobolev spaces. function spaces encountered in the study of partial differential equations. The Lipschitz nature of the . boundary of N in Stokes' theorem is exactly what is needed to make this approximation process work. The key ingredients are approximation properties in M (which are obtained from those in R") and the Calderon extension theorem to reduce approximations in N to those in R". (Proofs of these facts may be found in Stein [1970]. Marsden [1973]. and Adams [1975].) Exercl...
7.2A
Let M and N be oriented n-manifolds with boundary and f: M ..... N an orientation-preserving diffeomorphism. Show that the change-ofvariables formula and Stokes' theorem imply that rod = d r.
7.28
Let M be a manifold and X a smooth vector field on M. Let CI E Ok ( M). We call CI an invariant k-form of X iff Lx ex = O. Prove the following theorem of Poincare and Cartan: Let X be a smooth vector field on an-manifold M with flow F,. and let CI E Ok( M). Then ex is an invariant k-form of X iff for all oriented
0
compact k-manifolds with boundary (V. aV) and C<X:J mappinKs 'P: V ..... M. such that the domain of F, contains 'P( V). 0 ~ t ~ T. we have
f v( F,
0
'P )*CI =
f v'P*CI.
(Hint: For converse show that the equality between integrals holds
for any measurable set and conclude that ( FA cp )*CI "" cp*ex; then let V be a portion of various subspaces in local representation.) 0
7.2C Let X be a vector field on a manifold M and CI. II invariant forms of X. (See Exercise 7.28) Prove the following: (i) iXCl is an invariant form of X. (ii) dCl is an invariant form of X. (iii) LxY is closed iff dy is an invariant form. for any y E O*(M). (iv) CI /\ P is an invariant form of X. (v) Let f.£x denote the invariant forms of X. Then (:I'x is a /\ subalgebra of ll( M). which is closed under d and i x. 7.2D Let X be a vector field on a manifold M and ClE Ok(M). Then CI is called a relatively inf)ariant k· form of X if L XCI is closed. Prove the following theorem of Poincare and Cartan.
STOKES' THEOREM
423
Let X be a vector lield with flow F, on n-manilold M. and leI .. e O· - I ( M). Then .. is a relatively invariant (k - I )-Iorm 01 X ifflor
all oriented compact k-manilolds with boundary (V. ;W ) and cae map.f cp: V .... M such that the domain 01 F, contains cp( II ) lor 0 .e;; t .e;; T. we have
where i: 7.2E
av .... V is the inclusion map.
If X e ~X (M). let tf x be the set of all invariant forms of X, ~ x the set of all relatively invariant forms of X, the set of all closed forms in O( M). and f, the set of all exact forms in O( M). Show that
e
e
(i) (f x C ~'R x' t;, c c ~,~ x' ff x is a differential subalgebra of O( M), but ~ x is only a real vector subspace. . I· x
;
"
(ii)
the sequence 0 .... (fx"" O(M)"" O(M) .... O( M)jlm(L x ) ~ 0 is exact.
(iii)
0~
e ....; I3t x ....4 tf x ....." Cf. x If, n (f. x .... 0 IS. exact.
(iv) d«(fx)C tf. x and ix«(f.x)C (ix' Let M be a compact (n + l)-dimensional manifold with boundary,/: aM .... N a smooth map and", e N) where d", = O. Show that if I extends to M, then faM/*'" ""' O. 7.20 Let M be a boundaryless manifold and I: M .... R a COO mapping having a regular value a. Show that 1«( a, oo() is a manifold with boundary I(a). 7.2H Let I: M .... N be a COO mapping. M having boundar\' aM. and let PeN be a submanifold of N, where P and N are houndaryless. Assume I is transverse to P. Show that I-I(p) is a manifold with boundary arl(p)- rl(p)naM and dimM-diml I(p)= dimN -dimP. (Hint: At the boundary. work with a boundary chart with the technique of the proof of 3.S.II; then use the preceding exercise.) 7.21 Let M be a paracompact manifold with boundary. Show that there exists a positive smooth function I: M .... [0. oo[ with 0 a regular 1(0). (Hint: First do it locally and then value, such that aM patch the local functions together with a partition of unity.) 7.21 (Col/an). Let M be a manifold with boundary. A col/or for M is a diffc:omorphism of aM x [0, I[ onto an open neighborhood of aM in M that is the identity on aM. Show that a manifold with boundary and admitting partitions of unity has a collar:. (Hint: Via a partition 7.2F
one
r
r
r
424
INTEGRATION ON MANIFOLDS
of unity, construct a vector field on M that points inward when restricted to aM. Then look at the integral curves starting on aM to define the collar.) 7.2K (71re boundlJlyless double). Let M be a manifold with boundary. Show that the topological space oblained by identifying the points of aM in the disjoint union of M with Itself is a boundaryless manifold in which M embeds, called the boundar),less double of M. (Hint: Glue together the two boundary charts.) 7.2L (L. Loomis and S. Sternberg (1968). and R. Rasala). Let M be a compact Riemannian manifold with boundary. not necessarily oriented. Let X be a vector field on M. let p be the Rjema~nian density and let Px be a density on aM defined by Px( v,,",, vn - I) = p( v,,",, v,,-I' X) and let,x(x) = I if X points out of M. 0 if X is tangent to aM. and -I if X points into M. Show that
J. (divpX)p =1 M
'xPx
aM
and that this reduces to the usual divergence theorem if M is oriented. 7.2M Let M be a compa,·' orientable boundaryless n-manifold and oE 0" - I( M). Show that do vanishes at some point. 7.3 THE CLASSICAL THEOREMS OF GREEN, GAUSS, AND STOKES
This section shows explicitly how to obtain these three classical theorems as consequences of Stokes' theorem for differential forms. We begin with Green's theorem. Green's theorem relates a line integral along a closed piecewise smooth curve C in the plane R 2 to a double integral over the region D enclosed by C. (Piecewise smooth means that the curve C has only finitely many comers.) Recall from advanced calculus that the line integral of a one-form w - Pdx + Qdy along a curve C parametrized by y: la. b) -0 R2 is by definition equal to
{w = {
TH~ THEOREMS
OF GREEN, GAUSS, AND STOKES
425
(Positively oriented means the region D is on your left as YOII traverse the curve in the positive direction.) Suppose P: D -+ Rand Q: 0 --+ Rare e l . Then Pdx + Qdy = iJQ - iJ~) dxdy.
1c
if.n( ax
a..
e
Proof. We assume the boundary = iJO to he smooth. (The piecewise smooth case follows from the generalization of Stokes' theorem outlined in Box 7.2B.) Let Co) = P(x. y)dx + Q(x. y)~ E nl(O). Since dill ~ (iJQliJxaplay) dx 1\ dy and the measure associated with the vplume ,,," 1\ til' on R2 is the usual Lebesque measure dxdy. the formula of the theorem is a restatement of Stokes' theorem for this case. • This theorem may be restated in terms of the divergence and the outward unit normal. More preCisely. if e is given parametrically by ( ..... (x(l). y(t». then the outward unit normal is (I) and the infinitesimal arc-length (the volume element of e). is ds = r;(-t)2 + ~'W2 dt. (See Fig. 7.3.1.) If X = piJ I ax + QiJ I iJy E ,;,( (D). recall that div X = • d. X~ = iJPliJx + iJQ liJy. .
7.3.2. Corollary. Let D he a region in R 2 bounded by a closed piecewise smooth curve C. If X E :'X (D). then {(X',,)ds= ffn(diVX)dxdy ,
n
Figure 7.3.1
426
INTEGRA TION ON MANIFOLDS
where Ie/ ds denotes the classical line integral 0/ the function / over the positively oriented curve C and X·" is the dot product. Using formula (I) for" we have
Prvof.
1(X·,,) ds == j"[P(x(t). y(t»y'(t)-Q(x(t}. y(t»x'(t» dt C
a
=
f/d.v-Qdx
by the definition of the line integral. where x. y: [a. hI -> R. But this latter integral equals
ffJ
~: + ~;) dxd.v = fIr}divX) dxd.v
by 7.3.1. • laking P(x. y) = x and Q(x. y) = y in Green's theorem, we get the following.
7.3.3 Corollary. Let D be a region in R2 hounded hy a closed piecewise smooth curve C. The area
1
0/ D is given by ! x dy - }' dx. C
The classical Stokes theorem relates the line integral of a vector field around a simple closed curve C in R3 to an integral over a surface S for which C = as. Recall from advanced calculus that the line intexral of a vector field X in R 3 over the curve a: [a, hI -> R) is defined hy
l X 'd., "
=
j"X(a(t))'a'(t)dt II
The surface integral of a compactly supported two-form (a) in R 3 is defined to be the integral'of the pull-back of (a) to the oriented surface. If S is an oriented surface. " is called the outward unit normal at xES if " is perpendicular to TxS and {II. el' e2} is a positively oriented hasis of R) whenever {el,e2} is a positively oriented basis of 1~S. Thus S is orientable iff the normal bundle to S -which has one-dimensional fiber-is trivial. In particular. the area element tIS of S is given by 6.5.8. That is.
THE THEOREMS OF GREEN, GAUSS, AND STOKES
427
"th.
We want to express J... '" in a form familiar from vector calculus. Let,., - Pdy" dz + Qdz" dx + R dx" dy so that,., - • Xb. where ~nd IIY - dx " dy
x-
a
P ax
a
a
+ Q ay + R az .
Recall that CI" P ... (CI, P) I' where I' is the volume form. Letting CI = ·"b. Xb and I' - IIY. we get lIb " • Xb - (X·,,) IIY. Applying both sides to (",VI'Oz) and ':Ising (2) gives .
P-
(3) (Jhe base point x is suppressed). The left side of (3) is • Xb( VI' Oz) since n b is one on n and zero on VI and Oz. Thus (3) becomes • Xb= (X'n)IIS.
(4)
Therefore. is identified with a surface integral familiar from vector calculus. A physical interpretation of fs( X·,,) dS may be useful. Think of X as the velocity field of a fluid. Then X is pointing in the direction in which the fluid is moving across the surface S and X· n measures the volume of fluid p,assing through a unit square of the tangent plane to S in unit time. flence the integral fs( X· n) dS is the net quantity of flUid flowing acrou the sruface per unit time, i.e .• the rate of fluid flow. Accordingly, this integral is also called the flux of X across the surface.
Theo.......
7.3.4 Classical Stok. .' Let S be an oriented compact sruface in R3 and X a C l vector field on S and its bountlmy. Then f.<curIX)'ndS=
s
1asx· •. ·
where n is the oulward unil normal to S (Fig. 1.3.2). Proof. First extend X via a bump function to all of R 3 so that the extended X still has compact support. By definition. fils X •tis = fasXb where b denotes the index lowering action defined by the standard metric in RJ. But flXlI - .(curl X)II (see 6.4.3 (C» so that by (2) and Stokes' theorem,
1asx·. -: 1 Xb_f.4Xb =f. •(curl X)b ... f.(curl X'n )dS. ilS
S
S
S
•
428
INTEGRATION ON MANIFOLDS
x
Figure 7.3.2
7.3.5 EX8mplft A. The historical origins of this formula are connected with Faraday's law, which is discussed in Chapter 8 and Example 8 below. In nuid dynamics, this theorem is related to Kelvin's circulation theorem, to be discussed in Section 8.2. Here we concentrate on a physical interpretation of the curl operator. Suppose X represents the velocity vector field of a nuid. Let us apply Stokes' theorem 10 a disk D, of radius T at a point P E R3 (Fig. 7.3.3). We get
/, X·lb= /, (curIX)'nds=(curIX'n)(Q)'lTT 2 , liD,
liD,
the last equality coming from the mean value theorem for integrals: here QED, is some point given by the mean value theorem and 'lTT2 is the area of D,. Thus «curIX)·n)(P)= lim
~1aD,X·lb.
,-0 'ITT
Figure 7.3.3
THE THEOREMS OF GREEN, GAUSS, AND STOKES
429
The number leX·. is called the circulation of X around the closed curve C. It represents the net amount of turning of the nuid in a counterclockwise direction around C. The preceding equality gives the following physical interpretation for curl X, namely: (curl Xl'" is the circulation 0/ X per unil area on a sur/ace perpendicular to ,.. The magnitude of (curl X)·,. is clearly maxi~ when ,. - (curl X)/Ucurl XU. Curl X is called the vorticity vector. B. A basic law of electromagnetic theory is that if I:.' (t. x. y. z) and H(t, x, y, z) represent the electric and magnetic fields at time t. then V X E = - aN/at, where V x E is computed by holding t fixed. and aH/at is cdmputed by holding x. y, and z constant. Let us use Stokes' theorem to determine what this means physically. Assume S is a surface to which Stokes' theorem applies. Then
{ E·.=l(VXE)'dS'""-l aH. dS ..l-!...l N . dS .
JliS
S
S
(The last equality may be justified if N is
f
ifS
E·tIs =
at
at
S
e l .) Thus we obtain
at iSN . dS . .
-.!
This equality is known as Faraday's law. The quantity lasE·tIs represents the" voltage" around as, and if as were a wire, a current would now in proportion to this voltage. Also IsH·dS is called the flux of N, or the magnetic nux. Thus, Faraday's law says that the voltage around a loop equals the negative 0/ the rate 0/ change 0/ magnetic flUX through the loop. C. Let X E 'X (R'),'Since R' is contractible. the proof of the Poincare lemma shows that curl X = 0 iff X - grad/ for some function / E ~(R3). This in turn is equivalent (by Stokes' theorem) to either of the following: (i) for any oriented simple closed curve e, leX' tis = 0, or (ii) for any oriented simple curves cl.ez with the same end points. Ic,X·. = fe,X· ... The function / can be found in the following way: /(x,y,z)=l x X I (t,0.0)dt+1 Y XZ(x,t,O)dt+ o 0
l'·x (x.y,t)dt. 3
0
.
Thus, for. example, if a a a X= Y + YCOS(YZ)-aZ . . Yax +(ZCOS(YZ)+X)-a
curl X = 0 and.so X = grad/, for some f. Using the foregoing formula, one
430
INTEGRATION ON MANIFOLDS
finds
f(x. y. z) ... xy +sinyz D. The same arguments apply in R 2 by the use of Green'S' theorem. Namely if
lhen.X'" gradf, for some f E ~1(R2) and conversely. E. The following statement is again a reformulation of the Poincare lemma: div X - 0 iff X - curl Y for some Y E ~x. (R 3).
7.3.8 CluelCliI Gau.. Theorem. Let n be a compact set with nonempty interior in R3 bounded by a surface S that is piecewise smooth. Then for X a c' vector field on 0 U S, fo(diVX) dV= L(x'") dS,
where dV denotes the standard l'olume elemenl in R) (Fig. 7.3.4). Proof.
Either use 7.2.10 or argue as in 7.3.4: By (4)
• .... ~.'.' -:':' ": '.: .::-:.: ::<; '::. . ••
,','
••
:
'0
s-av FIgu.. 7.3.4
x
THE THEOREMS OF GREEN. GAUSS. AND STOKES
~1
By Stokes' theorem we get
Iud. X"= !o(divX) dV. since d. X" = (div X) dJ'. • 7.3.7 Example. We shall use the preceding theorem
1ao r'" dS= {4.". O. r3
10
prtlve Gauss' law
if Oen if o~n
where n is a compact set in R3 with nonempty interior. an is the surface bounding n. which is assumed to be piecewise smooth. " is the outward unit normal. 0 E an and
r-(x,y,z). ·IfO ~ n. apply 7.3.6 and the fact that div(r/r 3 ) - 0 to get the result. If oE n, surround 0 inside n by a ball D. of radius, (Fig. 7.3.5). Since the orientation pf aD, induced from n\D is the opposite of that induced from D, (namely it is given by the inward unit normal). Gauss' theorem gives
"
BO,
F.... 7.3.5
432
INTEGRA TION ON MANIFOLDS
since OEO\D, and thus on 0\0,. div(r/rJ)=O. But on iJD,. r=E and " ... - r/e. so that
since
1aD,dS ... 4."e
2•
the area of the sphere of radius E.
•
In electrostatics Gauss' law is used in the following way. The potential due to a point charge q at 0 E R J is given by q/(4."r). r = (x 2 + y2 + Z2)1/2. The corresponding electric field is defined to be minus the gradient of this potential; i.e.. E = qr/(4."r J ). Thus Gauss' law states that the total electric nux l.JoE· n dS equals q if 0 EO and equals zero. if 0 EO. A continuous charge distribution in 0 described by a charge clemity pis related to E by p = divE. By Gauss' theorem the electric nux 1,'IlE·ndS":' lop dV. which represents the total charge inside O. Thus. the relationship div E = p may be phrased as follows: The flux out of a surface of an electric field equols the total charge inside the surface.
Exercl... 7.3A.
Use (i) (ii) (iii)
Green's theorem to show that The area of the ellipse x 2/a 2 + y2/b 2 = I is ."ob. The area of the hypocycloid x = acosJ8. y = bsin3 8 is 3.,,0 2/8. The area of one loop of the four-leaved rose r" 3sin28 is
9.,,/8.
. ydx/(x 2
7.3B
Why does Green's theorem fail in the unit disk for y2)+xdy/(x2 + y2)?
7.3C
For an oriented surface S and a fixed vector II. show that
+
21.11 .,. dS = 1as(II X r)· dS . ... 7.3D
7.3E
7.3F
Let the components of the vector field X be homogeneous of degree one; i.e.. Xi(tx. ~v. tz) = tXi(X. y. z). i = 1.2.3. Show that if curl X = 0, then X == grad f. where f = !(xX' + yX 2 + zX J ). Let S be the surface of a region 0 in R3. Show that volume(O) = H.. r·"dS. Give an intuitive argument why this should be so. (Hint: Think of cones.) Let S be a closed. i.e.. compact boundaryless. oriented surface in R3. (i) Show in two ways that fs(curl X)·,. dS = O.
INDUCED FLOWS ON FUNCTION SPACES AND ERGODICITY
(ii)
433
Let X E ':X (S) and f E ~'t( S). Make sense of and show that
1.s(gradf)X·dS
= -
1.s(f l:url X) tiS
where grad, curl. and div are taken in HI. 7.3G
Let X and Y be smooth vector fields on H·I at least one with compact support. Show that
1 Y.curIXdv=l X'curlYdV R'
A'
(Hint: Show that y. curl X - X· curl Y
7.3H
7.31
If
~
=
dive X
X Y).)
is a closed curve bounding a surface S show that
where f and g. are C 1 functions. (A. Lenard). Faraday's law relates the line integral of the electric field around a loop C to the surface integral of the rate of change of the magnetic field over a surface S with boundary C. Regarding the equation V X E = - aH/ at as the basic equation. Faraday's law is a consequence of Stokes' theorem. as we have seen in Example 7.3.5B. Suppose we are given electric and magnetic field~ in space that satisfy V X E = - aH/ al. Suppose C is the boundary of the Mobius band shown in Fig. 6.5.1. Since the Mobius band cannot be oriented. Stokes' theorem does not apply. What becomes of Faraday's law? Resolve the issue using the results of Box 7.2A or a direct reformulation of Stokes' theorem for nonorientable surfaces. If aH/ at is arbitrary. in general does a current flow around C or not?
1.4 INDUCED FLOWS ON FUNCTION SPACES AND ERGODlaTYt Flows on manifolds induce flows on tangent bundles. tensor bundles. and spaces of tensor fields by means of push-forward. In this section, we shall be concerned mainly with the induced now on the space of functions. tThis section may be omitted without loss or continuity.
434
INTEGRA TION ON MANIFOLDS
This induced flow is sometimes called the LiOUf)ille flow. This section requires some results from functional analysis. Specifically we shall require a knowledge 0/ Stone's theorem and self-adjoint operators. The requircd results may be found in Appendices C and D. Let M be a manifold and IL a volume element on M; i.e., (M. IL) is a volume manifold. If F, is a (volume-preserving) now on M. then F, induces a linear one-parameter group (of isometries) on the Hilbert space H = L 2 (M,,,) by
The association of U, with F, replaces a nonlinear finite-dimensional problem with a linear infinite-dimensional one. There have been several theorems that relate properties of F, and u,. The best known of these is the result of Koopman [1931]. which shows that U, has one as a simple eigenvalue for all t if and only if F, is ergodic. (If there are no other eigenvalues. then F, is weakly mixing.) A few basic results on ergodic theory are given below. We also refer the reader to the excellent texts of Halmos (1956), Arnold and Avez (1967). and Bowen [1975]. We shall first present a result of Povzner (1966). which relates the completeness of the flow of a divergence-free vector field X to the skewadjointness of X as an operator. (The hypothesis of divergence free is removed in Exercises 7.4A-C.) We begin with a lemma due to Ed Nelson:
L.........
7.4.1 Let A be an (unbounded) self-adjoint operator on a complex Hilbert space H. Let Do C D(A) (the domain 0/ A) be a dense linear subspace 0/ H and mppose U, == e i / A (the unitary one-parameter group generated by A) leaves Do invariant. Then Ao = (A restricted to Do) is essentially self-adjoint; that is, the closure 0/ A 0 is A.
Proof. Let.fa denote the closure of Ao. Since A is closed and extends Ao. A extends A-;'. We need to prove that.fa extends A. For A > 0, A - iA is surjective with a bounded inverse. First of all. we prove that A- iAo has dense range. If not, there is a ., E H such that (."Ax-iAox)=O
In particular, since Do is U,-invariant.
so
for all xEDo'
INDUCED FLOWS ON FUNCTION SPACES AND ERGODICITY
435
Since Do is dense. this holds for all x E H. Since IIU,II = 1 and A> O. we conclude that ., - O. Thus (A - iAO)-1 makes sense and (A - iAr- 1 is its closure. It follows from Appendix C that A is the closure of Ao. •
7.4.2 Propoaltlon. Let X be a Coo diverKen("(!~free I'c'clor /it·/J on ( M. 1') with a complete flow F,. Then iX is an essentiu/~v selj-tl((;oilll operator on CcOO = the Coo functions with compact support in the' complex lIi1bert spa('e L2( M. II). Proof.
Let U,f -= f
0
F _I be the unitary one-parameter group induced from
F,. A straightforward convergence argument shows that U,f is continuous in t·in L2(M. II). In Lemma 7.4.1. choose Do = Coo functions with compact support. This is clearly invariant under If f E Do. then
u,.
so the generator of U, is an extension of - X
7.4.3 Theorem. Let M be a manifold with a volume elemellf " and let X be a Coo divergence-free vector field on M. Suppose that. as an operallJr 01/ L2( M. 1'), iX is essentially self-adjoint on the Coo fun('tion.f with compact support. Then. except possibly for a set of points x of measure zero, the flow F,(x) of X is defined for all t E R. Actually we will prove more than this. Namely. if the defect index of iX is zero in the upper half-plane [i.e.• If (iX + i)(C.?") is dense in L 2 ]. we shall show that the flow is defined, except for a set of measure zero. for all t > O. Similarly, if the defect index of iX is zero in the lower half-plane. the flow is essentially complete for t < O. The converses of these more general results can be established along the lines of the proof of 7.4.1.
Proof. Suppose that there is a set E of finite positive measure such that if x
E
xE
E, F,(x) fails to be defined for t sufficiently large. Let Er be the set of E for which F,(x) is undefined for I ~ T. Since E = U7=-IET' some'Er
436
INTEGRATION ON MANIFOLDS
has Positive measure. Replacing E by ET • we may assume that all points of E "move to infinity" in a time ..,. T. If I is any function on M, we shall adopt the convention that I( F,(x» - 0 if F,(x) is undefined. For any x E M, if t < - T. F,(x) must be either in the complement of E or undefined; otherwise it would be a point of E that did not move to infinity in time T. Hence we must have XE( F,( x» = 0 for 1< - T, where XE is the characteristic function of E. We now define a
function on M by
g(x) =
fIX; e-Txd FT(x» dT. -00
Note that the integral converges because the integrand vanishes for I < - t. In fact. we have 0.;; K(X)';; f'XTe 'dT = e T. Moreover. K is in L2. Indeed. because F, is measure-preserving. where defined, we have IIx,: f~II!.;; IIx/JI2 (where II 112 denotes the L2 norm), so that 0
IIgll2 410 f~e 'lixE £.112 dT';; IIXf:lbe T • 0
The function g is nonzero because E has positive measure. Fix a point x EM. Then F,(x) is defined for I sufficiently small. It is easy to see that in this case £,(F,(x» and FT+/(x) are defined or undefined together, and in the former case they are equal. Hence we have XE( FT ( F,(x» - XE( F.+/(X» for I sufficiently small. Therefore, for t sufficiently small
= e'g(x).
Now if 'P is Coo with compact support. we have
· = I1m
Ig(F I(X»-K(.")-
= lim
e I-I I --g(x)q>(x)dl1
I ~ II
1-0
= -
I
I
I g(x)'P(x)dl1
q>
( x )dP-
INDUCED FLOWS ON FUNCTION SPACES AND ERGODICITY
437
(These equalities are justified because on the support of fJ' the flow F, exists for sufficiently smallt and is measure-preserving.) . Thus g is orthogonal to the range of X + I. and therefore the defect index of iX in the upper half-plane is nonzero. The case of completeness for t < 0 is similar. • Methods of functional analysis applied to L2(M. f&) can. as we have seen. be used to obtain theorems relevant to flows on M. Related to this is a measure-theoretic analogue of the fact that any automorphism of the algebra ~t( M) is induced by a diffeomorphism of M (see Box 4.2A.). This result. due. to Mackey (1962). states that if U, is a linear isometry on 1.2( M. f&). which is multiplicative (i.e .. U,(fg) = U,f·U,g. where defim;d). then U, is induced by some measure preserving flow F, on M. This may he used to give another proof of the Povzner-Nelson theorem. 7.4.3. A central notion in statistical mechanics is that of ergodicity; which is intended to capture the idea that a flow is random or chaotic. In dealing with the motion of molecules. the founders of statistical mechanics. particularly Boltzmann and Gibbs. made such hypotheses at the outset. One of the earliest precise definitions of randomness of a dynamical system was minimality: the orbit of almost every point is dCIl<,e. In order to prove useful theorems. von Neumann and Birkhoff in the early 1930s required the stronger assumption of ergodicity. defined as follows.
7.4.4 Definition. Let S he a measure space and F, a ( measllrable ) flow on S. We call F, ergodic if the only invariant measurahle sels are 0 and all of S. Here, invariant means F,( A) = A for all t E R and we agree to write A ,.. B if A and B differ by a set of. measure zero. (It is not difficult to see that ergodicity implies minimality if we are on a second countable Borel space.) A function f: S -+ R will be called a conslant of Ihe mol ion if 1 F, = f a.e. (almost everywhere) for each IE R. . 0
7.4.5 Proposition. A flow F, on S is ergodic ilf the only constants oj the motion are constant a.e. Proof. If F, is ergodic and I is a constant of the motion. the sets (x E SI/(x) ~ a}.{x E Slf(x) E; a} are invariant; it follows thatf must he con-
stant a.e. The converse follows by
t~king
f to be a characteristic function .
•
The first major step in ergodic theory was taken by J. von Neumann (1932). who proved the mean ergodic theorem 11 remains as one of the most
438
INTEGRA TION ON MANIFOLDS
important basic theorems. The setting is in Hilbert space. but we shall see how it applies to flows of vector fields in 7.4.7. (Consult Appendix 0 for background on one-parameter groups.) 7 ....8 M_n Ergodic Theorem. Let H be a Hilbert space and u,: H ..... H a strongly continuous one-parameter unitary group (i.e .• U, is unitary for each t. is a flow on H and for each x e H.t ..... U,x is continuous). Let the closed subspace Ho be defined by Ho = {x E HI u,x = x for all t
E
R}
and let P be the orthogonal projection onto Ho. Then for any x
Il'U.xds
I,1m t
I-±OO
=
E
H.
Px
0
The limit in this result is called the time average of x and is customarily denoted x.
Proof of 1.4.6 (F. Ria: [19441). We must show that
II~~ 11+ fo'u. xds - pxll -0. If Px = x. this means x E Ho. so U.( x) = x; the result is clearly true in this case. We can therefore suppose that Px = 0 by decomposing x = Px +(xPx), We remark that {U,Y-."lyEH.tER}l =Ho
where .1 denotes the orthogonal complement. This is an easy verification using unitarity of U, and U, '= U _I' It follows from this remark that ker P is the closure of the space spanned by elements of the form u.y- y. Indeed kerP= HoJ.. and if A is any set in H. and B - A 1.. then Blois the closure of the span of A. Therefore. for any e> O. there exists ', •...• '" and x, •...• x" such that
It follows from this. again using unitarity of u,. that it is enough to prove our assertion for x of the form u,. y - y. Thus we must establish that lim! I-OC
('u.(u,.y- y)ds=O.
1)0
INDUCED FLOWS ON FUNCTION SPACES AND ERGODICITY
439
For t > to we may estimate this integral as follows:
I +10'[ u.( u,. y) - u. y] tlsll-ll- +fo'°u.< y) tis + +1,' '0U.( y) tlsil < !t 1'il yll tis + ! f' "'I dr +
I
0
= 2t ollyll_ 0 t
,
+
J'II
ast--+oo.
•
To apply 7.4.6 to a measure-preserving flow F, on .\', we consider the unitary one-par~meter group U,(/)= 1 0 F, on 1.2(S. fi). We only require a minimal amount of continuity on F, here. namely. we a~sume that if tn-t,F,(x)-F,(x) for a.e. xES. We shall also 'assume ",(S) <00 for convenience. Under these hypotheses. U, is a strongly continuous unitary one-parameter group. Again we will leave the verification as an exercise in the use of the dominated convergence theorem. 7~4.7 Corolla". To the hypotheses just described. add the condition that F, be ergodic. Then lor I E L 2( S. fi)
lim
.!. 1'01
I-t-x>t
0
F.ds = -1-f.ldP.' p.(s) s
the limit hei,,/{ in the mean. Proof. By 7.4.5. the space No of 7.4.6 is one-dimensional. consisting of constants. It is readily checked that
PI- isldP./p.(S) so 7.4.6 gives the result. • Thus. if F, js ergodic. the time average I of a function is constant a.e. and equals its space average. A refinement of this is the individual ergodic theorem qf O. D. Onoft (1931), in which one obtains convergence almost everywhere. Also. if p.(S) ... 00 but IE LI(S)() L 2(s). one still concludes a.e. convergence of the time average. (If I is only L 2• mean convergence to zero is stiD assured by 7.4.5.)
440
INTEGRATION ON MANIFOLDS
Exercl... (Exercises 7.4A-C form a unit.) . 7.4A Given a manifold M. show that the space of half-densities on M carries a natural inner product. Let its completion be denoted X(M). which is called the intrinsic Hi/bert space of M. If " is a density on M. define a bijection of L 2 (M.,,) with X(M) by' ..... ,{V. .Show that it is an isometry. 7.48 If F, is the (local) flow of a smooth vector field X. show that F, induces naturally a flow of isometries on X ( M) (make no assumption that X is divergence-free). Show that the generator iX = I." of the induced flow on :K ( M) is
iX(J/~)
=
(X(j]+ Hdiv"X)/)/"
and check directly that X is a symmetric operator on half-densities with compact support. 7.4C Prove that F, is complete a.e. if and only if X is essentially selfadjoint. 7.4D Consider the flow in R2 associated with a reflecting particle: for t> O. set
F. = { -q-tp q + tp ,(q.p)
if if
q>O. q>O.
q+tp>O q+tp
and set
F, ( - q. p) - - F, ( q. p) and F_, - F,- I .
7.4E
What is the exact generator of the induced unitary flow? Is it essentially self-adjoint on the Coo functions with compact support away from the line q = O? Let M be an oriented Riemannian manifold and L 2 (A'(M» the space of L 2 k-forms with inner product (a. P) = fa /I. • p. If X is a Killing field on M with a complete flow F,. show that iL" is a self-adjoint operator on L 2( Ak ( M».
7.5 INTRODUCTION TO HODGE-DeRHAM THEORY AND TOPOLOGICAL APPLICATIONS OF DIFFERENTIAL FORMS Recall that a k-form a is called closed if da = 0 and exact if a = dP for some k -1 form p. Since d 2 = O. every exact form is closed. but the
INTRODUCTION TO HODGE-DeRHAM THEORY
441
converse need riot hold. Let
(where dIe denotes the exterior derivative on k-forms). and call it the kth d,eRham cohomology group of M. (The group structure here is as a real vector space). The celebrated deRham theorem states that for a finite-dimensional compact manifold. these groups are isomorphic to the singular cohomology groups (with real coefficients) defined in algebraic topology; the isomorphism is given by integration. For "modern" proofs. sec Singer and Thorpe (1967) or Warner (1971). The original books of Hodge (1952) and deRham (1955) remain excellent sources of information as well. A special but important case of the deRham theorem is proved in'Box 7.5A. The scope of this section is to informally discuss the Hodge decomposition theory based on differential operators and to explain how it is related to the deRham cohomology groups. In addition. some topological applications of the theory are given. such as the Brouwer fixed-point theorem. and the degree of a map is defined. In the sequel. M will denote a compact oriented Riemannian manifold. and 3 the codifferential oper
- v 2f. where V 2 is the Laplace-Beltrami operator. This minus sign can be a source of confusion and one has to be careful. Recall that the L2-inner product in O"(M) is defined by (a. P) =
1a" • P M
and that d and, 8 are adjoints with respect to this inner product. That is. (da.p)
=
(a. lIP)
for a EO" - I( M). P e 0"( M). Thus it follows that for a. p E 0"( M). we have (Aa,P) = (dBa, P) + (&da. P) = (Ba.lIP) + (da. dP) =
(a. d3P) + (a. 8dP) = (a. AP).
442
INTEGRATION ON MANIFOLDS
and thus 6 is symmetric. This computation also shows that (£1ex.ex)." 0 for
an exE 0·( M).
7.5.2 ProposlUon. Let M be a compact (boundaryless oriented Riemannian.) manifold and ex E 0" ( M). Then 6cx == 0 iff dex = 0 and 8ex = o.
Proof. It is obvious from the expression £1ex == d8 ex + 8 dex tbat if 8ex == 0 and da- 0, then 6cx - O. Conversely, the previous computation shows that 0- (6cx. ex) - (da. da) + (8a, 8a), so the result follows. •
7.5.3 The Hodge Decomposition Theorem Let M be compact, boundarylen, oriented, Riemannian and let Co) E 0"( M). Then there is an a E O"-I( M), tJ E OA:+ I( M) and y E 0"( M) such that Co) = da + 811 + Y and £1( y) = O. Furthermore, da. 811, and yare mutually L 2 orthogonal and so are uniquely determined. That is,
We can easily check that the spaces in the Hodge decomposition are orthogonal. For example. dO A I( M) and 80" . I( M) are orthogonal:
(da.8II) = (ddex,tJ) == O. since 8 is the adjoint of d and d 2 ... O. The basic idea behind the proof of the Hodge theorem can be abstracted as follows. We consider a linear operator T on a Hilbert space E and assume that T2 - O. In our case T = d and E is the space of L 2 forms. (We ignore the fact that T is only densely defined.) Let T* be the adjoint of T. Let X - {% E £1 T% = 0 and T*%"" O}. We assert that .
£ ... range T $ range T* $X, which, apart from technical points on differentiability an~ so on, is the essential content of the Hodge decomposition. To see this, note that the ranges of T and T* are orthogonal because
(T%. 7* y) - (T 2%. y) = o. Let e be the orthogonal complement of range T$range T*. Certainly Xc e. But if %E e, then (Ty. %) ... 0 for all y implies 7*%'" O. Similarly, T% ... 0, so c X and hence ~l(.
e
e ""
The complete proof of the Hodge theorem requires elliptic estimates and may be found in Morrey (1966). For more elementary expositions. consult Flanders (1963) and Warner (1911).
INTRODUCTION TO HODGE-DeRHAM TI1EORY
443
7.1.4 Corollary. LAt X" denol, th, space of harmonic k-/orms. Then the veclor spaces X" and H" ( - kerd"/range d" - I) are isomorphic. Proof. Map X" - ker"" by inclusion and then to Hl hy projection. We need to show that this map is an isomorphism. Suppose y E 'l( , and (y 1= 0 where fy] E H" is the class of y. But (y] = 0 means that y is ellact; y = liP. But since ay = O. 'Y is orthogonal to liP; i.e.• y is orthogonal to itself. so y - O. Thus the map y .... [y] is one-ta-one. Next let [wJ E H". We can. by the Hodge theorem. decomposew = da.+ ¥ + y. wherey EX". SincedwO;,,¥-O so O-(P."¥)-(¥.¥). so ¥-O. Thus w==da.+y. Thus [w] = fy]. so the map y .... (yl is onto. •
. The space X" a; HI. is finite dimensional. Again the proof relies. on elliptic theory (the kernel of an elliptic operator on a compact manifold is finite dimeAsional). The Hodge theorem plays a fundamental role in incompressible·hyd.rodynamics. as we shall see in Section 8.2. It allows the introduction of the pressure for a given fluid state. It has applications to many other areas of mathematical physics and engineering as well; see for example. Fischer and Marsden [1979] and Wyatt et al. [1978]. Below we shall state a generalization of the Hodge theorem for some decomposition theorems for general elliptic operAtors (rather than the special case of the Laplacian). However. we first pause to discuss what happens if a boundary is present. This theory was worked (lut by Kodaira [1949]. Duff and Spencer [1952]. and Morrey [1966. Ch. 7). Differentiahility across the boundary is very delicate. but important. The hest possible results in this regard are due to Morrey. Note that d and a may not be adjoints in this case. because boundary terms arise when we integrate by parts (see Exercise 7.5E). Hence we must impose certain boundary conditions. Let a.E O"(M). Then ai is called parallel or tangent to aM if the normal part. defined by na == i*( * a.) is zero where i: aM - M is the inclusion map. Analogously. a. is perpendicular to aM if its tangent part. defined by ia- i*( a) is zero. Let X be a vector field on M. Using the metric. we know when X is tangent or perpendicular to aM. Now X corresponds to the one-form X" and also to the (n -I)-form ;xl' == • X" (I' is the Riemannian volume form). One checks that X is tangent to aM if and only if X" is tangent to aM iff ixl' is normal to aM. Similarly X is normal to aM iff ;xl' is tangent to aM. Set O:(~) - {aE 0"( M») a is tangent to aM} O!(M) - {aE O"(M)I a. is perpendicular to aM}.
X"(M) - {aEO"(M)1 "a,-o.aa-O}.
and
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INTEGRATION ON MANIFOLDS
The condition that ti• ... 0 and 8. = 0 is now stronger than 6. = O. One calls elements of X" harmonic fields, after Kodaira (1949]. 7.5.5 The Hodge Theorem for Manifolds with Boundary. Let M be a compact oriented Riemannian manifold with boundary. The following decomposition holds:
One can easily check from the following formula (ohtained from Stokes' theorem): (ti., p) = ( •• 8P) +
1 .". P ilM
(see Exercise 7.5E), that the summands in this decomposition are orthogo-
nal. Two other closely related decompositions of interest are
(i) where
D," = {.e 0:( M)18. = O} are the co-closed k-forms tangent to
aM and, dually.
(ii) where
C,," are the closed k-forms normal to aM.
To put the Hodge theorem in a general context. we give a hrief discussion of differential operators and their symhols. (See Palais (1965a], Wells (1980], and Marsden and Hughes (1983) for more information and additional details on proofs). Let E and F be vector hundles over M and let COO( E) denote the Coo sections of E. Assume M is Riemannian and that the fibers of E and F have inner products. A kth-order differential operator is a linear map 0: Coo(E) .... C""(F) such that, iffeC""(E) and f vanishes to k th order at x e M, then D(f)( x) = O. It is not difficult to see that vanishing to k th order makes intrinsic sense independent of charts and that o is a k th-order differential operator iff in local charts 0 has the form
D(f)=
E ",a llif ax ax"
0" III" k
J ••••
,
INTRODUCTION TO HOOGE-DeRHAM THEOR,(
445
wherej - (jl •...•i.) is a multi-index and "j is a COO matrix-valued function
of x (the matrix corresponding to linear maps of E to F).
The operator 0 has an adjoint operator 0* given in charts (with the standard Euclidean inner product on fibers) by
where pdx l 1\ ••• 1\ dxn is the volume element ~n M and aJ' is the transpose of ",. The crucial property of D* is
where (.) denotes the L2 inner product. g E C,:x'( E). and hE C,«J( F). That is. g and h are Coo sections with compact support. For example. we have the operators
8:
COO(A") .... C«J(A"·I) COO(A") ..... Coo(A"-I)
(first order)
~:
COO(AIt) ..... COO(A")
(second order)
d:
where d* = 8.8* = d and ~* = ~. The symbol of 0 assigns to eaeh
( fi rst order)
EE T: M. a linear map
It is defined by
where g E COO( M.R). dg(x) = E and IE C OO ( E)./(x) = e. By writing this out in coordinates one sees that at so defined is independent of g and I and is a homogeneous polynomial expression in E of degree k obtained hy substituting each (, in place of ai ax) in the highest order terms. For example. if
O(J) = then
Ea
a1 + (lower order terms) ax' ax' .
2 i } - - .- .
446
INTEGRATION ON MANIFOLDS
("i1 is for each i. j a map of Ex to Fx )' For real-valued functions. the
classical definition of an elliptic operator is that the foregoing quadratic form be definite. This can be generalized as follows: 0 is called elliptic' if 17, is an isomorphism for each E. O. To see that 4: C OC ( Ak ) .... ("'''( AA) is elliptic one uses the following facts: .
I. The symbol of II is 17, - f 1\ . 2. The symbol of is 17, - i, •. 3. The symbol is multiplicative: 17,(0,0 O2 ) "" 17,(0,)0 17,(02 )"
a
From these, it follows by a straightforward calculation that the symbol of 4 is given by efta-nEI12., so 4 is elliptic. (Compute E1\ O,.a)+ i,.( E1\ a) applied to (0, .... , Ok)' noting that all but one term cancel.)
7.5.8 EllipUc Spliting Theorem or Fredholm Altematlve.
Let 0 be an
elliptic operator as above. Then
Indeed this holds true il il is mere(v a.vsumed that either 0 or 0* hav injective' symbol.
The proof of this leans heavily on elliptic estimates that are not discussed here. As in the Hodge theorem, the idea is that the L 2 orthogonal complement of range 0 is ker 0*. This yields an L 2 splitting and we get a eGO splitting via elliptic estimates. The splitting in case 0 (resp. 0*) has injective symbol relies on the fact that then 0*0 (resp. 00*) is elliptic. For example, the equation Ou = I is soluble iff f is orthogonal to kerO*. More specifically, 4u = I is soluble if f is orthogonal to the constants; i.e., Ildp. '"'" O. The Hodge Theorem is derived from the elliptic splitting theorem as follows. Since 4 is elliptic and symmetric,
eGO ( Ak (M») -
range4e ker 4 - range4e X
Now write a k-form .., as .., = 4p + y = dip + Blip + y. so to get 7.5.3, we can choose a- Ip and II = dp. Now let us turn our attention to some topological applications of differential forms. Stricdy speaking, these do not require Hodge theory, but they are in the same spirit since a crucial point is the global distinction between closed and exact one-forms.
INTRODUCTION TO HODGE-DeRHAM THEORY
447
7.5.7 DeIInlUon. Two smooth mappings f, g: M ~ N. are called C'-homotopic if there exists a C'-map F: [0, I] X M ~ N such that F(O. m) = f( m) and F(I, m)- g(m).for all m EM. A key fact for the iollowing applications is eontllincd in the next proposition.
7.5.. Propoaltlon. Let M and N be orientable II-mall;/i,lds. with M boundaryless and ""E O"(N) with compact support andf. 8: M smooth homotopic maps. Then
4
N be proper
A direct proof of this fact is based on Exercise 6.4K. Indeed, since f and g g*", = d .... for some ... e 0" - I( M) and has compact are ·homotopic. support since f and g are proper. Thus. by Stokes' theorem
r", -
An alternative proof of 7.5.8 is based on the following lemma.
7.5.9 Lemma. Let V and N be orientable manifolds. dim( V) = n + I and dim( N) = n. Iff: av ~ N is a smooth proper map that extends to a smooth map of V to N. then for every '" e 0"( N) with compact .~Ilpp(lrt. favr", = O. Proof.
Let F: V ~ N be a smooth extension of f. Then hy Stokes' theorem
.1 r", 1av ;IV
==
p", = 1.dP4l) = 1. Pd",
v
~.
~ O.
sinced", .. O. • To apply this lemma to V = [0. I] x M, a natural orientation on the product of two oriented manifolds must be introduced. In general. if M and N are orientable manifolds (at most one of which has a boundary). then M x N is a manifold (with boundary). which is orientable in the following way: Let''''I: M x N ~ M and "'2: M x N ~ N be the canonical projections and ["'].(,,] orientations on M and N. respectively. Then the orientation of M x N is defined to be Alternatively. if vi •.... 0", e 1"x M and "'I ..... "''' e r;.N are positively oriented bases in the respective tangent spaces. then (VI.O)..... (v"'.O).(O. "'I)'" .• (0. "'II) e 7i .. ,1')( M x N) is defined to be a positively oriented basis in their product. .
["'r", " ",:. . ].
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INTEGRATION ON MANIFOLDS
Thus. for (O.I]X M. a natural orientation will be given at every point (t.m)E(O.I]xM by (1.0).(0• .")•... ,(0• .,,,). where v, ..... v"ET"'M is a positively oriented basis.
Proof of 7.5.& Since f and g: M --+ N are homotQpic. there exists a smooth map' F: (O.I)X M --+ N such that F(O. m) = f(m) and F( I. m) "'" gem) for all mEM. Since a«o.I)X M) = ({O}X M)U({I}X M). every element of this union is oriented by the orientation of M. On the other hand. this union i~ oriented by the boundary orientation of (O.I]X M. Since the outward normal at (I. m) is (1.0). we see that a positively oriented hasis of T.l,m,({I} X M) is given by (0• ." )..... (0. VII) for """ .. v" E T",M a positively oriented basis. However. since the outward normal at (0. m) is (- 1.0). a positively oriented basis of 1(0. ml({O} X M) must consist of elements (0. "" )..... (0. "'II) such that (-1.0).(0. "" )•...• (0. "'II) is positively oriented in (O.IJx M -i.e.• defines the same orientation as (1.0).(0. V,) •.•. •(0. v,,). for v, ..... .", E T",M a positively oriented basis. This means that "', ..... "'" E T", M is negatively oriented (see Fig. 7.5.1). Now apply the lemma to [0. I) x M by noting first that
1
/*w - -
(O)X M
1. /*w and f M
(I)x
Figure 7.5.1
g*w ==
M
1. g*w. M
INTRODUCTION TO HODGE-DeRHAM THEORY
449
so that
19-..,-11*.., .. 1 M
M
'(FliI([O.I]XM»-..,-O . •
iI«(O.I(><MI
This theorem is useful in three important applications. The first concerns vector fields on spheres (the" hairy ball theorem").
7.5.10 Theorem.
Every l'e('lor field on an el'en
dimen.~i(Jn(/1
sphere
h/J,~
(/
crili('al point. Proof. Let S2" he emhedded as the unit sphere in A 2" I and X E ~'\ (S2,,). Then X defines a map f: S2" ..... A 2n I I with componentsf(x) = (f'(x) •...• f2n + '(x» satisfying ~ I/,( x)x' = O. Assume that X has no critical point. Then replacingfhy flllfil we can assume that f: S2" ..... S2". The map I
1::::
F: [0. I] X S2" ..... s2n.
F(I. x) =
(cos WI)X + (sin WI) I( x)
is a homotopy between F(O, x) = x and F( I, x) = - x. That is, the identity Id is homotopic to the antipodal map A: S2" ..... S2". A(x) = - x. Thus by 7.5.8.
HoweveJ:, since the Jacohian of A is - I (this is the place where we use evenness of the dimension of the sphere), A is orientati(ln-reversing; i.e .. fs1.A-.., = - 1S2."', Consequently 1... = 0 for any.., E 0 2,,( S2,,). which is clearly false (take.., to be the standard volume element). •
2.'"
The second application is to establish the existence of fixed points for maps of the unit ball to itself. '
Brouwer'. Fixed-Point Theorem. Any smoolh mapping of Ilie closed unit ball of R" into itself has a'fixed point.
7.5.11
Proof. Let B denote the closed unit ball in R" and S" '= ilB its houndary. the unit sphere. Iff: B -+ B has no fixed point, define g(x) E sn-' to be the intersection of the line through the pointsf(x) and x, with S,,-I. It is easy to see that g: B ..... S" - , so defined is smooth and th'at for xES" - '. we have g(x) = x. If n = I this already gives' a contradiction. since g must map B = [-1.1) onto {- I.I} = So, which is disconnected. For n ~ 2. define a
450
INTEGRATION ON MANIFOLDS
homotopy F: [0, I] X S" - I -+ SIt - I by F(I, x) = g(lx). Thus F is a homotopy between the constant map c: S .. -I.:-+ S .. - I, c(x) = g,(0) and the identity of S .. -I. But clearly c·Co) - 0 for any Co) E O,,-I(S,,-I), so that by 7.5.8,
for any Co) E O,,-I(S .. -I). This, however, is clearly false. • lbe Brouwer fixed point theorem is also true for continuous mappings and is proved in the following way. If f has no fixed points. then by compactness there exists a positive constant K > 0 such that IIf(x)- xII> K for all x E B. Let r < min( K,2) and choose 8 > 0 such that 28/(1 + 8) <,; i.e., 8 < ,/(2 - r). By the Weierstrass approximation theorem (see, for example, Marsden [1974a» there exists a polynomial mapping q: R" -+ R" such that IIf(x)- q(x~1 < 8 for all x E B. The image q( B) lies inside the closed ball centered at 0 of radius 1+8, so that p. q/(l + 8): B -+ Band 28
which contradicts the choice, < K. The reader should note that Brouwer's fixed point theorem is false in an open ball, for the open ball is diffeomorphic to R" and translation provides a counterexample. The proof we have given is not "constructive". For example, it is not clear how to base a numerical search on this proof, nor is it obvious that the fixed point we have found varies continuously with f. For these aspects. see . Chow, Mallet-Paret and Yorke [1978]. A third application of 7.5.8 is a topological proof of the fundamental theorem of algebra, which normally requires complex function theory. 7.5.12 The Fundamental Theorem of Algebra. Any polynomial p: C -+ C of degne n > 0 has a root.
hoD/. Assume without loss of generality that p(z)=z"+a"
IZ,,-I
+ ... + ao, where Q j E C, and regard p as a smooth map from R2 to R2. If
INTRODUCTION TO HODGE-DeRHAM THEORY
451
p has no root, then we can define the smooth map f(z)= p(z)/lp(z)1 whose restriction to Sl we denote by g: Sl -+ Sl. Let R > 0 and define for (E (0, I) and Z E SI, p
p,(z)-(Rz)"+/[a,,_I(Rz)"- 1+ ... + ao]. Since p,(z)/(Rz)"=I+/(a,,_I/(Rz)+ ... +an/(Rz)") and the coefficient of I converges to zero a! R -+ 00; we conclude that for sufficiently large R, none of the p, has zeros on S I. Thus
F: [0. I]XSI
p (z)
-+
SI defined by F(/. z} = ~( )
./ Ill, z I
is a homotopy of d,,(z) == zIt with g. On the other hand, G: [O.I)X SI -+ Sl defined by GU, z) == i(lz) is a homotopy of the constant mapping c: Sl -+ Sl, c(z) - /(0) with g. Thus d" is homotopic to a constant map. Hence if Ca) = d8 E (}1(SI) is the volume form of SI. then by 7.5.8.
f d:Ca) f c*Ca) =
.')'
=
0
:;-'
since c*Ca) = O. On the other hand if we parametrize Sl by ar~ length 8, 0", 8 '" 2'17. then d" maps the segment 0 '" (J '" 2'1T/n onto the segment 0", (J '" 2'17 since d" has the effect e;' >-+ e''''. Using this fact and the change of variables formula. we get
J..'), d:Ca) == nJ.s' Ca) = 2'1Tn . Thus for n ... 0 we get a contradiction. •
BOX 7.U ZERO AND n-DIMENSIONAL COHOMOLOGY AND THE DEGREE OF A MAP In this box we shall compute HO( M) and H"( M) for a connected n-manifold M. Recall that Hk(M)= kerdk/ranged k - I • where d k : (}k( M) -+ (}k+ I( M) is the exterior differential. Thus HO(M)-{fEIj(M)ldf=O}aR since any locally constant function on a connected space is constant. If M were not connected, then HO( M) = R e , where c is the numher of connected components· of M. By the Poincare lemma, if M is contractible. then H'( M) ... 0 for q .. O.
452
INTEGRATION ON MANIFOLDS
The rest of this box is devoted to the proof and applications of the following special case of deRham's theorem.
7.5.13 Theorem. ut M be a boundaryless connected. compact nmanifold. (i)
If Mis orientahle. then H"( M) ~ A. the isomorphism being given bv integration: [Co)) ..... fMCo). In particular Co)EO"(M) is e.'Wct iff ~Co)=Q
(ii)
•
If Mis nonorientahle. then H"( M) -
o.
Before starting the actual proof. let us discuss (i). The integration mapping 1M: 0"( M) - A is linear and onto. To see that it is onto. let Co) be an n-form with support in a chart in which the local expression is Co)'" fdx' 1\ ••• 1\ dx" withfa bump function. Then fMCo) = fR"f(x)dx > O. Since we can multiply Co) by any scalar. the integration map is onto. Any Co) with nonzero integral cannot be exact by Stokes' theorem. This last remark also shows that integration induces a mapping. which we shall still call integration. fM: H"(M) -A. which is linear and onto. Thus. in order to show that it is an isomorphism as (i) states. it is necessary and sufficient to prove it is injective. i.e.• to show that if IMCo) - 0 for Co) E 0"( M), then Co) is exact. The proof of this will be done in the following lemmas; the main technical ingredients are Exercise 6.4K. Stokes' theorem. and the Poincare lemma.
7.5.14 Lemme. The theorem holds for M
=
S I.
Proof. Letp: A -Sl be given by p(t)~eil and Co)E01(SI). Then p.Co) = fdt for f E ~(A) a 2.".-periodic function. Let F be an antiderivative of f. Since
0=
1s· == f ' Co)
+2"
f( s ) d~ - F( t
+ 27T ) -
F( t )
I
for all tEA. we conclude that F is also 2.".-periodic. so it induces a unique map G E ';'t(SI). determined by p·G = F. Hence p.Co) == dFp·dG implies Co) ... dG since p is a surjective submersion. ...
7.5.15 Lemme. The theorem holds for M
= S". n >
1.
Proof. This will be done by induction on n. the case n = I being the previous lemma. Write S" ... NUS. where N ... {or E S"lx"'" I .. O} is the closed northern hemisphere and S = {x E S"lx"+ 1 "O} the closed
INTRODUCTION TO HODGE-DeRHAM THEORY
453
southern hemisphere. Then N n S = S" - I is oriented in two different ways as the boundary of Nand S, respectively. Let ON = {x E S"lx"+ I> - t}, Os = {x E S"lx"+ I < t} be open contractible neigh~ borhoods of Nand S, respectively. Thus by the Poincare lemma, there exists an aN EO" I( ON)' as EO" I( P.. ) such that daN = w, on ON' da s '= W on Os. Hence by hypothesis and Stokes' theorem,
o=f w=fw+fw=fdaN+fdas = ( aN+f as
sn
-1.
.~"
N
aN I
S
1.
,fi"
as'" I
N
1.
JilN
S
.'i"
,IS
(aN - as); I
the minus sign appears on the second integral becaus~ th~ {lrientations of S,,-I and as are opposite. By induction, aN - a.\ E U" I(S" I) is exact. Let 0 = ON n Os and note that the map r: 0· ... S" I, sending each xES to r( x) E S" - I, the intersection of the meridian through x with ~e equator S"-I, is smooth. Let j: SrI I -+ S" he the inclusion of S" - I as the equator of S". Then r 0 j is the identity on S" I. Also, j 0 r is homotopic to the identity of 0, the homotopy being given by sliding x E 0 along the meridian to r( x). Since d( aN - as) = w - w = 0 on 0, by Exercise 6.4K we conclude that (aN - a s )- r*j·( aN - Us) is exact on O. But we just showed that j*( aN - as) E O,,-I( S" .. I) is exact, and hence· r*j*(u N - as) E 0,,-1(0) is also exact. Hence aN - as E 0,,-1(0) is exact. Thus, there exists p E 0" 2(0) such that aN - as = dP on O. Now use a bump function to extend p to a form yEO" 2( SrI) so that on 0, p = y and y - 0 on S"\V, where V is an open set such that cl U c V. Now put xEN xES
and note that by construction
~
is C'" and
d~ =
w.
•
7.5.16 Lemma. A compactly supported .n.-form wE O"(R") is the exterior derivative of a compactly supported (n - I )-form on R" iff f.ow = o.
Proof. Let 0: S" -+ R" be the stereographic projection from the north pole (O..... I)E S" on R". By the previous lemma, o·w = do., for some aE O,,-I(S") since 0 = f.ow = fsoo*w by the change-of-variables for-
454
INTEGRA TlON ON MANIFOLDS
mula. But o·w = dCl is zero in a contractible neighborhood. V of the north pole, so that by the Poincare lemma. CI = dP on V. where peO,,-2(v). Now extend P to an (n-2)-form yeO,,-2(S") such that P== y on V and y == 0 outside a neighborhood of cl( V). But then 0.( CI- dy) is compactly supported in R n and do.( CI- dy) = o.dCl == w. "
7.5.17 L....
m..
Let M he a compact connected If-manifold. Then H n ( M) is at most one-dimensional.
Proof.
Let
{( u,. cr, n. i =
I ..... p, where cr,: U,
--+
B = {x
E
R" I II xII
<
I}
be a finite atlas of M. Let 10) E 0"( M). satisfying supp 10) C VI' he the pull-back of a form fdJe'/\ ... /\ dx" E O"(B) where f~ 0 and JR"/(X)dJc "" I. To prove the lemma, it is sufficient to show that for every "e 0"( M) there exists a number C e R such that ,,- CIo) = tit for some t e O,,-I( M). First assume" E 0"( M). where supP(,,) c U; is compact and let U;" U;2"'" U;, be a finite covering of a curve starting in VI == U;, and ending in U;-U;, such that U;,rlU;,.,.(lJ. Let CI,eO"(U;,),I=I •... , k - I be non-negative n-forms such that supp (CI,) c U;,. supp( CI,)rl U;,., .... (lJ, and JR"fJ>i,.( CI,) ... I. Let Clo '" 10), and CI.... ". But then
1= I •...• k by the change-of-variables formula. so that with c, = -l/fR"CJ'".( CI, I) we have JR"fJ>i,.( CI, - C,CI,_ I) = O. Thus by the previous lemma. CJ'".( CI, - C,CI,_ I) is the differential of an (If - 1)-form supported in B. That is, there exists p, e 0" - I ( M). p, vanishing outside U;, such that
1=1. ...• k. Put
and
P = P. + (c.P._ 1)+ (C.Ck_ IPk
2) + ..•• + (CA
I'"
('2PI ) Eon· I (
Then " - cw ., Clk
-
cCIo == Clk
-
CkClk _ I
+ Ck (Clk - I -
Ck
ICIA - 2 )
+(Ck '" C2)(Cl1 - CiCIo) -dP,+C,dPk-l+ ... +(C,···C2)dPl=dP.
+ ...
M).
fNTRODUCTION TO HODGE-DeRHAM THEORY
455
Let 'IIeon(M) be arbitrary and {X,li=I •...• k} a partition of unity subordinate to the given atlas {(U;.cp;)I;=I •...• k). Then X;'II is compactly supported in U, and hence there exist constants· c; and forms a, eon - I( M) such that Xi'll - C;W = dai. If k
c= E c;
,-I
and
"a= Ea;. i-I
then A
k
'II-Cw= E (X,'II-c,w)= Eda,=du . . . ;-1
Proof of the theofV!m. (i) By the previous lemma. HR( M) is zeroor one-dimensional. We have seen that 1M : HIf( M) - A is linear and onto so that necessarily H"( M) is one-dimensional; i.e.• IMw >= 0 iff w is exact. (ii) Let M be the oriented double covering of M and '1/": M --+ M the canonical projection. Define '1/"': H"( M) --+ W'( M) hy 'I/"'(al = ('II"·a). We shall first prove that '1/"' is the zero map. Let {U,li = I •.... k} be a finite open covering of M by chart domains and {X,ii = I•...• k} a subordinate partition of unity. Let '1/"- I(~) = u,1 U u,2. Then {U,JI j ... 1.2. i = I •...• k} is an open covering of M by chart domains and the maps'~/=hi°'ll": V/-A, i=I, ... ,k,j=I.2 form a subordinate partition of unity on M. Let a e 0"( M). Then
each term vanishing since their push-forwards by the coordinate maps coincide on RIt and 0;1 and u,2 have opposite orientations. By (i). we conclude that 'II" "'a = 4 for some ~eO"-I(M); i.e.. 'I/"'lul=('II"·a)"'; (0] for all (~]e H"(M). We shall now prove that'll"' is injective, which will show that H"(M)=O. Let aeO"(M) be such that 'II"·a=4 for some ~e on - I( M) and let ,: M - M be the diffeomorphism associating to (m,[w)) eM the point (m, -(w]H= M. Then clearly, 0 'II" = 'II" so that d(,"'~)" ,"'(4)" '·'II"·a= ('II" 0 ,)"'a:z 'II""'a - 4. Define ye Olt-I( M) by y .. !(~+ ,.~) and note that ,"'y= y and dy-! (d~+ dr"~)= 4= 'I/"·u. But y projects to a well-defined form yeO" I( M) such that 'I/".y = y, since ,.y .... y. Thus 'II"·a'" dy= d'll"·y ... 'II"*dy, which implies that a'" dy, since'll" is a suJjective submersion. •
456 . INTEGRATION ON MANIFOLDS 7.5.18 Corollary. Let M and N be oriented compact connected n-manilolds and I: M -. N a smooth map. (i) There exists an integer deg( /) called the deglW
of I such that
for any,. E O"(N). II x E M ;s a regplar point 011. let sign( TJ) be I or - I depending on whether the isomarphism TJ: T"M ..... ~(tlN preserves or retll!rses orientation. (ii) The integer deg(/) i.' given by
where y is an arbitrary regular value 01 f. (iii) II M. Nand P are oriented. compact. connected N-man;folds and I: M ..... N. g: N ..... P. then deg(g 0 I) = deg(/) deg(R).
Recall that the isomorphism H"( N) ~ R is given by Thus the linear map (,.1 .... IMr,. of H"(N) to R must be some multiple of this isomorphism. The multiple is called deg( I); i.e.• Proof.
(i)
(,.1 .... IN'"
We still have to prove that deg(/) is an integer and that the formula (ii) for deg( /) is independent on the regular value y. Both facts will be shown together. Let y be any regular value of I and I( y). Then there eltist compact neighborhoods V of y and V of xE x such that II V: V ..... V is a diffeomorphism. Sincer I(y) is compact and discrete. it must be finite. say l (y)={x l ••••• X4}. This shows that I( V) == VI U ... U V4 with all If; disjoint and the sum in the degree formula is finite. Shrink V if necessary to lie in a chart domain. Now choose 1J E 0"( N) satisfying supp( 1J) C V. Then
r
r
r
INTRODUCTION TO HODGE-DeRHAM THEORY
457
by the change-of-variables formula in Hn. so the degree f()rmula in (ii) is proved. Statement (iii) follows from the definitions. • Notice that by construction. if deg( f)'" O. then I is onto. 7.5.19 Corollary. Let V and N be orienlable manifolds with dim( V) = n + I and dime N) = n. II av and N are compact and f: av ..... N extends to V. then deg( I) = o.
This is a reformulation of 7.5.9. Everything done in this hox required C(lmpal·tnc~~, One can generalize everything to noncompact manifolds hy working with proper maps and cohomology with compact supports. For instanl'c. one has the following. 7.5.20
CorOllary. Let M. N be connected orientab/e n-mtlllilold~ and = deg( I{).
1.1{: M ..... N be smooth proper homotopic maps. Then deg( f)
The proof is straightforward from proposition 7.5.8. Degree theory can he extended to infinite dimensions as well and has important applications to partial differential equations and bifurcations. This theory is similar in spirit to the above and was developed by Leray and Schauder in the I 930s. See Chow and Hale [1982\. Choquet-8ruhat et al. (1977]. Nirenberg (1974]. and Elworthy and Tromba (I970b] for modern accounts.
Exercl...
7.5A (Poincare duality.) Show that. induces an isomorphism.: H k
.....
H"~k.
7.58
7.5C
(For students knowing some algebraic topology). Develop some basic properties of deRham cohomology groups such as homotopy invariance. exact sequences. Mayer- Vietoris sequences and excision. Use this to compute the cohomology of some standard simple spaces (tori. spheres, projective spaces). ' (i) Show that any smooth vector field X on a compact Riemannian manifold (M. g) can be written uniquely as
x = Y + gradp where Y has zero divergence (and is parallel to aM if M has boundary).
458
INTEGRA TlON ON MANIFOLDS
(ii) Show directly that the equation
Ap = -divX (grad p )." = X·" is formally soluble using the ideas of the Fredholm alternative. 7.SD Show that any symmetric two-tensor II on a compact Riemannian manifold (M.g) can be uniquely decomposed in the form II=Ld+ k •
7.5E
where &k ... 0, IS being the divergence of g, derined by Bk = (L( )g )*k. where (L( .,g)* is the adjoint of the operator X ..... L xg. (See Berger and Ebin [1969) and Cantor (1981) for more information). Let aeO"-I(M),peO"(M), where M is a compact oriented Riemannian manifold with boundary. Show that (du. P) - (a, 8P) = (
(i) (Hint: Show that 7.2.13.)
(ii)
u
JilM
* c5P =
(-
I )kd -
P and
(dBa. P) - (Ba. ap)
=
A- p.
use Stokes' theorem or
1 Bu
A
,1M
(da.dP)-(a.BdP)
=1
ul\
ilM
*P
-dP·
(iii) (Green's formula)
(Aa, P) - (a. AP)
=
1 (Ba
A -
P - 8P 1\ • U
ilM
(Hint: Show first that
(Aa,P)- (du, dP) - (Ba. ap)
=
1 (Ba ilM
7.SF
A-
P-
P 1\ * da).)
(For students knowing algebraic topology) Define relative cohomology groups and relate them to the Hodge decomposition for manifolds with boundary.
I~TRODUCTION TO HODGE-DeRHAM THEORY
459
7.5G Prove the local formula (8«)1, ... /, ,'" -ldet-I-1/2g. . .. g',I, .. -.L(g.io1' ... gil .. ',10 axm
,I. 'am/'ll l'I
1,···/, ,I,
Idet 811/2) •
where i l < ... < i. __ 1./1 < ... < I. I < I,. and exf~ la'( M) according to the following guidelines. Work in a chart «(I. ",) with ",( U) = BJ(O) = open ball of radius 3. and prove the formula 1111 '" I( 8 1(0». For this. choose a function X on R" with supp( x) C B)(O) and xIBI(O)!IE 1.. Then extend XCP.ex to R". denote it hy tit and consider the set 8 4 (0). (i) Show from 7.5E(i) that (tIP. ti) = (P. Bti) for any P E DIc + I(B4 (0» (ii) In the explicit expression for (tIP, ii). perform an integration by parts and justify it. (iii) Find the expression for aii by comparing (P. aii) with the expression found in (iii) and argue that it must hold on , 'I'-I(BI(O»_ 7.5H Let '1': M - M be a diffeomorphism of an oriented Riemannian manifold ( M. g) and let a. denote the codifferential corresponding to the metric g and (.). the inner product on DIc( M) corresponding to the metric g. Show that (i) (ex. P). = ('I'*ex. 'I'*P)~ •• for ex. P'E glc( M) and (ii) a" .•('I'*ex) = '1'*( 8.ex) for ex E DIc(M) (Hint: Use the fact that d and Bare adjoints.) 7.51 (i) Let c i and c 2 be two differen1iably homotopic curves and II) E DI( M) a closed one-form. Show that
(ii)
Lei M be simply connected. Show that HI ( M ) = O.
(Hint: For mo EM. let c be a curve from m" to m EM. Then
f(m)
= /...., is well defined by (i) and df = 11).) Show that H I( S I) * 0 by exhibiting a closed
one-form that is not exact. 7.5J The Hopf degree theorem states that / and g: Mil --+ S" are homotopic iff they have the same degree. By consulting references if necessary. prove this theorem in the context of Box 7.5A. 7.5K What does the degree of a map have to do with Exercise 7.2D on integration over the fiber? Give some examples and a discussion.' (iii)
460
INTEGRATION ON MANIFOLDS
7.SL
Show that 7.5.10. 7.5.11. alid 7.5.12 follow from considerations of degree. 705M Show that the equations +sin{lzI2)z' +3z" +2 = 0 Zl +cos(lzI 2)Z5 + Slog(lzl 2 )z .. + 53 "'0 Zl3
have a root. 7.5N Let f: M .... Ii where M and N are compact orientable boundaryless manifolds and N is contractible. Show that deg( f) - O. Conclude that the only contractible compact manifold (orientable or not) is the one-point space. (Hi"t: Show that the oriented double covering of a contractible nonorientable manifold is contractible.) 7.50 Show that every smooth map f: sn -> Tn, I has degree zero (Hi"t: Show that I is homotopic to a constant map.) Conclude that S" and T" are not diffeomorphic if " > 1.
,,>
CHAPTER
8
Applications
This chapter presents some applications of manifold theory and tensor analysis to physics and engineering. Our selection is of limited scope and depth. with the intention of providing an introduction to the techniques. There are many other applications of the ideas of this book as well. especially when combined with Lie groups and Riemannian geometry. We list below a few selected references for further reading in the same spirit. 1.
Arnol'd [1978]. Abraham and Marsden [1978]. Chernoff and Marsden (1974]. Weinstein (1977]. and Marsden (1981] for Hamiltonian mechanics.
2.
Marsden and Hughes (1983] for elasticity theory.
3.
Flanders [1963]. von Westenholtz (1981]. and Schutz (1980] for diverse applications
4.
Hermann (1980]. Knowles (1981]. and Brockett (1983] for applications to control theory.
5.
Bleecker (1981] for Yang-Mills theory (this requires background in Lie groups and connections in addition to what is given in this book).
6.
Misner. Thorne. and Wheeler (1973). Hawking and Ellis [1973]. and Burke (1980] for general relativity (this requires more hackground in Riemannian geometry than is given in this book). 461
462
APPLICATIONS
.8.1
HAMILTONIAN MECHANICS
Our starting point is Newton's second law in Rl. which states that a particle of mass m > 0 moving in a potential V( x) where x E R J. moves along a curve x( t) satisfying m.f - - grad V( x). I f we introduce the momentum p =- mk and the energy H(x, p) "" (1/2m~lpIl2 + V( x) then Newton's law becomes Hamilton's elfUlltions:
{
JcI=aH/aPi . Pi = - aH/ax'
i=I,2,3.
To study this system of first-order equations for given H we introduce the matrix
where' is the 3 x 3 identity; note that the equations become t = J grad H( E> where E""(x,p). In complex notation, setting z=x+ip, they may be written as t- 2iaH/iJ't. Suppose we make a change of coordinates, w-/(E), where I: R6 .... R 6 is smooth. If E(t) satisfies Hamilton's equations, the equations satisfied by w(t) are w- At- AJgrad,H(E) == AJA-grad .. H(E(w», where Aj(aw'/ a~J) is the Jacobian matrix of I, A- is the transpose of A and E( 11') denotes the inverse function of f. The equations for II' will be Hamiltonian with energy K( 11') == H(E( 11'» if AJA- = J. A transformation satisfying this condition is called canonical or symplectic. The space R 1 X R 1 of the E's is called the phase space. For a system of N particles we would use R1N XRJN. Many fundamental physical systems have a phase space that is a manifold rather than Euclidean space. For example, the phase space for the motion of a rigid body about a fixed point is the tangent bundle of the group S0(3) of 3 x 3 orthogonal matrices with determinant + I. This manifold is diffeomorphic to RpJ and is topologically nontrivial. To generalize the notion of a Hamiltonian system to the context of manifolds, we first need to geometrize the symplectic matrix J. In infinite dimensions a few technical points need attention before proceeding. Let E be a Banach space and B: E x E .... R a continuous bilinear mapping. Then B induces a continuous map BI>: E .... E-. e .... BIt( e) defined by BIt( ~ )./= B(~, /). We call B weakly nondegenerate if Bb is injective: i.e., B( ~, /) = 0 for all/ E E implies ~ == O. We call B nondegenerate or .~tmngly nondegenerate if Bit is an isomorphism. By the open mapping theorem it follows that B is nondegenerate in B is weakly nondegenerate and Bb is onto.
HAMIL TONIAN MECHANICS
463
If E is finite dimensional there is no difference hctWl'l'1l strong. and weak nondegeneracy. However, in infinite dimensIons the llistinction is important to bear i.n mind, and the issue does come up in hasi~' examples, as we shall see in Box S.IA. Let M be a Banach manifold. By a weak Riemannian structure we mean a· smooth assignment g: x .... (,)..... g(x) of a weakly nondegenerate inner product (not necessarily complete) io each tangent space T. M. Here smooth means that in a local chart U c E, the map x .... (,). E L2(E, E; A) is smooth where L2(E. E; A) denotes the Banach space of bilinear maps of E x'E to A. Equivalently. g is a smooth section of the vector bundle L 2(TM.TM; A) whose fiber at xEM is L 2(T.M.T,M; AI. Bya Riemannian manifold we mean a weak Riemannian manifold in which. (.), is nondegenerate. Equivalently. the topology of (.). is complete on T, M.. so that the model space E must be isomorphic to a Hilbert space. For example the Li inner product (t. g) = Mf(:{ )g( x) dx on E = CO«O.I).A) is a weak Riemannian metric on E but is not a Riemannian metric. 8.1.1 DefInition. leI P be a manifold modeled on a Banach space E. By a sympkctk form we mean a two-form II) on P such thaI (i) II) is closed. dll) - 0; (ii) for each x E P.II).: T.P X T.P -. A is
weak~v
nondcgenerllte.
If 11). in (ii) is nondegenerate. we speak of a stl'Oltg .fympledic form. If (iiI is dropped we refer to II) as a pruympkctic form. (For thi' moment the reader may wish. 10 assume P is finite dimensional. in which (me the weak - strong distinclion vanishes.) The first result is referred to a~ Darhoux's theorem. Our proof follows Moser 11965) and Weinstein 11969)•
D.-boux'. Theorem. Let II) be a strong symplectic form Oil the Banach manifold P. Then for each x E P Ihere is a local ('"ordillate chart t100111 x in which II) ;s ·conslanl.
••1.2
can
Proof. We assume P = E and x = O.E E. Let 11). he the constant fQrm equaling tOo ... 11)(0). Let II) =- 11). - II) and 11), = II) + tw. for 0 <EO; t <EO; I. For each t. 11),(0) - 11)(0) is nondegenerate. Hence by openness of the set of linear isomorphisms of E to E*. there is a neighborhood of 0 on which 11), is nondegenerate for all 0" 1';;1. We am assume that this neighborhood is a ball. Thus by the Poincare lemma. w= da for some one-fllrm a. We can suppose a(O) -= o.
464
APPLICA nONS
Define a smooth vector field X, by ix,w, = - a. which is possible since
w, is strongly nondegenerate. Since X,(O) = O. from 4.1.5 (and i1ssociated remarlts the local existence theory for ordinary differential equations). there is a sufficiently small ball on which the integral curves of X, will be defined for a time at least one. Let F, be the now of X, starting at Fo = identity. By the Lie derivative formula for time-dependent vector fields (Theorem 5.4.4.) we have
=F,*(d(-a)+w)=O. Therefore. rawl =- Pow,) = w. so FI provides the chart transforming W to the constant form WI' • We note without proof that such a result is not true for Riemannian structures unless they are nat. Also. the analogue of Darboux's theorem is known to be not valid for weak symplectic forms. (For the example. see Abraham and Marsden (1978). Exercise 3.2H and for conditions under which it is valid. see Marsden (1981).)
8.1.3 Corollary. (i) (ii)
If P is finite dimensional and W i.f a symplectic form. then
P is even dimensional. say dim P ,. 2n. Locally about each point there are coordinates that w=
L"
,-1
XI ••••• XH.
y' •...• y" such
dx i " dyi.
Such coordinates are called ctIIIOIIicfll. Proof. By elementary linear algebra. any skew symmetric bilinear form that is nondegenerate has the canonical form
where I is the n x n identity. (This is proved by the same method as 6.2.9.) This is the matrix version of (ii) pointwise on P. The result now follows from Darboux's theorem. • As a bilinear form.
W
is given in canonical coordinates by (YI' Xz). In complex notation with
w« XI' YI)' (x z• yz» = ( Yz. XI) -
HAMIL TONIAN MECHANICS
465
E - x + iJ' it reads ..,(EI' E2) - -Im(EI' E2)' This form for canonical coordinates extends to infinite dimensions (see Cook (1966) and Chernoff and Marsden [1974] for details). . Now we are ready to consider canonical symplectic forms.
8.1.4 DetinlUon. LeI Q be a manifold modeled on a Banach space E. Let. 1'*Q be its cotangent bundle. and T*: 1'*Q -+ Q the projection. Define the ctIIIOIfd OM-form 8 on 1'*Q by
:":.r~
:t.E T;Q and ET.. } 1'*Q). The H'
In a chart U c E. the formula for
CIIIIOIIica/
two-form ;s defined by
e becomes
8(x,CI)·(e.p) = CI(e). where (x. CI) E U x E* and (e. P) E E x E*. If Q is finite dimensional. this formula may be written
where ql •.. .• qn. P ...... p" are coordinates for 1'*Q and the summation convention is enforced. Using the local formula for d from 6.2.4.
or. in the finite-dimensional case. w=dqi
A
dp,.
In the infinite-4imensional case one can check that.., is weaklv nondegenerate and is strongly nondegenerate iff E is renexive (Marsden I I968b». If (.).• is a weak Riemannian (or pseudo-Riemannian) metric on Q. we have a smooth vector bundle map", = gb: TQ -+ 1'*Q defined by cp( ". )ev. = (0...... ) •• x EQ. which is injective on fibers. If (.) is a strong Riemannian metric. then", is a vector bundle isomorphism of TQ onto 1'*Q. In any case. set 0 = ",*.., where Co) is the canonical two-form on 1'*Q. Clearly 0 is exact since 0 .. - de where e = ",*e. 'In the finite-dimensional case. the formulas for 8 and 0 become
466
APPLICA TIONS
and··
0 = gijdqi " dil +
::;: il dqi " dq4.
where ql, .... q". " 1•... ,"" are coordinates for TQ. This follows by substituting P, .. B;JII into.., - dq' " (}p;. In the infinite-dimensional case. if (.). is a weak metric. then.., is a weak symplectic form and one uses the local formulas
where D, denotes the derivative with respect to x. One can also check that if (.). is a strong metric and Q is modeled on a renexive space. then D is a strong symplectic form.
8.1.5 DefInition. Let (P • ..,) be a symplectic manilold. A (smooth) map I: P .... P is called etIIfOIIiaII or symp/«lic when j*.., = ..,. It follows that j*(..,,, ... " ..,) ... .., " ... " .., (k times). If P is 2ndimensional. then" .. .., " ... " .., (n times) is nowhere vanishing so is a volume form; for instance by a computation one finds" to be a multiple of the standard Euclidean volume in canonical coordinates. In particular, note symplectic manifolds are orientable. We call" the phase volume or the Liouville lorm. Thus a symplectic map preserves the phase volume. and so is necessarily a local diffeomorphism. A map I: PI .... P2 between symplectic manifolds (PI''''I) and (P2''''2) is calledsympl«licifj*"'2 = WI' As above. if PI and P2 have the same dimension, then I is a local diffeomorphism and preserves the phase volume. We now discuss symplectic maps induced by maps on the base space of a cotangent bundle.
8.1.6 Propo....on. Let I: QI .... Q2 be a diffeomorphism; deline the cottlll-
.-t lift 011 by
where q e Q2.Gq e r;Q2 and "e Ii '(q,QI; i.e. 1'*1 is the pointwise adjoilll 01 Tf. Then 1'*1 is sympl«tic and in lact (1'*1 ).'. - ~ where 8; is the canonical one-Iorm on Q;. i ... 1.2.
HAMIL TONIAN MECHANICS
Proof. By definition. for
467
wE T...(1'*Q2)
(1'*/).8 1(a q )( w) = 81(1'*/( a q })(T1'*/. K') ... 1'*/(aq}'(T"3,~/'w) =1'*/( aq).( T( "3,01'*/ ).w) =
a q • ( T/· T( "3, 0 1'*/ ) . w )
=
a".( T( / 0 "Q, 0 1'*/).",)
=
a q • (TTO'.: N')
= ~(aq)'w
since. by construction. / 0 TQ, 0 1'*/ = TO'2' • In coordinates. if we write/(ql ..... q") = (QI ..... Q"). then 1'*/has the effect . where
(ql ..... q". PI ..... P") ..... (QI ..... Q". PI'" .. P,,). iJQ'
-P PJ =iJql;
(evaluated at the correct points). That this transformation is always canonical and in fact preserves the canonical one-form may be verified. directly:
Sometimes one refers to canonical transformations of this type as .. point transformations" since they arise from general diffeomorphisms of QI to Q2' Notice that lifts of diffeomorphisms satisfy / 0 "3, =
"Q, 0 1'*/;
that is. the following diagram commutes: 1'*/ 1'*Q2--- 1'*QI
"Q11
1"Q,
Q2
QI
/
468
APPLICATIONS
Notice also that P(f 0 X) = Pg 0 Pf
and compare with T(f 0 g) = Tf 0 Tg.
8.1.7 Corollary. If QI and Q2 are Riemannian (or pseudo-Riemannian) manifolds and f: Q • ..... Q2 is an isometry, then Tf: TQI ..... TQ2 is symplectic, and in fact (Tf)·8 2 - 8 •.
Proof. This follows from the formula Tf = "
0 (
P f ) - logy.
All maps in the composition here are symplectic and hence so is Tf. • So far no mention has been made of Hamilton's equations. Now we are ready to consider them.
8.1.8 DefinlUon. Let (P,,,,,) be a symplectic manifold. A vector field X: p ..... TP is called Hllntilton;an if there is a C l function H: p ..... R such that ix"" = dH. We say X is locally Hamilton;an if ix"" is closed.
We write X = X H because usually in examples one is given H and then one constructs the Hamiltonian vector field X H • If "" is only weakly nondegenerate, then given a smooth function H: p ..... R, X H need not exist on all of P. Rather than being a pathology, this is quite common and essential in infinite dimensions, for, as we shall see, the vector fields then correspond to partial differential equations and are only densely defined. The condition ;xHW - dH is equivalent to
for x E P and., E T.P. Let us express this condition in canonical coordinates (ql, ... ,q", PI"" ,Pn) on a 2n-dimensional symplectic manifold P, i.e., when w=dqiAdp/. If X=Aia/aqi+Bia/ap" then ;x"dqi=A' and ;X"dPi = B'. so that
;x,,"" = ;X,,(dqi A dpi) = (;X"dqi) dp, - (ix"dp,) dqi
HAMIL TONIAN MECHANICS
469
This equals
aH aH dH = - . dq' + -dD.
aq'
ap,
ow
iff
Ai = aH ap,'
XII =
and
8'
aH ;
= _
a L" (aH ap -a' ' q
,-I
aq'
I.e.,
aH a ) -a' up. . q "
If
where I is the n X n identity matrix, the formula for XII can be expressed as XII = ( aH ,_ aH) = J.grad H.
ap,
aq'
Thus (q'( I), p,( t)) is an integral curve of XII iff Hamilton's equations hold;
., q =
aH.
aPi'
aH
p, = - aqi'
We now give a couple of simple properties of Hamiltonian systems. The proofs are a bit more technical for densely defined vector fields, so for purposes of these theorems. we work with COO vector fields.
8.1.9 Theorem. Let XII be a Hamiltonian vector field on the (weak) symplectic manifold (P. w) and let F, be the flow of XII' Then (i) F, is symplectic; i.e .• p,w = w. (ii) energy is conserved; i.e. H 0 F, = H.
Proof. (i) Since f(, = identi~v. it suffices to show that (d/dt)p,w = O. But by the basic connection between Lie derivatives and nows (Sections 5.4 and 6.4): . d dt p'w(x) = p,(Lxl/w)(x) = P, (dix w)( x)+
"
P,u." dW)( x). "
470
APPLICA TIONS
The first term is zero because it is dllH and the second is zero because w is closed. (ii) By the chain rule. d
dt (Ho F,)(.t) = dH( F,(x»·X,,( F,(x»
= W',I ,) ( X" ( F, ( x
». X" ( F, (x»).
but this is zero in view of the skew symmetry of w.
•
An immediate corollary of (i) in finite dimensions is Liouville's theorem: F, preserves the phase volume. This can also be seen directly in canonical coordinates by observing that XIf is divergence-free. We shall now generalize (ii). Define for any functions f. g: u ..... R. U open in p. their Poisson bracket by
Since
LxI If - ix, dg - ixIix.w - w( XI' XII) - - w( XII' XI) - - Lx f. II
we see that
{j.g}=-Lxf=Lxg· • I If 'P: PI ..... P2 • is a diffeomorphism where (PI,WI) and (P2 .w2 ) are symplectic manifolds. then by the property 'P.( Lxcx) = L.,oxcp·cx of pull-back. we have
and
Thus 'P preserves the Poi.uon bracket of any two functions defined on some open set of P2 iff 'P. XI = X"0l for 01/ Coo functions f: U ..... R, U open in P2' This says that 'P preserves the Poisson bracket irf it preserves Hamilton's equations. We have
so that by the (weak) nondegeneracy of wand the fact that any ., E T, P equals some X.(x) for a Coo function h defined in a neighborhood of x. we
, HAMIL TONIAN MECHANICS
471
conclude that cp is symplectic iff cp* X, .. X••, for aU Coo functions f: U -R.U open in Pz' We have thus proved the following.
8.1.10 Proposition. Let and cp: PI (i)
-+
(PI.W I ) and (PZ ,W2) be symplectic manifolds P2 a diffeomorphism. The following are equivalent: .
'I' is symplectic.
(ii) cp preserves the Poisson bracket of any two locally defined functions. (iii) '1'. Xf = X""f for any local f: U -+ R, U open in Pz (i.e., cp locally preserves Hamilton's equations).
Conservation of energy is generalized in'the following way.
8.1.11 Corollary. Let X H be a Hamiltonian vector lield lin the (weak) symplectic manifold (P, w) with (local) flow F,. Then for any C'Xl function I: U -+ R, U open in P, we have d
dt (f 0 F,) = {I. H}o F, = {lo
F"
H}.
Proof.
d
d
-(1 F) == -·P.f= P.Lv f dt ' dl' ''''11 0
=
F:{I. H} = {F:f. H}
by the formula for Lie derivatives and the previous proposition. • Two functions I. g: P -+ R are said to be in involutiol/ or to Poisson commute if {/ .. g} = O. A.,v function Poisson commuting with the Hamiltonian of a mechanical system is by 8.1.11 necessarily constant along on 'the now of the Hamiltonian vector field. This is why such functions are called con.Uants of the motion. A classical theorem of LiouvilJe states that in a mechanical !lystem with a 2n-dimensional phase space admitting k con!ltants of the motion in involution and independent almost everywhere (that is. the differentials are independent on an open dense set) one can reduce the dimension of the phase space to 2(n - k). In particular. if k == n. the equations of motion can be "explicitly" integrated. In fact. under certain additional hypotheses. the trajectories of the mechanical system are straight lines on high-dimensional cylinders or tori. If the motion takes place on tori. the explicit integration of the equations of motion goes under the name of finding action-angle variables. See Abraham and Marsden (1978. pp. 392-400) for details and Exercise 8.10 for an example. In infinite-dimensional systems the situation is considerably more complicated. A famous
472
APPLICA TlONS
example is the Korteweg-deVries (KdV) equation; for this example we also refer to Abraham and Marsden (1978. pp. 462-72) and references therein. The following box gives some elementary but still interesting examples of infinite-dimensional Hamiltonian systems.
BOX B.1A
TWO INFINITE-DIMENSIONAL EXAMPLES
8.1.12 Example: The Wave EquaUon as a Hamiltonian System. The wave equation for a function u( x. ,). where x E R" and, E R is given by
(where m ~ 0 is a constant). with u and
u= au/a, given at' = O. The energy is
We define H on pairs (u.
u)
of finite energy by setting
where HI consists of functions in L2 whose first (distributional) derivatives are also in L 2 •t Let
and define XH: D -. P by X,,(u. u)
=
(u. V 2 u + m 2 u).
Let the symplectic form be associated with the L 2 metric as in the discussion following 8.1.4. namely II)
«u. u). ( t'. t'» f i'u dx - f UV dx. =
It is now an easy verification using integration by parts. that XII' II) and H are in the proper relation. so in this sense the wave equation is The Sobolev spaces H' defined this way are Hilbert spaces that arise in many problems involving partial differential equations. We only treat them informally here.
t
HAMIL TONIAN. MECHANICS
473
Hamiltonian. That the wave equation has a now on P follows from (the real form of) Stone's theorem (see Appendix D).
8.1.13 Example: The SchrOcllnger Equation Let P = :l( a complex Hilbert space with Co) = a self-adjoint operator with domain D and let
-
Im(.). Let Hetp be
X,,(cp)=iHap'cp and
H(cp) - (H•..,.cp.cp)/2.
cpE D.
Again it is easy to check that Co). X" and H are in the corrcct relation. Thus. XII is Hamiltonian. Note that ",(t) is an integral curve of XII if I d",
ldi =
H...,."'.
which is the abstract SchrOdinKer equation of quantum mCl'hanics. That X" has a now is another case of Stone's theorem. We know from general principles that the now e""'" will be symplectic. The additional structure needed for unitarity is exactly complex linearity.
Turning our attention to geodesics and to Lagrangiiln systems. let M be a (weak) Riemannian manifold with metric (.), on the tangent space T,M. The spray S: TM ..... T2M of the metric (.), is the vector field on TM defined locally by Sex. v) = «x. v).( v. Y( X.l'))). for (x . •,) E 7: M. where y is defined by (I) and D.(v. v) .. ·w means the derivative of (v. v).• with respect to x in the direction of .... If M is finite dimensional. the Christoffel symbols are defined by putting yi( x. v) - - rj. (x) viv·. Equation (1) is equivalent to
. i. I ilK". A il1(". . A - f' v'v w = - -v'v'w - -iliA -V'W'll' i.e .. ,£ , 2 ih: A •
The verification that S is well defined independent of the charts is not too difficult. Notice that y is quadratic in v. We will show below that S is the
474
APPLICA TlONS
Hamiltonian vector field on TM assOciated with the kinetic energy t (v, v). The projection of the integral curves of S to M are called geodesic.v. Their local equations are thus
.i=Y(X,x), which in the finite-dimensional case becomes i == I, . .. ,n.
The definition of y in (I) makes sense in the infinite as well as the finite-dimensional case, whereas the coordinate definition of riA makes sense only in finite dimensions. This then provides a way to deal with geodesics in infinite-dimensional spaces. Let t - (x(t), v(t» be an integral curve of S. That is,
x(i)=v(t) and V(t)=y(x(/),v(t)).
(2)
As we remarked. these will shortly be shown to he Hamilton's equations of motion in the ahsence of a potential. To include a potential. let V: M -+ R, M. and we be given. At each x. we have the differential of V, dV( x) E define grad Vex) by
r:
(grad V(x). ",). = dV(x )·w.
(3)
(In infinite dimensions, it is an extra assumption that grad V exists. since the map l'xM --. T:M induced by the metric is not necessarily bijective.) The equations of motion in the potential field V are given by
X(/) "" V(/);
V(/)
=
y(x( I), v(t»-grad V( X(/».
(4)
The total energy, kinetic plus potential, is given by H(v.)= !lIv.1I 2 + V(x).' The vector field X H determined by H and the symplectic structure on TM induced by the metric, is given by (4). This will be part of a more general derivation of Lagrange's equations given below.
BOX 8.18 GEODESICS Readers familiar with Riemannian geometry can reconcile the present approach to geodesics based on Hamiltonian mechanics to the standard one in the following way. Define the covariant derivative V: ~'\ ( M) X ~'( ( M ) ..... ~X ( M) locally by (VxY)(x) =Y.( X(x), Y(x»+ DY( x)·X('d.
HAMIL TONIAN MECHANICS
475
where X(x) and Y(x) are the local representatives of X and Y in the model space E of M and Y.. : Ex E.-. E denotes the symmetric bilinear continuous mapping defined hy polarization of the quadratic form y(x.v). In finite dimensions. if E=R". then y(x.v) is an R"-valued quadratic form on R" determined hy the Chri~loffel symbols fJ,c. Th~ defining relation for VxY heCllmes
where locally
a and ax' .
X-=X'-
a ax'
y ... y ' - .
It is a straightforward exercise to show that the foregoing definition of V x Y is chart independent and that V satisfies the following conditions:
(i) V 'is R-bilinear. (ii) for /: M.-.R smooth. V,xY=/VxY and Vx/Y=/VxY+ X(/)Y.
Moreover. (VxY- VyX)(x) =- DY(x),X(x)- DX(x)'Y(x) - [X. Y](x)
by the local formula for the bracket of two vector fields. If c( t) is a curve in M and X E ~X ( M). the cOt'llrianl tieritlative 0/ X along c is defined by
where c is a vector field coinciding with t( t) at the points Locally. using the chain rule. this formula becomes
('(1).
DX d -d (c(t»= -y,.(,,(X(c(t».X(c(t»)+-d X(dl». t
•
I
which also shows that the definition of DX/dt depends only on c(t) and not on how t is extended to a vector field. In finite dimensions. the coordinate form of the preceding equation is
where t( t) denotes the tangent vector to the curve al c( I).
476
APPLICA TIONS
The vector field X is called autoparallel or is parallel-transported along c if DX/dt- O. Thus i: is autoparallel along c irr in any coordinate system we have
or. in finite dimensions.
That is. i: is outoparal/el along c iff c is a geodesic. There is feedback between Hamiltonian systems and Riemannian geometry. For example, conservation of energy for geodesics is a direct consequence of their Hamiltonian character but can also be checked directly. Moreover. the fact that the flow of the geodesic spray on TM consists of canonical transformations is also useful in geometry, for example in the study of closed geodesics (cr. Klingenberg (1978)). On the other hand Riemannian geometry raises questions (such as parallel transport and curvature) that are useful in studying Hamiltonian systems. We now generalize the idea of motion in a potential to that of a Lagrangian system; these are. however. still special types of Hamiltonian systems. We begin with a manifold M and a given function L: TM -+ R called the Lagrangian. In case of motion in a potential, take L( v.) = !(v., v.. ) - V(x).
which differs from the energy in that - V is used rather than + V. The Lagrangian I- defines a map called the fiber derivative, FL: TM ..... T* M as follows: let t'. N' E T\ M, and set FL(v)'N'E .!{L(V+'N,)I dl ,-0 .
That is. FL(v)· ... is the derivative of L along the fiber in direction .... In the
c:ase of L(~.) == !( v... v.. ). - V(x). we see that FL( v. ) ...... = (v... ..... ) ... so we
..
recover the usual map ,b: TM -+ T* M associated with the bilinear form (.) As we saw before. r M carries a canonical symplectic form w. Using FL we obtain a closed two-form W,. on TM by setting
w,.... (FL )*w.
HAMIL TONIAN MECHANICS
477
A straightforward pull-back computation yields the following local formula ror w,.: if M is modeled on a linear space E. so locally TM looks like U x E where U c E is open, then (0),.( u, e) for (u, e) e U x E is the skew symmetric bilinear form on E x E given by Io)/.
(u. e )·«e •• e 2 ). (el' e4» "=
D.(D2L(u,e)·e.)·e3 - D.(D2L(u.e)·e3)·e.
+ D2( D2L( u.e )·e. )·e4 - D2( D2L( lI.e )·eJ )·e2'
(5)
where D. and Dz denote the indicated partial derivatives of L. In finite dimensions this reads
a 2L .
aiL.
w - -.-.dq'" dq'+ -.-dq'"
,.
aq'aq'
aq'8i/'
dq'.
where (qi, qi) arc the standard local coordinates on TQ. Il is easy to sec that w,. is (weakly) nondegenernte if /)! D2 L( II. e) is (weakly) nondegenerate. In the case of motion in a potential. nondegeneracy of (0),. amounts to nondegeneracy of the metric (.),. The action of L is defined by A: TM - R. A( v) = FL( v)·v. and the enerl{V of L is E = A - L. ,In charts.
and in finite dimensions E is given by the expression E(q.q)= 8L;l_L(q.J.).
aq'
.,
Given L. we say that a vector field Z on TM is a Lagrallgian. l'e(·tor field or a Lagrangian sy.stem for l. if the LaRranRian condition holds: (0),. ( v)( Z( v). ",) = dEC v)· It'
for allveT"M. and It'eT.:(TM). Here. dE denotes the differential of E. We shall see that for motion in a potential. this leads til th,,' ,arne equations of motion as we found before. If (0),. were a weak symplectic form there would exist lit most one sUl'h Z. which would be the Hamiltonian vector field for the Hamiltonian E. The dynamics is obtained by finding the integral curves of Z: that is. the curves
478
APPLICATIONS
I ..... v(t) E TM satisfying (dv/dl)(l) = Z(V(I». From the Lagrangian condition it is easy to check that energy is conserved '(even though L may be degenerate).
1.1.14 PropMltlon. IAI Z be a Lagrangian veclor field for L and leI, v(1)E TM be an inlegral curve of Z. Then E(v(l» is constant in I.
Proof. By the chain rule. d dl E( v(
I» = dE( v( I »..,'( I)
= dE( v( t ))·Z( v( I»
... w,.(v(t»(Z( V(I». Z(v(t» by the skew symmetry of w,..
=
0
•
We now want to generalize our previous local expression for the spray of a metric. and the equations of motion in the presence of a potential. In the general case the equations are called Lagrange's equations.
1.1.15 Propo8ItIon. IAI Z be a Lagrangian syslem for Land suppme Z is a second-order equalion (Ihal is. in a charI UxE for TM.Z(u.e)= (u. e. e, Z2( u •• for some map Z2: U X E ..... E). Then in Ihe charI U x E. an inlegral curve (u( I). v(l» E U X E of Z salisfies Lag"",p's ..."tioru: Ihal is.
»
{
~~ (I) = v( t )
».,.,
; (D2L( u( t). v( 1
= D. L( U(I). V(I
».,.,
(6)
for all ,., E E. If D2 D2 L or equil1alently w,. is weakly nondegenerale. then Z is aUlomatically second order and if D2 D2 L is nondegenerale. then
In case of motion in a potential. (7) reduces to the equations we found pRViously defining the spray and gradient.
Proof. From the definition of the energy E we have locally DE( lI,e )·(e •• e2) == D.( D2L(II, e )·e )·e. - D.L(u.e)·e.
+ D2( D2L(u. e )·e )·e2
HAMIL TONIAN MECHANICS
(a term
D2L(u'~)'~2
479
has cancelled). Locally we may write
Using formula' (5) for
WL'
the condition on Z may be written.
- D.( D2L( u,
D.(D2L.(u,~ )·Y.(u,~ ».~.
+ D2( D2L( u, ~). Y.( u,~ »'~2
~ ).~. ).y.( u,~)
- D2( D2L( U,~ ).~.). Y
= D.( DaL( U,~ )·e ).~. - D.L( u, e ).~.
2(
u,e)
+ Da( D2 l.( u, e ).~ )'e2'
(8)
Thus if w,. is a weak symplectic form, then D2 D2 L( u, e) is weakly nondegenerate, so setting e.=O we get Y.(u,e)=e; i.e.• Z is a second-order equation. In any case. if we assume that Z is second order. then condition (8) becomes . D.L(u.~)·e. = D.(D2L(u.~)·e.)·e + Da(D2L(u.e)·e. )·Y2(u.e).
t»
for all ~. E E. If (u( t)• .,( is an integral curve of Z and using dots to denote time differentiation, then;' = .., and U'" Y2(u, ;'), so D. L( u. ;,).~. = D. (D2L( u,;, ).~. ).;,+ D2( D2 [,( u. il)·e. )·u
=
!D
2 L(u,iI)'e.
by the chain rule. • The. condition of being second order is intrinsic; Z is second order iff TT,., 0 Z = identity. where T,.,: TM - M is the projection. See Exercise 8.10. In finite dimensions Lagrange's equations (6) take the form
dqi
=
Ii
dl
d(aL) aqi' aL
dt aqi
=
i= l ..... n
For L the kinetic energy of a Riemannian metric, these equations reduce to . the geodesic ~uations by a straightforward calculation. Exercl...
8.IA Let (M,w) be a symplectic manifold with w=d8 and/: M-M a local diffeomorphism. Prove that / is a symplectic iff for every
480
APPLfCA TIONS
compact oriented two-manifold B with boundary in M.
f
8-f
i/B
8.1 B
8.1 C
8
IIi/B)
Use the method of proof of Darboux's theorem to prove that if M is a compact manifold, I" and , are two volume forms with the same orientation. and
then there is a diffeomorphism f: M ..... M such that f*v = flo. (IIint: Since I", - I •. I" - , - da (see Box 7,SA): put v, "" tv +(1 - I)", and i x ,., = a. Let cp, be the flow of X, and set f = cpd . On T"R). consider the periodic three-dimensional Toda lattice Hamiltonian.
(i) Write down Hamilton's equations. (ii) Show that
(iii)
(iv)
are in involution and are independent everywhere. Prove the same thing for
Can you establish (iii) without explicitly computing the Poisson brackets? (Hint: Express g •• g2' g3 as polynomials of f •• f2' f,·) 8.10 (Second-order equations). A second-order equation on a manifold M is a vector field X on TM such that TTM 0 X = Id TM • Show that (i) X is a second-order equation iff for all integral curves c of X in TM we have (TM 0 c)' = c. One calls TM 0 C a base integral curoe.
FLUID MECHANICS
481
(ii) X is a second-order equation iff in every chart the local representative of X has the form (u.~) .... (u.~.~. JI(u.~». (iii) If M is finite dimensional and X is a second-order equation. then the base integral curves satisfy d 2x(t) --2- =
dt
8.1 E
~.I
F
. JI(x(t). x(t)).
where (x. x) denotes standard coordinates on TM. Prove the following Noether theorem: Let.Z be a Lagrangian vector field for L: T/II . ~ R lind suppose Z is a second-order equation. Let lit,. be a one-parameter group of diffeomorphi.mu of M generated by Ihe veclor field Y: M - TM. Suppose Ihal for each real number I. L 0 Tilt, ... L. Then the funclion P( Y): TM - R. defined by P( Y)(.,) - FL( t ) . Y ;s constanl along integral curves of Z. Use Exercise 8.1E to show conservation of linear (resp.. angular) momentum for the motion of a particle in R 3 moving in a potential that has a translation (resp.• rotational symmetry).
8.10 Consider R 2 ,,+2 with coordinates (ql ..... q".E.PI ..... p".t) and let W =.dqi 1\ dpi + dE 1\ dl. Consider a function P(". p. E. I) = I!(". p. 1).- E. Show that the vector field XI' = l/iJ/iJqi + p,aiap, + EiJ / iJl +,tiJ / iJ~ defined by ;](pw = dP reproduces familiar equations for q. P. I and E. 8.1 H Show that the wave equation (see Box 8.1A) may be derived as a Lagrangian system. 8.11 Refer to Example 8.1.13 on the SchrOdinger equation. Let A and B be self adjoint operators on :K and let fA: ~1( - R be given by fA("') = A",> (the expectation value of A in the state ",). Show that Poisson brackets and commutators are related hy
<"',
8.2· FLUID MECHANICS We shall present a few of the basic ideas concerning the motion of an ideal fluid from the point of view of manifolds and differential forms. The same thing is usually done in purely classical terms using vector calculus. For the latter approach and additional details, the reader should consult one of the sfandard texts on the subject such as Batchelor [1967). Chorin and
482
APPLICA TlONS
Marsden [1979]. or Gurtin [1981]. The use of manifolds and differential forms can give additional geometric insight. The present section is by necessity somewhat superficial and is intended only to show how to use differential forms and Lic deriv.llivcs in fluid mechanics. Once the basics are understood. more sophisticated questions can be asked. such as: In what sense is fluid mechanics an infinitedimensional Hamiltonian system? For the answer. see Arnol'd [1978]. Abraham and Marsden [1978]. Marsden and Weinstein [1983]. and Marsden. Ratiu. and Weinstein [1982]. For analogous topics in elasticity. see Marsden and Hughes [1983]. and for plasmas. see Marsden and Weinstein [1982]. Let M be a compact. oriented finite-dimensional Riemannian n-manifold. possibly with boundary. Let the Riemannian volume form be denoted '" e 0"( M). and the corresponding volume element dp.. Usually M is a bounded region with smooth boundary in two- or three-dimensional Euclidean space. oriented by the standard basis. and with the standard Euclidean volume form and inner product. Imagine M to be filled with fluid and the fluid to be in motion. Our object is to describe this motion. Let x e M be a point in M and consider the particle of fluid moving through x at time t = O. For example. we can imagine a particle of dust suspended in the fluid; this particle traverses a well-defined trajectory. Denote by cp,( x) = cp( x. t) the trajectory of the fluid particle that is at position x at time t = O. Let II(X. t) denote the velocity of the particle of fluid moving through x at time I. Thus. for each fixed time. II is a vector field on M. See Fig. 8.2.1. We call II the ()(!/oci(y field of Ihe fluid. Thus the relationship between II and cp, is
dcp,(x) -----;Jt
«(»
= II cp, X • 1 :
that is. II is a time-dependent vector field with evolution operator cp, in the same sense as was used in Section 4.1.
trajectory of nuid particle moving through x at time ( = O. .
Figure '.2.1
FLUID MECHANICS
483
For each ~ime I. we shall assume that the fluid has a well-defined mass density p/(x)=p(x.t). Thus if W is any subregion of M, we assume that the mass of fluid in W at time t is given by m(W,t)'=j p,dll w
Our derivation of the equations is based on three basic principles: I. 2. 3.
Mass is neither created nor destroyed. The rate of change of momentum. of a portion of the fluid equals the force applied to it (Newton's second law). Energy is neither created nor destroyed.
Let us treat these in turn. 1. Corueroa,;olf of mass. This principle says that the total mass of the fluid, which at time t - 0 occupied a nice region W remains unchanged after time t; i.e.,
(We call a region W "nice" when it is an open subset of M with smooth boundary.) Let us recall how to use the transport theorem (7.1.12) to derive the continuity·.equation. Using the change-of-variables formula, conservation of mass may be rewritten as
for any nice region W in M, which is equivalent to
where J( 'P/) is ·the Jacobian of CP"~ This in turn is equivalent
= 'Pi { (,,[p/J+ p/div" + ~),,) = 'Pi { ( div( p/" ) +
~ )" }
\0
484
A PPLICA TIONS
by the Lie derivative fOmHIla and 6.5.17. Thus
~ + dive P,u ) == 0 is the differential form of the law of conservation of mass. also known as the continuity equation.
Because of shock waves. p and u may not always be smooth enough to justify the steps leading to the differential form of this law; the integral form will then be the one to use. Also note that the Riemannian metric has as yet played no role; only the volume element of M was needed. 2. Balllllce 0/ momelltum. Newton's second law asserts that the rate' of change of momentum of a portion of the nuid equals the total force applied to it. To see how to apply this principle on a general manifold. let us discuss the situation MeR 3 first. The momentum of a portion of the nuid at time t that at time t = 0 occupied the region W is !.",(w)pudp.. Here and in what follows the integral is R 3-valued. so we apply all theorems on integration componentwise. For any continuum. forces acting on a piece of material are of two types. First there are forces of stress. whereby the piece of material is acted on by forces across its surface by the rest of the continuum. Second there are external. or body forces. such as gravity or a magnetic field. which exert a force per unit volume on the continuum. The clear formulation of surface . stress forces in a continuum is usually attributed to Cauchy. We shall assume that the body forces are given by a given force density b; i.e .• the total body forces acting on Ware! wpb dp.. In continuum mechanics the forces of stress are assumed to be of the form !awo(x. t )'11 do. where do is the induced volume element on the boundary. II is the outward unit normal. and o( x. t) is a time-dependent contravariant symmetric two-tensor. called the Cauchy stress tensor. The contraction o( t. X )'11 is understood in the following way: If (J has components 0') and II has components n A• then 0'11 is a vector with components (0'11)' = 8;Aai)nA' where g is the metric (in our case 8,~ = 6,4)' The vector 0 '11. called the Cauchy traction vector. measures the force of contact (per unit area orthogonal to II) hetween two parts of the continuum. t Balance of momentum is said to hold when ddt
1
'P,(W)
pudp.
=
1
",,(W)
pbdp. +
f
o·nda
iI",,(W)
t A theorem or Cauchy states that ir one postulates the existence or a continuous Cauchy traction vector T( x. I. ,,) satisfying balance or momentum. then it must be of the rorm G'" for a two-tensor G; moreover ir hahnn' or moment of momentum holds. G must be symmetric. See Chorin and "! u. >
FLUID MECHANICS
485
for any nice reaion Win M - RJ. If diver denotes the vector with compo. nents (div(erli).div(er2i).div(er3i». then by Gauss' theorem
1
er'lIda =
1
",,(WI
iI",,(WI
(diver) d".
By the change-of-variables formula and Lie derivative formula. we get d' -1 dt
"".WI
pu'dp. =
=
d fw -",.( pu' dp.) dt a(pu') . ) 1 (-a-+ ( L.p ) u' + pL.u' + pu'div u dp.. ",,(WI
I
so that the balance of momentum is equivalent to
~ u i + p aa~i + (dp' U )u i + pL.u' + pu'div" =
,)h'
But dp'" + pdiv" = div(p,,) and by conservation of mass. - O. Also. I •• u i - (iluilaxi)u i ... (,,·v )u i • so we get
~ (diver)' "pial +div(pu)
a. ,,·v,,'" b+i-d 'Iver p'
- + ill
which represents the basic equations 01 motion. Here the quantity a"l at + is usually called the material derivative and is denoted by D" I dt. These equations are for any continuum. be it elastic or nuid. An Ideal fluid is by definition a nuid whose Cauchy stress tensor er is given in terms of a function p(x. t) called the pressure, by a" = - p~". In this case. balance of momentum in differential form hecllmes the E,uler equalions lor an ideal fluid:
" .v"
au - + at
"·v,, = b -
I -grad p. p
The assumption on er in an ideal nuid means that if S is any fluid surface in M with outward unit normal II. then the force of stress per unit area exerted across a surface element S at x with normal" at time t i!. .. p(x. I)" (see Fig. 8.2.2). Let us return to the context of a Riemannian manifold M. First. it is not clear what the vector-valued integrals should mean. But even if we could make sense out of this. there is a considerably more serious problem with
486
APPLICA nONS
s
• force across S is - JIll.
Figure 8.2.2
the integral form of balance of momentum as stated. Namely, if one changes coordinates, then balance of momentum does not look the same. One says that the integral form of balance of momentum is not covariant. Therefore we shall concentrate on the differential form and from now on we shall deal only with ideal fluids. t Rewrite Euler's equations in R) with indices down; i.e., take the flat of these equations. Then the i th equation, i = 1,2.3 is
au, au, au, au, I ap -+u -+u --+u -=b - - at I axl 2 ax 2 3 ax) , p a."(' We seek an invariant meaning for the sum of the last three terms on the left-hand side. For fixed i this expression is
That is. Euler's equations can be written in the invariant form
aub + L
-
at
I
•
I
ub- -d{ub(u») = - -dp + bb 2 p
We postulate this equation as the balance of momentum in M for an ideal fluid. The reader familiar with Riemannian connections (see Box 8.1 B) can t For a detailed discussion of how to fonnulate the basic integral balance laws of continuum mechanics covariantly. see Marsden and Hughes (\983].
FLUID MECHANICS
487
easily prove that this form is equivalent to the form
au
at + v•u =
-
-
I -grad p + b p
by showing that
The boundary conditions that should be imposed come from the physical significance of ideal fluid: namely. no friction should exist between the fluid and aM; i.e.. u is parallel to aM on ,1M. Summarizing. the equations of motion of an ideal fluid on a compact Riemannian manifold M with smooth boundary aM and outward unit normal n are
a:, + dive pu ) = 0 We also have boundary conditions:
uliaM. i.e:. u'" = 0 on aM; and initial condition.r:
,,(x.O)=uo(x) givenon M. In many applications b == 0 and we shall assume this from now on for simplicity. . 3. COIIHI'WI,iOll oj ewrgy. A basic problem of ideal fluid dynamics is to solve the initial-boundary-value problem. The unknowns are u. P. and p. i.e.• n + 2 scalar unknowns. We have. however. only n + I equations. Thus one might suspect that to specify the fluid motion completely. one more equation is needed. This is in fact true and .the law of conservation of energy will supply the necessary extra equation in fluid mechanics. (The situation is similar for general continua; see Marsden and Hughes [19113].) for a fluid moving in M with velocity field II. the kinetic energy of the fluid is
where '1111 2 == (II.
"> is the square length of the vector fu!"ction u. We assume
488
APPLICATIONS
that the total energy of the nuid can be written Elolal
= Ekinelic + EinlcmaJ'
where EinlcmaJ is the energy that relates to energy we cannot "see" on a. macroscopic scale and derives from sources such as intermolecular potentials and molecular vibrations. If energy is pumped into the nuid or if we allow the nuid to do work. EIOIal will change. Next we describe two particular examples of energy equations that are useful. A. Assume that Einlernal'" constant. Then we ought to have EkilWlic as a constant of the motion; i.e.•
To deal with this equation it is convenient to use the following. 8.2.1 Transport Theorem with Ma.. Density. Let I he a time-dependent smooth lunction on M. Then il W is any nice open set in M.
where DI/ Dt =
a1/ at + L.f.
Proof. By the change-or-variables formula. the Lie derivative formula. div(pII) = II(pJ+ pdiv(II). and conservation of mass, we have
... fw"'*( a(;[) "+ L.( PI"»)
- {CWI( ~ I" + P :{" + II[p]/" + pL./" + pIL.,,) .. {,CWI{ (~ + II[p]+ pdivII )/1' + p( ~~ + L./)I'} =
{,CWI{ I( ~ +diV(PII»)+ p( :{ + L.I )}"
=
1",unp(
.
CIaI
t
+ L.I )". •
· FLUID MECHANICS
489
il. Using L.OII,2) ""' L~(III/(II» = (L."I/)(II) = d(III/(II»("), the transport " lemma. and Euler's equations, we get
= - l,.,dP ·,lId". =
(Lcihnil rule for I.• )
I,.,{(div II )PIL - L.( pP.n
I,.,{( div II) pp. - d('i. pp.)} . , f,.,(div,,)pp..
(L.
=
=
dill
+ i.d)
The last equality is obtained by Stokes' theorem and the houndary conditions 0 = (".,,) do = i.p.. If we imagine this to hold for the same fluid in all conceivable motions. we are forced to postulate one of the additional equations div" ,.. 0 or p - o. The case div" = 0 is that of an incompressible fI"id. Thus in this case the Euler equations are all b
- at + L
II
"b -
I
I
2 ... - -dp -dl"1 2 p
ap + dp'" =
at
0
div,,=O with the boundary condition i.p. = 0 on aM and initial condition II(X.Q) = The case P -= 0 is also possible but is less interesting. For a homogeneous incompressible fluid. with constant density p. Euler's equations can be refOf''''' '1 led in terms of the Hodge decomposition theorem (see section 7.4),t ; .hat anyone-form a can he written in a
.II( x).
! Nonhomogeneous incompressible now requires a weighted Hodge decomposition (see Marsden (1976).
490
APPLICA TIONS
unique way as CI- lIP + y, where 8y .. O. Define the linear operator P: OI(M) -+ (y e Ol( M)lay - 0) by P(CI) - y. We are now in a position to reformulate Euler's equations. Let O~_o be the set of Coo one-forms y with ay" 0 and tangent to aM; i',e.• • ylilM = O. Let T: 01_0 -+ 01_0 be defined by
P(L."b) Thus Euler's equations can be written as a"bI iJt + T("b) = 0, which is in the T("b) =
"standard form" for an evolution equation. Note that T is nonlinear. Another important feature of T is that it is nonloeal; this is because P( 11)( x) depends on the values of II on all of M and not merely those in the neighborhood of x e M. B. We postulate an internal energy over the region W to he of the form Eintemal =
1. pK'dp. If'
where the function w is the internal energy density per unit mass. We as...ume that energy is balanced in the sense that the rate of change of energy in a region equals the work done on it:
:,r (1
.,,(W)
~1"12dp + 1
.",(W)
PWdP ) =
1a",(w)p"·,,da.
By the transport theorem and arguing as in our previous re...ults. this reduces to 0=
. DDW) dp. 1 (pdlv"+P t ",(It')
Since W is arbitrary, we get the identity
Dw 0 · P d IV" + PDt = . Now assume that W depends on the fluid motion through the density; i.e., the internal energy depends only on how much the fluid is compressed. Such a fluid is called ideal isentropic. The preceding identity then becomes 0= pdivu + p(
.
=
~; + "..,.,,)
aw
pdlv" + P ap
= pdiv" + P
CJp
aw
at + pTpdp '"
~: ( -
pdiv,,)
FLUID MECHANICS
491
using the .equation of continuity. Since this is an identity and we are not restricting div II. we get
If p is a given function of p note that w = - fp d(l/p). Another simple identity that will be useful below is that dp /p == d(,., + p /p). This follows from p'" p2,., by a straight forward calculation in which p and ,., are regarded as functions of p. Thus Euler's equations for compressible ideal isentropic now are
aaP + div( pu ) t
II(X.0)=1I0(X) onM
.
=
0
and U'II=O on
aM.
where p = p2 w'( p) is a function of P. called an equation of state. which depends on the particular fluid. It is known that these equations lead to a well-posed initial value problem (i.e.• there is an existence and uniqueness theorem) only if p'(p) > O. This agrees with the common experience that increasing the surrounding pressure on a volume of fluid causes a decrease in occupied volume and hence an increase in density. Many gases can often be viewed as satisfying our hypotheses. with p = ApT where A and y are constants and y ~ I. Cases A and B above are rather opposite. For instance. if p == Po is a constant for an incompressible fluid. then clearly p cannot be an invertible function of p. However. the case p'" constant may be regarded as a limiting case p'( p) -0 00. In case B. p is an explicit function of p. In case A. p'is implicitly determined by the condition div II == O. Finally, notice that in neither case A ~r B is the possibility of a loss of total energy due to friction taken into account. This leads to the subject of viscous fluids. not dealt with here. Given a fluid flow with velocity· field lIe x. t). a streamline at a fixed time is an integral curve of II; i.e.• if xes) is a streamline parametrized by s at the instant t. then xes) satisfies dx
ds =u(x(s),t).
t fixed.
On the other hand. a trajectory is the curve traced out by a particle as time
492
APPLICA TIONS
progresses, as explained at the beginning of this section: i.e., is a the differential equation dx dt =u(x(t),t)
~olution
of
with given initial conditions. If u is independent of t (Le., au/at = 0), then, streamlines and trajectories coincide. In this case, the flow is called stationary or steady. This condition means that the "shape" of the fluid flow is not changing. Even if each particle is moving under the flow, the global configuration of the fluid does not change. The following criteria for steady solutions for homogeneous incompressible now is a direct consequence of Euler's equations, written in the form au~ / at + P( L.u~) = 0, where P is the Hodge projection to the co-closed I-forms.
'.2.2 Proposition. Let u, be a solution to the Euler equations for homogeneous incompressible flow on a compact manifold M and cP, its flow. Then the following are equivalent: (i)
_, is a steady flow (i.e., (a_/at),. 0).
(ii) cp, is a one-parameter group: CPt+. == cP, 0 CPs' (iii) is an exact I-form. (iv) is an exact I-form.
L.o-t ;.o""t
It follows from (iv) that if U o is a harmonic vector field; i.e.•
Uo
satisfies
aut == 0 and ""t '"' O. then it yields a stationary flow. Also. it is known that
there are other steady flows. For example. on a closed two-disk, with polar coordinates (r.O).u==f(r)(a/aO) is the velocity field of a steady flow because u·Vu=-vp.
where p(r,O)= io'f 2 (s)sds.
Cleatiy such a _ need not be harmonic. We have seen that for compressible ideal isentropic flow. the total energy fM( ipl_1 2 + pw)tip is conserved. We can refine this a little for stationary flows as follows.
1.2.3 Bernoulli'. Theorem. For stationary compressible ideal isentropic flow. with p a Junction of P.
ilul 2 + f!!£. p
==
ilul 2 + w + I!..p
is constant along streamlines where f dp / f' = w + p / p denotes a potential for the one form dp/p. The same hold.f for statIOnary' homogeneous (p == constant
FLUID MECHANICS
·493
in space'" Po) incompressible flow with Jdp /p rep/aced by p /Po. If body forces deriving from a potential U are present i.e .. bb ... - dUo then the conserved . qutlntity is
Proof. Since (L."b)." =
d("b(,,».,,:
for stationary ideal compressible or incompressible homogeneous nows we have
so that
=
iJ"b I.s, _·,,(x(s»ds=O iJs S2
since x'(s)'" ,,(x(s». • The two-form II) = tbl b is called vorticity. (In R) we can identify II) with curl ".) Our assumptions so far have precluded any tangential forces and thus any mechanism for starting or stopping rotation .. Hem:e. intuitively. we might expect rotation to be conserved. Since rotation is intimately related to the vorticity. we can expect the vorticity to be involved. We shall now prove that this is so. . Let C be a simple closed contour in the nuid at t = 0 and let C, be the contour carried along the now. In other words.
where fIi, is the nuid now map. (See Fig. 8.2.3.) The circulation around C, is defined to be the integral
494
APPLICATIONS
c,
-,----...., Figure •• 2.3
8.2.4 Kelvin Circulation Theorem. Let M be a manifold and I c M a smooth closed loop. i.e., a compact one-manifold. Let II, solve the Euler equtltions on M for ideal isentropic compressible or homogeneous incompressible flow and I( t) be the image of I at time t when each particle moves under the flow 'P, of "'; i.e., 1(1) = 'P,(/). Then the circulation is constantin time; i.e .•
df
-
dt
IIb=O
1(/1'
•
/'roof. Let 'P, be the now of ",. Then I(I)-'P,(I). and so changing variables.
which becomes, on carrying out the differentiation,
However, L."b + a"b/ at is exact from the equations of motion and the integral of an exact form over a closed loop is zero. • In practical nuid mechanics. this is an important theorem. One can obtain a lot of qualilative information about specific nows by following a • closed loop in time and using the fact the circulation is constant. We now use Stokes' theorem. which will bring in the vorticity. If E is a surface (a two-dimensional submanifold of M) whose boundary is a closed
FLUID MECHANICS
495
"
Figure 8.2.4 contour C. then Stokes' theorem yields
See Fig. 8.2.4. Thus. as·.a corollary of the circulation theorem. we can conclude:
8.2.5 Helmholtz' Theorem.
Under the hypotheses of 8.2.4, the flux of vorticity across a surface moving with the fluid is constant in time. We shall now show how 10) and'll = Io)/p are Lie propagated by the flow.
8.2.8 Propoaltlon. For isentropic or homogeneou.r incompressible llow. we have (i)
iIlo)
-+L 10)=0 and
al
•
ii'll +L ... - ... div(u)=0
at
,,""
called the vorticity-stream equatiOll and
(ii) where 'II,(X) = 'II(x. t) and J( 'P,) is the Jacobian of 'P" . Proof. Applying d to Euler's equations for the two types of nuids we get , the vorticity equation: iIlo)
iii + L.Io)=O.
Thus
-a'll + L ilt
•
'II = -I ( -alo) + L 10) ) - -10) ( -iIp + dp' u ) p ilt • p2 ilt = - -'II (il -p
P iIt
='II divu by conservation of mass.
+ d P . u + Pd'IV.II) + 'II d'IV U
496
A PPLICA TlONS
From aw/at + L.w - 0 it follows that (a/at)(tp;ow,) - Wo. Since tp;op, - Po/J(tp,) we also get tp,."" = J(tp,)",o' •
0 and so tp,·w,
In three dimensions we can associate to '" the vector field equivalently) ie!' ==",. Thus t = (curl ,,)/p.
t=
*'" (or
'.2.7 Corolla". If dim M - 3. then t is transported as a wctorby tp,: i.e .•
t, = tp,.to or Proof. co
t, (tp,( x») =
Dtp, (x H, (0).
tp;o1l, = J( tp, )110 by 8.2.6, so tp;oi,,!' = J( tp, )i,,,!'. But tp;oi,,!' = ;"~,,tp;o!, Thus i.,~,,!, = i,,, .... which gives tp,·t, = to. •
i,,~e/( tp, )....
Notice that the vorticity as a two-form is Lie transported by the flow but as a vector field it is vorticity /P. which is Lie transported. Here is another instance where distinguishing between forms and vector fields makes an important difference. The flow tp, of a fluid plays the role of a configuration variable and the velocity field" plays the role of the corresponding velocity variable. In fact. to understand fluid mechanics as a Hamiltonian system in the sense of Section 8.1. a first step is to set up its phase space using the set of all diffeomorphisms tp: M --+ M (volume preserving for incompressible flow) as the configuration space. The references noted at the beginning of this section carry out this program (see also Exercise 8.21). Exercl...
"·V"
In classical texts on fluid mechanics, the identity = ! v(,,· ,,)+ x ")X,, is often used. To what does this correspond in this section? 8.2B A flow i!o ~alled potential flow if "b = dIP for a function tp. For (not necessarily stationary) homogeneous incompressible or isentropic flow prove Bernoulli's law in the form aIP/at+11"12+ Jdp/p= constant on a streamline. 8.2C Complex variables texts "show" that the gradient of tp(r. 8) = (r + l/r)cos8 describes stationary ideal incompressible flow around a cylinder in the plane. Verify this in the context of this section. 8.2D Translate 8.2.2 into vector analysis notation in R J and give a direct proof. 8.2E Let dim M = 3. and assume the vorticity w has a one-dimensional kernel. (i) Using Frobenius' theorem. show that this distribution is integrable. (ii) Identify the one-dimensional leaves with integral curves of t (see 8.2.7)-these are called l'Ortex line.~. (iii) Show that vortex lines are propagated by the flow. 8.2A
(V
FLUID MECHANICS
497
8.2F
Assume dim M - 3. A vortex tube T is a closed oriented two-manifold in· M that is a union of vortex lines. The .~trenRth of the vortex tube is the nux of vorticity across a surface }: inside T whose boundary lies on T and is transverse to the vortex lines. Show that vortex tubes are propagated by the now and have a strength that is constant in time. 8.2G Let f; R) -+ R be a linear function and g: S2 -+ R its restriction to the unit sphere. Show that dg gives a stationary solution of Euler's equations for now on the two-sphere. 8.2H (Stream Functions) (i) For incompressible now in R 2, show that there IS a function ~ such that u l = iNliJy and u 2 = - iJI/Jlih. One calls ~ the stream function (as in Batchelor (1967» (ii) Show that if we let • I/J = I/J dx 1\ dy be the associated two form. then u b = ,; • y,. (iii) Show that u is a Hamiltonian vector field (see Section 8.1) with energy I/J (i) directly in R 2 and then (ii) for arbitrary twodimensional Riemannian manifolds M. (iv) Do stream functions exist for arhitrary fluid flow on T 2? on S2?
8.21
(v) Show that the vorticity is '" = fl. I/J. (Clebsch Variables; Clebsch (1359]). Let '!tbe the space of functions on a compact manifold M with the dual space ,?r*. taken to be densities on M; the pairing between f E ~ and p E \~* is (f. p) =
/"dp·
(i) On the symplectic manifold ':t x \~* x \'f x \'f* with variables (a. J\. p.. p). show that Hamilton's equations for a given Hamiltonian Hare . 8H ~= - 8a . wher~
. 8H p= - 8p. .
6H 16~ is the functional derivative of H defined by 6H . . ( 8~ ,~) = DH(~)'~'
(ii)
In the ideal isentropic compressible nuid equations: set M
=
pub, the momentum density. If dx denotes the Riemannian volume form on M, identify the density o( x) dx E ~* with the
function O(X)E~~ and write M=-(pdp.+~da)dx. For momentum densities of this form show that Hamilton's equations for the variables (a, ~,p., p) imply the Euk'r equation and the equation of continuity.
498
APPLICA TIONS
8.3 ELECTROMAGNETISM Classical electromagnetism is governed by Maxwell's field equations. The form of these equations depends on the physical units chosen. and changing these units introduces factors like 4fT, c = the speed of light. EO = the dielectric constant and Po = the magnetic permeability. This discussion assumes that EO' Po are constant; the choice of units is such that the equations take the simplest form; thus c = EO = Po = I and factors of 4fT disappear. We also do not consider Maxwells equations in a material, where one has to distinguish E from D. and B from H. Let E. B, and J be time dependent CI-vector fields on R J and p: R} x R ..... R a scalar. These are said to satisfy Maxwell's equation.f with charge density p and current density J when the following hold: divE = p
(Gauss's law)
(I)
divB = 0
(no magnetic sources)
(2)
(Faraday's law of induction)
(3)
(Ampere's law)
(4)
aB at
curIE+-=O
BE
curIB--=J
at
E is called the electric field and B the magnetic field. The quantity Igp dV = Q is called the charge of the set Sl c R 3. By the classical Gauss theorem, (I) is equivalent to
{ E'ndS= IpdV=Q
Jill!
(5)
1/
for any (nice) open set Sl c R'; i.e .. the electric flux out of a dosed slIrface equals the total charge inside the .~Ilrface. This generalizes Gauss' law for a point charge discussed in Section 7.3. By the same reasoning. Eq. (2) is equivalent to
(6)
{ B·IIdS=O. Jill!
That is. the magnetic flux out of any closed surface is zero. In other words there are no magnetic sources inside any closed surface. By the classical Stokes theorem, (3) is equivalent to
/, E·ds = l(curIE)'"dS = as
s
~ at iB,"dS
(7)
ELECTROMAGNETISM
499
for any closed loop as bounding a surface S. The quantity !,1SE·tb is called the voltage around as. Thus, Faraday's law of induction (3) says that the voltage around a loop equals the negative of the rate of change of the magnetic flux through the loop. Finally, again by the classical Stokes theorem, (4) is equivalent to
1 il·tb= !.(curIB)'nds=: !.E.ndS+ !.J.ndS. as
t s
s
s
(8)
Since !sJ'ndS has the physical interpretation of currelll, this form states that if E is constant in time, then the magnetic potential difference !;lS B· d, around a loop equals the current through the loop. In general. if E varies in time. Ampere's law states that the magnetic potential difference around a loop equals the total current in the loop plus the rate of challge (If electric flux through the loop. We now show how to express Maxwell's equation!> in terms of differential forms. Let M = R4 -= {(x. y, z. t)} with the Lorentz metric If on R4 of diagonal form (1.1.1. -I) in standard coordinates (x. y. z,l).
8.3.1 PropolHlon. There is a unique two-form F on R 4, ('alled the Faraday two-form such that (9) Bb= - iii. F.
(10)
ii,
(Here the b iJ Euclideall ill R 3 and the. iJ Lorentzi~n in R4.) Proof
If
F= F,\.dx
1\
+ F"dx
dv 1\
+ ~,dz 1\ dx + F..,dy 1\ dz
dt + Fv,dy
1\
dt + ~,dz
1\
dt,
then (see Example 6.2.14E)• • F = F,\.dz " dt + F,..,dy " dt + Fv:dx
1\
dt
- F"dy" dz - F..,dz" dx - F,.,dx" d.l'
and so - if,F= F"dx
+ F,.,dy + F,.,dz
500
APPLICA TIONS
and
i~
-
• F= Ftldz +
ii,
~,d.v
+ F,.:dx
Thus, F is uniquely determined by (9) and (10), namely F,.., E' dx
1\
dt
+ E 2 d.v 1\ dt + E 3 dz
+ B) dx 1\ dy + B2 dz
1\
1\
dt
dx + B' dy
1\
dz.
•
We started with E and B and used them to construct F, but one can also take F as the primitive object and construct E and B from it using (9) and (10). Both points of view are useful. Similarly. out of p and J we can form the source one -form j = - pdt + J. dx + J 2 dy + J) dz; i.e.,j is uniquely determined by the equations - iil/il') .. p and iil/il,. j = • Jb; in the last relation. J is regarded as being defined on R4.
'.3.2 Propoeltlon.
Maxwell's equations (1)-(4) are equivalent to the
equations
dF= 0 and 3F= j on the manifold R4 endowed with the Lorentz metric.
A straightforward computation shows that
Proof.
dF ... (curl E +
~~), d.v 1\ dz 1\ dt + (curl E + ~~) ,. dz 1\ dx 1\ dt
+ (curl E +
a;:): dx
1\
dy
1\
dt
+ (div B) dx 1\ d.v 1\ dz.
Thus dF = 0 is equivalent to (2) and (3). Since the index of the Lorentz metric is I, we have Ii = • d •. Thus 3F= .d. F= .d( - E,dy 1\ dz - E 2dz
1\
dx - E)dx
+ B. dy 1\ dt + B2 d.r 1\ dt + B)dz = • [ -
(div E) dx
+ (curl B =
(curl B -
1\
dy
1\
dz + ( curl B -
~~
dy
dy
1\
dt)
1\
dz
1\
dt +
~~ ),. dz 1\ dx 1\ dt + (curl B - ~~): dx 1\ 4y 1\ dt]
aE) dx + (curl B - aE) dy at .. ilt ...
+ (curl B -
L
1\
~~): dz -
Thus 3F= j iff (I) and
(4)
(div E) dl
hold. •
ELECTROMAGNETISM
501
As a skew matrix, we can represent F as follows
F-
x
y
[- ~,
8
82 -£'
z 3
0
-8' -£2
- 82 8' 0
- £3
x
E'l £2
Y
£3
;:
0
Recall from Section 6.5 and Exercise 7.5G, the formula
Since Idetgl = I. Maxwell's equations can be written in terms of the Faraday two-form F in components as
(II) and F'·'A
(12)
= -}'.
where F".A ,., 8F,,/8x·, etc. Since a2 ... 0, we obtain
0= a21" = 6j = = • [(
•
d • j = • d( - p dx /\ dy /\ dz + ( • j~) /\ dt)
a;: +diVJ) dx /\ dy'/\ dz /\ dt] = a;: +divJ;
a,
i.e.• 8p/ +divJ = 0, which is the continuity equation (see Section 8.2). Its integral form is, by the classical Gauss theorem dQ dl
=~ipdV=f dt n
J·ndS
,III
for any bounded open set D. Thus the continuity equation says that the flux of the currenl density oul of a closed surface equals the ratc of change of the lotal charge inside the surface. Next we show that Maxwell's equations are Lorentz invariant, i.e., are special-relativistic. The Lorentz group ~ is by definition the orthogonal group with respect to the Lorentz metric If; i.e.,
502
APPLICA TIONS
Lorentz invariance means that F satisfies Maxwell's equations withj iff A*F satisfies them with A*j. for any A E e. But due to Proposition 8.3.1 this is clear since pull-back commutes with d and orthogonal transformations commute with the Hodge operator (see Exercise 6.2D) and thus they commute with 8. As a 4 x 4 matrix, the Lorentz transformation A acts on F by F ...... A* F "" AFAT. Let us see that the action of A E f mixes up E's and B's. (This is the sOurce of statements like: "A moving observer sees an electric field partly converted to a magnetic field.") Proposition 8.3.1 defines E and B intrinsically in terms of' F. Thus, if one performs Ii Lorentz transformation A on F. the new resulting electric and magnetic fields, E' and B' with respect to the Lorentz unit normal A *( a/ at) to the image A (R 3 x 0) in R 4 are given by
(E,)b - - i,. • .!..A*F, 1(B,)b - - i,. • .!.. *A*F ii,
ii,
For a Lorentz transformation of the form x'== {
x-l'I .• r'= l'.Z'=Z.
1- v 2
'
,
t,'= {1-ll . '2 t'\'
(the special-relativistic analogue of an observer moving uniformly along the' x-axis with velocity v) we get
E2 - vB 3 E3 + vB2 ) E'= ( E'
. ';1- v 2
'
';1- v 2
and
We leave the verification to the reader. By the way we have set things up, note that Maxwell's equations make' sense on any Lorentz manifold: i.e., a four-dimensional manifold with a pseudo Riemannian metric of signature ( + , + . + . - ). Maxwell's vacuum equations (i.e.• j = 0) will now be shown to be conJormally invariant on any Lorentz manifold ( M. ,). A diffeomorphism cp: ( M.,) .... (M.,) is said to be conformal if 'cp*, -f'" for a nowhere vanishing function f.
8.3.3 PropoaHlon.t Let FE 02( M). where (M.,) is a Lorentz manifold, salisfy dF = 0 and 8F = j. LeI cp be a conformal diffeomorphism. Then cp* F tSce Fulton. Rohrlich. and Witten (1962) for a review of conformal invariance in physics and the original literature references.
ELECTROMAGNETISM
503
satisfies drp*F= 0 and arp*F= f2rp*j. Hence Maxwell's vacuum equations (with j == 0) are conformally invariant; i.e. if F satisfies them, so does cp* F.
Proof. Since rp* commutes with d. dF = 0 implies d'P* F = O. The second equation implies rp*&F= rp*j. By Exercise 7.5H. we have 8q .• lI'rp*P = rp*l\P. Hence &F= j implies &"'ttrp*F= rp*j= 8"II'
shows that when one replaces g by f2g. \\e get
an
Let us now discuss the energy equation for the electromagnetic field. Introduce the energv density of the field by ~; = !(E·E + B·B). and the Poynting energy -flux vector S = E X B. Poynting's theorem statelr that
at.;
- -
at
=divS+ E·J
This is a straightforward computation using (3) and (4). We shall extend this result to R 4 and. at the same time. shall rephrase it in the framework of forms. Introduce the stress-energy-momentum tensor (or the Maxwell stre.H tensor) T by ( \3) (or intrinsically, T = F-F -1 (F. F)g, where F· F denotes a single contraction of F with itself). A straiglttforward computation shows that the divergence of T equals
F;,/
where = fa F;,q/ axle )g,le. Now taking into account 8F = j written in the form (12). it follows that (14)
504
APPLICATIONS
For; - 4, the relation (14) becomes Poynting's theorem.t It is clear that Tis a symmetric 2-tensor. As a symmetric matrix,
[
(I
T= (EXB)T
IEx B] f,
where (I is the stress tensor and f, is the energy density. The symmetric 3 x 3 matrix (I has the following components all
==![ _( EI)2_ (8 1)2+ (E2)2+ (82)3 +(E3)2+( 83)2]
17 22
==!l< EI )2+ (8 1)2 _ (£2 )2_ (82)2 + (£3)3 + (8·\)2]
1733
== ! [( £1)2 + (81)2 + (E2)2 + (82)2 _ (8.1)2 _ ( 8 3)2]
+ 8 18 2
17 12 ...
EIE2
all ...
EIE3 + 8 18 3
a 2l == E 2£3
+ 8 28 3 •
We close this section with a discussion of Maxwell's equations in terms of vector potentials. We first do this directly in terms of E and B. Since div B = 0, if B is smooth on all of A 3, there exists a vector field A, called the tlector potential, such that B == curl A, by the Poincare lemma. This vector field A is not unique and one could use equally well A' ... A + grad f for· some (possibly time-dependent) function f: A 3 ... A. This freedom in the choice of A is called gauge free·dom. For any such choice of A we have hy (3),
iI (ilA) o == curl E + -aB at = curl E + -at curl A = curl E + -a, , so that again by the Poincare lemma there exists a (time-dependent) function tp: A} ... A such that ilA E+Tt=-gradtp.
(8)
tPoynting's theorem can aJso be understood in terms of a Hamiltonian formulation-see Box S.3A below. The Poynting energy-nux vector I~ the Noether con- . served quantity for the action of the diffeomorphism group of R) on T*~, where ~ is the space of vector potentiaJs A defined in the following paragraph, and Poynting's theorem is just conservation of momentum (Noether's theorem). We shaJl not dwell upon these considerably more advanced aspects and refer the interested reader to Abraham and Marsden (IIJ78).
ELECTROMAGNETISM
505
Recall that the Laplace-Beltrami operator on functions is defined by V 21 - div(grad On vector fields in R 3 this operator may be defined componentwise. Then it is easy to check that
n.
curl( curl A) = grad( div A ) - V 2A . Using this identity. (8). and B = curl A in (4). we get J= curlB - -aE = curl(curl A)- -a (aA - -grad 'P )
at
a,
a,
. alA t1 =grad(dlvA)-3 2A + 2 + ~(gradcp).
a,
and thus
"t
( a-:l).
a2A
v 2A - -
= - J+grad divA +
a,2
rr, .
,at
( 16)
From (I) we obtai"n as before
a, -gradcp)=-v2cp-~(diVA) . a,'
p=diVE=div(- aA that is.
V 2cp = - P - :, (divA).
or. subtracting
a21(>/fIt 2 from hoth sides. V
a (divA .
2 fI21(> I(> - - , = -
p- -
at"
at
al(>- ). + -.
at
It is apparent that (16) and (17) can be considerably simplified if one could choose. using the gauge freedom. the vector potential A and the function cP such that divA +
~~ =0.
So. assume one has Ao. CPo and one seeks a function f such that A = An + gradl and cp='Po-afla, satisfy divA + acp/al=o. This becomes. in terms of I. 0= div( At) + grad f) + :, ( CPo _ . acpt) 2 - divA" + -a + V 1I'
a 1.
2 -2 •
at
:~)
506·
APPLICATIONS
( 18)
i.e.,
This equation is the classical inhomogeneous wave equation. The homogeneous wave equation (right-hand side equals zero) has solutionsf(t. x. y, z) -1/I(x - t) for any function 1/1. This solution propagates the graph of 1/1 like a wave-hence the name wave equation. Now we can draw some conclusions regarding Maxwell's equations. In terms of the vector potential A and the function cpo (I) and (4) become
( 19)
which again are inhomogeneous wave equations. Conversely. if A and cp satisfy the foregoing equations and divA + acp / at = 0, then E = - grad cp aA / at and B - curl A satisfy Maxwell's equations. Thus in R 4, this procedure reduces the study of Maxwell's equations to the wave equation, and hence solutions of Maxwell's equations can be expected to be wavelike. We now repeat the foregoing constructions on R4 using differential forms. Since dF- 0, on R4 we can write F= dG for a one-form G. Note that F is unchanged if we replace G by G + df. This again is the gauge freedom. Substituting F .. dG into 8£ = j gives 8dG = j. Since 6 = d8 + 8d is the Laplace-DeRham operator in R 4 , we get 6G = j - d8G. (20) Suppose we try to choose G so that 8G = 0 (a gauge condition). To do this, given an initial Go' we can let G = Go + df and demand that
0= 8G = 8Go + 8df = 8Go + 6f so f must satisfy 6f = - 8Go. Thus, if the gauge condition
6f= -8Go
(21 )
holds, then Maxwell's equations become
6G=j.
G-
(22)
Equation (21) is equivalent to (18) and (22) to (19) by choosing Ab + "dl (where b is Euclidean in R 3 ).
EL ECTRt;JMAGNE TISM
507
BOX B.3A MAXWELL'S VACUUM EQUATIONS AS AN INFINITE-DIMENSIONAL HAMILTONIAN SYSTEM
We shall indicate here briefly how the dynamical pair of Maxwell's vacuum equations (3) and (4) with J = 0 are a Hamiltonian system (see Box 8.1 A). (A proper understanding of the "constraints" div E = p.divB = 0 requires a procedure called reduction; we refer the interested reader to Marsden and Weinstein (1982].) As' the configuration space for Maxwell's equations. we tak,e the space ~I of vector potentials. (In more general situations. one should replace ~I hy the set of connections on a principal hundle over configuration space.) The corresponding phase space is then the contangent bundle .,..~ with the canonical symplectic structure. Elements of .,..~ may be identified with pairs (A. Y) where Y is a vector field density on R). (We do not distinguish Yand Yd.t.) The pairing between A's and Y's is given by integration. so that the canonical symplectic structure Co) on .,..~ is given by
with associated Poisson bracket (2) where SF/SA is the vector field defined by D... F(A. Y)·A'=
f :~ ·A'dx
with the one-form SF/SY defined similarly. With the Hamiltonian (3) Hamilton's equations are easily Computed to be
ay =
-
a,
-curlcurlA
and
aA
-
at
... Y.
(4) .
If we write B for curl A and E for· - Y. the Hamiltonian becomes the
508
APPLfCA TIONS
usual field energy (5)
Equation (4) implies Maxwell's equations
aE = curl Band -aB = - curl E at at'
(6)
-
and the Poisson bracket of two functions F(A. E).G(A. E) is . (F,G}(A,E)=-
8F 6G 6G 8F) 1 (6A' 8E- 6A' 6E d:c
(7)
H'
We want to express this Poisson bracket in terms of E and B = curl A when F and G are functions of Band E. For this. we need to compute 8F/8A·8G/8E in terms of Band E. Let f be a function of Band E and let F(A.E)=Fc.B.E). where B=vxA. Let L he the linear map A ...... V x A. We have Df'( A). 6G =
6E
f 6A6F . 6A
6(; /. t.\
by definition of 6F/6A and. by the chain rule.
f BF 6B
8G = DF(A)· -BG = (DF{B)o(DL(A»)·6E 6E
-
. curl ( -BG
6E
) dx
since DL(A) = Land 8G/6E = M1/6E. Thus the Poisson bracket (7) becomes
--
{F.G}(B. E) = -
-
f( 8f 6G 6B ·curl 6E -
6G f( 6f 6E ·curl 6B -
6G 6B
6f) ·curl 6E dx
6G
6f ) 6E ·curl 6B dx
(8)
(Using integration by parts we see that f X· curl Y dx = Jy. curl X dx). This bracket was found by Born and Inreld [1935] by a different method. With respect to the Hamiltonian
f
II ( B. E) = ! (II BII2 + II EII2 ) dx.
ELECTROMAGNETISM
509
(3) and (4) are easily verified to be equivalent to the Poisson bracket equations
f- (f. if) (For one implication use the chain rule. for the other. usc the functions Fe B, E) "7 IBi dx and Fe B. E5 = 1£' dx).
Exercf... Assume that the Faraday two-form F depends only on t - x. (i) Show that dF= 0 is then equivalent to B) = £2. B 2 ... - £1, BI ... 0. (ii) Show that IF= 0 is then equivalent to Bl = £2. B2 = - £1, £1 =0. These solutions of Maxwell's equations are called plane electromagn~tic waves; they are determined only by £2. E) or B2. B). respectively. . , IUR Let" = ii/ ilt. Show that the Faraday two-form F E 02(R 4) is given in terms of E and B E ','X (R 4) by F ... lIb A £~ - • (u b A B~). 8.3(' Show that the Poynting vector satisfies
8.3A
where" =
a/at and E. B E ~·X (R4)
8.3D Let (M.,) be a Lorentzian four-manifold and" E ' \: ( M) a timelike unit vector field on M; i.e., ,(u. u) = -I. (i) Show that any CI E 02( M) can be written in the form
(ii) Show that if ;.CI = 0, where ClE 02( M)("CI is orthogonal to u"), then • CI is decomposable. i.e. • CI is the wedge product of two one-forms. Prove that CI is also locally decomposable. (Hint: Use the Darboux theorem.) 8.3E ,The field of a stationary point charge is given by
E=~
47Tr J '
B=O.
where r is the vector xi + yj + zk in Rl and r is its length. Use this
510
APPLICA TlONS
and a Lorentz transformation to show that the electromagnetic field produced by a charge e moving along the x-axis with velocity I' is
and. using spherical coordinates with the x-axis as the polar axis,
8.3F
(the magnetic field lines are thus circles centered on the polar axis and lying in planes perpendicular to it). (c. Misner. K. Thome. and J. Wheeler [I973J.) The following is the Faraday two-form for the field of an electric dipole of magnitude PI oscillating up and down parallel to the z-axis. E=Re
. . [2cos8 (I?-? iW) d,/\dt {Ple,wr-u.J/
. (1,J
2
iw - w ) ,d8/\dt +sm8 - - ,2 ,
(2)
iw +sin8 ( -?--;d,/\,d8
]} .
Verify that dE = 0 and 3E = O. except at the origin. 8.30 Let the Lagrangian for electromagnetic theory be
Check that ae; agi ) is the stress-energy-momentum tensor T" (see Hawking and Ellis (1973. sec. 3.3». 8.3H Show that the Bom-Infeld Poisson bracket{.} on functions of E and B defined by equation (8) in Box 8.3A defines a unique (weak) symplectic structure on the space {( E. B) I div E = 0 and div B = O}. Verify that the Hamiltonian vector field for the energy H gives the vacuum Maxwell equations for aE; ill and ilB; ill.
THERMODYNAMICS, CONSTRAINTS, AND CONTROL
. 511
8.4 THERMODYNAMICS, CONSTRAINTS, AND CONTROL The applications in this final section all involve the Frobenius theorem in some way. Each example is necessarily treated briefly. but hopefully in enough detail so the interested reader can pursue the subject further by utilizing the given references. We begin with a discussion of equilibrium thermodynamics. This subject has a rather bad reputation for being sloppily taught and full of mysterious jargon. This is partly because the basic physical laws involve notions of. irreversibility. whereas laws for particle motion are reversible. This apparent contradiction and the difficulty in properly explaining it contributes to the problem. On the matbematical side. the formalism can be made clean and precise. It is the latter aspect we wish to describe. For additional reading. three sources we find useful are Sommerfeld (1964). Truesdell (1969]. and Hermann (1973). To motivate the formalism we need a little physical background and intuition and so begin with it. A. thermodynamic system is usually thought of as a continuum or a set of particles. possessing internal energy and interacting with its surroundings by either delivering or absorbing heat and thereby doing work. For instance. the particles' could be molecules in an internal combustion engine. Let M represent the set of thermal equilibrium states of the given system. i.e.• the states at which the system does not exchange energy (and thus heat) w,ith its surroundings. In the mathematical formulation of thermodynamics the basic assumption is that M. called the phase space. is a given connected finitedimensional manifold. such as Rn. The first principle of thermodynamics was first formulated by Rudolf Clausius around 1850. building on earlier work of ('Iapeyron. Carnot. and Thompson (Lord Kelvin) as follows. "In all cases in which work is produced by the agency of heat: a quantity of heat is consumed that is proportional to the work done; and conversely by the expenditure of ~n equal quantity of work an equal quantity of heat is produced." Thus, the rust principle is just the principle of conservation of energy for a closed system (a totally isolated thermodynamic system). Let E denote the internal energy of the system. W the work done. and Q the heat delivered by the system. Regard W and Q as given one-forms on M and E a given function on M. Then they are said to satisfy the first principle when Q=W+dE.
(In many texts one sees this written as IJQ = IJW + 8E.) A thermodynamic system often utilizes three additional real-valued functions on M: the, volume V. the temperature T. and pressure P. The work is expressible in
512
APPLICATIONS
terms of the volume and pressure as W = P dV. An equation of state consists of functions relating P and E to Vand T: P = P( V. T). E = E( v. T). These equations of state are specific functions whose exact form depends on the nature of the particular substance under investigation. The second principle 0/ thermodynamics states that IjT is an integrating factor for Q. In other words, QIT is an exact one-form (at least locally). If Q IT - dS, the function S is called the entropy of the system. Thus the second principle states that locally I I dS- r Q - r(dE
+ PdV).
We shall now reconcile this formulation involving the Frobenius integrability condition on the Pfaffian equation Q = 0 to the classical formulation of Kelvin: "It is impossible. by means of an inanimate material agency, to derive a mechanical effect from any ponion of matter by cooling it below the temperature of the coldest of the surrounding objects." That is. it is impossible in a closed system to transfer heat from a hot to 'a cold body without making other changes as well. t This reconcilation will be done by means of the Caratheodory inaccessibility theorem. A path y: [0,1) -. M is called a quasi-static adiabatic path of the closed thermodynamic system if Q. dyldt,.. 0 (the one-form Q acting on the vector dyldt). This says that along y no heat is added. If m EM is a fixed thermal equilibrium state of the system and if all other points of M can be connected to m by a quasi-static adiabatic path-i.e .. any thermal equilibrium can be obtained from any other one without addin/{ heat. then the work done by a closed system would be entirely due to its intrinsic internal energy (such as by molecular movements. atomic vibrations. etc.). Thus one could get work done in a machine without any heat transfer: a perpetual motion machine. The following theorem of Caratheodory [1909] states that this is impossible, recovering Kelvin's formulation of the second principle. Here we see the fundamental difference between the first and second principle. The first principle allows all possibilities consistent with conserving energy, whereas the second principle gives serious restrictions to these possibilities.
1A.1 C8ralheodo..,'. lnecceulbillty Theorem. Let M be a smooth n-manifold without boundary and Q E Ol( M) a nowhere l'ani.vhing one-form.
t
Tbcrc is also a formulation of 1M second law for dynamic process that is related to
die popular statcllleDt that '"eatropy increases with time- or .. things tmel to become rmdom." The usual s..tement is called 1M Clausius- Duhem ineqvtllity, but it is IOl'IICWhat coatrovenial; lee Manden and Huahes (1983) for a discussion.
,THERMODYNAMICS, CONSTRAINTS, AND CONTROL
513
Then the following are equivalent: (i) (ii)
Q = 0 is an integrable Plalfian system; i.e .• local(v Q = TdS lor function.s Tand S. For each Xo e M there exists an open neighborhood V of Xo on M such that each neighborhood W of Xo. W c V. contains a point x e W that cannot be connected to Xo by a ( pieceWise smooth) quasi-static adiabatic path.
If Q - 0 is integrable. then the distribution defined hy its annihilator defines a foliation by the Frobenius theorem. (Sec Section 4.4 and Box 6.4A.) In particular. locally the submanifolds of the foliation are defined hy' S = constant. where S is the entropy of the system (recall that TdS = Q). But as ,Q is nowhere zero. these (n -I)-dimensional submanifolds do not intersect. so it is impossible to reach all points in the given neighborhood by curves lying only on these submanifolds. Thus statement (i) implies (ii). To prove the converse. let V be a neighborhood of xI) e M and let Proof.
E=
{"e T.MIQ(x)·,,= O. x e V}
be the annihilator of Q in TV. If we show that the subhundle E of TV is involutive. then by the Frobenius theorem. Q .. 0 is integrable in V and hence in M. Let E' be the smallest involutive subbundle of TV containing E and let A = (K e Vlx cannot be joined to Xo by a path y: [0. I) ..... V satisfying Q(x)·dy/dt .;. O}. By (ii). A .. V. so that E' is not equal to TV. In partiCular. the fiber dimension of E' is """ n - 1. But E':::> E and E has fiber dimension n - 1. so that E has fiber dimension n -I and hence E = E'. Consequently E is involutive. • • Thermodynamics has a connection with the symplectic geometry of Section 8.1 through the notion of a "Lagrangian submanifold." This treatment is beyond the scope of this book. The reader may consult Hermann (1973). Marsden and Hughes [1983]. Abraham and Marsden (1978) and Oster and PereISGn (1973) for inforniation and further references. We tum next to the subject of holonomic constraints in Hamiltonian systems. A Hamiltonian system as discussed in Section 8.1 can have a "~to available points in phase space. Such a condition imposed that lill' , condition is a constraint. hi! .:xample. a ball tethered to a string of unit length in R3 may be considered to be constrained only to move on the unit sphere S2 (or possibly interior to the sphere if the string is collllpsible). If the phase space is 1'*Q and the constraints are all derivable from cOnstraints imposed only on the configuration space (the q ·s). the constraints are ~lIed
514
APPLICA TlONS
holonomic. For example. if there is one constraintf(q) = 0 for f: M -+ R. the constraints on T* M can be simply obtained by differentiation: df = 0 on T* M. If the phase space is TM. then the constraints are holonomic iff the constraints on the velocities are saying that the velocities are tangent to some constraint manifold of the positions. A constraint then can be thought of in terms of velocities as a subset E C TQ. If it is a subbundle. this COIUlrainl is IhllS holonomic iff it is integrable in the sense of Frobenius' theorem. Constraints that are not holonomic. called nonholonomic constraints. are usually difficult to handle. Holonomic constraints can be dealt with in the .enle that one understands how to modify the equations of motion when the constraints are imposed. by adding forces of constraint. such as centrifugal force. See, for example Goldstein (1980. ch. I]. and Abraham and Marsden (1978. sec. 3.7]. We shall limit ourselves to the discussion of two examples of nonholonomic constraints. A classical example of a nonholonomic system is a disk rolling without slipping on a plane. The disk of radius a is constrained to move without slipping on the (x. y)-plane. Let us fix a point P on the disk and call fJ the angle between the radius at P and the contact point Q of the disk with the plane. as in Fig. 8.4.1. Let (x. y. a) denote the coordinates of the center of the disk. Finally. if" denotes the angle between the tangent line to the disk at Q and the x-axis. the position of the disk in spal:c is completely determined by (x. y, fJ. fJ». These variables form elemcnts of our configuration space M = R 2 X s' X st. The condition that there is no slipping at Q
..
Flgunt'.4.1
...
THERMODYNAMICS. CONSTRAINTS. AND CONTROL
515
means that the velocity at Q is zero; i.e.•
dx dB -- +a-coscp=O dl dl •
dB. -dy + asm
dt
dt
II
(total velocity = velocity of center plus the velocity due to rotation hy angular velocity dB/dl). The differential form expression of these constraints is WI = O. ~ = 0 where W I - dx + acoscpdB and ~ ... dy + asincpdll. We compute that W - WI" W2 -
dW I = dW I /\ W = -
dw,./\
W
dx /\ dy + acoscpdll /\ dy + asin",dx /\ dB.
asincpdcp" dB,
dW2 =
acoscpdcp /\ dB.
asincpdcp" dB" dx" dy.
= acoscpdcp /\ dB" dx /\ dy.
These do not vanish identically. Thus. according to Thcorem 6.4.20. this system is not integrable and hence these constraints are nonholonomic. A second example of constraints is due to E. Nelson (1967). Consider the motion of a car and denote by (x. y) the coordinates of the center of the front axle. cp the angle formed by the moving directiun of the car with the horizontal. and B the angle formed by the front whecls with the car (Fig. 8.4.2). Thus. the configuration space of the car is R 2 X T 2. parametrized hy (x. y. CPo B). We shall prove that the constraints imposed on this motion are nonholonomic. Can X == iJ / iJII the vector field steer. We want to compute a vector field Y corresponding to drive. Let the car be at the configuration point (x. y. CPo II) and assume that it moves a small distance h in the direction of the front wheels. Notice that the car moves forward and simultaneously turns. Then the neW configuration is
(x + hcos( cp + (1)+ o(h). y + hsin( cp + (1)+ o(h). cp + h sin II + o(h ).11). Thus the "drive" vector field is
Y ... cos( cp + B)
!
+ sine cp + II)
:y +
sinfJ ;cp .
A direct computation shows that the vector field wriggle.
w = lX, Y] = -sine cp + II)
!
+cos( cp + II)
:y
+ cos II :cp'
516
APPLICA TIONS
.t
Flgure ••4.2 and slide.
s= [W • YJ =
-sincp~ ax +coscp~ ay'
satisfy
[x.W)--Y. [s.X)-[s.y)-[s.Z) ... o; that is. {X. Y. W, S} span a four-dimensional Lie algebra. In particular the subbundle of T(R 2 x T 2) spanned by X and Y is not involutive and thus not integrable. By the Frobenius theorem. the field of two-dimensional planes spanned by X and Y is not tangent to a family of two-dimensional integral surfaces. Thus the motion of the car. subjected onl), to the constraints of "steer" and "drive" is nonholonomic. Next we tum our attention to some elementary aspects of control theory. We shall restrict our attention to a simple version of a local controllability theorem. For extensions and many additional results. we recommend consulting a few of the imp()f;.,,' papers and notes such as
THERMODYNAMICS, CONSTRAINTS, AND CONTROL
517
Brockett [1970,1983), Sussmann [1977], Hermann and Krener [19771. Russen (1979), Hermann [1980]. and Ball. Marsden, and Slemrod (1982) and references therein. Consider' a system of differential equations of the form
w(t) = X( we,
»+ pc, )Y( wet»~
(I)
on a time interval [0. T] with initial conditions w(O) = ~h where w takes values in a Banach manifold M. X·and Yare smooth vector fields on M and p: [0. T) ..... R is a prescribed function called a control. The ex.istence theory for differential equations gUClrantccs that (I) has a now that depends smoothly on Wo and on p lying in a suitahle Banach space Z of maps of [0. T] to R. such as the space of C' maps. Let the now of (I) be denoted . (2) F,(wo • p) = wet. p.wo ). We consider the curve w(t.O. wn ) = wo(t); i.e.• an integral curve of the vector field X. We say that (1) is locally controllable (at time T) if there is CI neighborhood U of wo(T) such that for any point hE U. there is II p E Z such that wU. p. wo ) = h. In other words. we can alter the end point of wo(t) in a locally arbitrary way by altering p. (Fig. 8.4.3). To obtain a condition under which local controllability can be guaranteed. we fix. T and Wo and consider the map
(3)
P:Z ..... M;
The strategy is to apply the inverse function theorem to P. The derivative of F,( Mh. p) in the direction P E Z is denoted
D,F,( wn.O)·p = L,P E TF,(M'IIO)M
... ---- ...... ... u
\.
....(T)
;
I
...
Figure •. 4.3
.....
__
I
...
""
I
518
APPLICATIONS
and by differentiating d dt wet. p) = X( wet. p
»+ p(t)Y( wet. p»
with respect to pat p = O. we find that in T2M.
To simplify matters. let us assume M - E is a Banach space and that X is a linear operator. so (4) becomes d dt L,P = X·L,P
+ pY( wo(t ».
(5)
This has the solution given by the variation of constants formula (6)
since woe t) = e'X Wo for linear equations.
•.4.2 PropoaWon. If the linear map L r : Z ..... E given by (6) ;s surjective•. then the Eq. (I) is locally controllable (at time T).
This follows from the "local onto" form of the implicit function theorem (see 2.5.9) applied to the map P. Solutions exist for time T for small p since they do for p = o. • Proof.
•.4.3 Corolla". Suppose E
=
R n and Y is linear as well. If
dimspan{Y( wo ). [X. Y]( wo). [X.[X. Y]]( wo) •... } ... n. then (I) is locally controllable.
Proof. We have the Baker-Campbell-Hausdorff formula 2
e-"xYe" x = Y
+ s[X. Y] + s2 [X.[X. Y]] + ....
obtained by expanding e" x = I + sX + (.f 2/2)X 2 + ... and gathering terms. Substitution into (6) shows that Lr is surjective. •
THERMODYNAMICS, CONSTRAINTS, AND CONTROL
519
For the case of nonlinear vector fields and the system (I) on finitedimensional manifolds. controllability hinges on the dimensiun of the space obtained by replacing the foregoing commutator brackets by Lie brackets of vector fields. n being the dimension of M. This is related to what' are usually called Chow's theorem and Hermes' theorem in control theory (see Chow [1947]).
To see that some condition ;nvolving brackets is necessary. suppose that the span of X and Y forms an involutive distribution of TM. Then by the Frobenius theorem. Wo lies in a unique maximal two-dimensional leaf ~M' of the corresponding roliation. But then the solution "r (I) can never le~~e t .., . no matter how p is chosen. Hence in such a situation. (I) would not be 1~lIy controllable; rather. one would only be able to move'in a two-dimensional subspace. If repeated bracketing with X increases the dimension of 'vectors obtained then the attainable states increase in dimension accordingly. '
Exercl... 8.4A Assume that a closed thermodynamic system has given equations of state E = E(V. T). P = P(V. T). Derive from the second principle of thermodynamics. the classical integrability condition
(Hint: Write out d(QIT) = 0 using Q"" dE + PdV.)
8.4B
(i)
Let T=T(S,v). P=P(S.V). E=E(S.T) be the temperature. pressure. and internal energy of a closed thermodynamic system with entropy S and volume V. Derive from the second principle the identity
(Hint: Multiply the relation TdS=dE+PdV by liP and ,differentiate.) (ii) Let T = T( p. S). S = SeT. P). P = peT. S) be equations of state. Prove Maxwell's identity aT ap as ap as aT =-1.
(Hint: Compute dT AdS.)
520 8.4C
APPLICA TIONS
Consider a thermodynamic system with two phases: phase I = liquid. phase 2 - vapor. Let N .. N, + N2 be the total amount of substance (constant) and V .. V, + V2 be the total volume. The .fpecific tlolumes are defined by
where
v, = v,( N,. T).
i = 1.2.
The latent heat of vaporization is defined by L = T as/ aN,. (i) Show that the second principle implies dT /\ dS = dP /\ dV. (ii) Assuming an equation of state P = P(T). show that as av dP aN, = aN, dT' (Hint: Multiply the equation for dS and dT and the equation for dV by (dP/dT)dT. considering T and N, as variables.) (iii) Derive from (ii) the Clausius-Clapeyron formula. dP L dT= T(V,-V2)' 8.40 Check that the system in Fig. 8.4.1 is nonholonomic by verifying that there are two vector fields X. Y on M spanning the subset E of TM defined by the constraints
x+ alJcosq> =
0 and
.v+ alJsinq> =
0
such that IX. Y] is not in E; i.e.• use Frobenius' theorem directly rather than using Pfaffian systems. 8.4E Justify the names wriggle and slide for the vector fields Wand S in the example of Fig. 8.4.2 using the product formula in Exercise 4.20. Use these formulas to explain the following statement of Nelson 11967. p. 35]: "the Lie product of Steer and Drive is equal to Slide and Rotate ( - a/ aq» on 8 - 0 and generates a now which is the simultaneous action of sliding and rotating. This motion is just what is needed to get out of a tight parking spot." 8.4F The word holonomy arises not only in mechanical constraints as explained in this section but also in the theory of connections
TH~RMODYNAMICS. CONSTRAINTS. AND CONTROL
521
(Kobayashi-Nomizu (1963. II. sec. 7.8]). What is the relation between' the two uses? 8.40 In linear control theory (I) is replaced by N
"~t)=X·"'(t)+
L
,-1
p;(t)Y,.
where X is a linear vector field on A" and Y, are mflStant vectors. By using the methods used to prove 8.4.2. rediscover for yourself the Kalman criterion for local controllability, namely. {X'Y,lk = 0.1, .... n - I, i = I, ... ,N} spans A".
APPENDIX
A
The Axiom of Choice and Balre Spaces
This appendix presents. for completeness. supplementary topics in topology that were used in the main text in a few technical proofs. For additional details. see Kelley (1975) and Choquet [1969). THE AXIOM OF CHOICE
One of the most widely used axioms in set theory is the axiom of choice. A.1 AxIom of ChoIce. Let S; be a collection of nonempty sets. Then there exists a function x: ~ -+ USE" S such that X(S) E S for every S E S;.
The function X chooses one element from each S E S; and is called a choice function. Even though this statement seems self-evident. it has been shown to be equivalent to a number of nontrivial statements using other axioms of set theory. To discuss them. we need a few definitions. An order on a set A is a binary relation. usually denoted by .. ~ .. satisfying a a
~
~
a~
522
a (renexivity); band b ~ a implies a = b (antisymmetry). and band b ~ c implies a ~ (" (transitivity)
APPENDix A: THE AXIOM OF CHOICE AND BAtRE SPACES
523
An ordered set A is called a chain if for every a, b EA. a '* h we have a ~ h or b EO a. The set A is said to be well ordered if it is a chain and if every nonempty subset B has a first elqnent; i.e., there exists an element bE B such that b ~ x for all x E B. An upper hound u E A of a chain C c A is an element for which c ~ u for all c E C. Finally a maximal element m of an ordered set A is an element for which there is no other /I E A such that m ~ a. a" m; in other words. x ..,; m for all x E A that are comparable to m. We state the (ollowing without proof.
A.2 Theorem.
Given other axioms of set theory. the lollowin?, statemellts
are equivalent: (i)
(ii) (iii) (iv)
The (/.lciom ol choke. If {A,}, E ' i.v a collection of nonemplY set.r then the product [1, E ,A, = {(x,)lx, E A,} is nonemplY (product axiom). Any set can be well ordered (Zermelo's theorem). If A is an ordered set for which every chain has all upper hound (i.e .• A is induct~/y ordend). then A has at least one maximal element ( Zorn's lemma).
BAIRE SPACES
The Baire condition on a topological space is fundamental to the idea of "genericity" in differential topology and dynamical systems.
· A.3 Definition. Let X be a topological space and A c X a subset. Then A · is called nsidluJl if A is the intersection of a ('Ountable famill' of open dense subsets of X. A space X is called a Bain space if every residual set is dense. A set B c.X is called a first category set if Be U jCn where (:, is c/o.red with int( Cn ) = 0 . A second category set is a sel not of the first category. A set Be X is called nowhere dense if int(cI(B» =0. so that X\A is residual iff A is the union of a countable collection of nowhere dense closed sets. i.e.• if X.\ A is of first category. Clearly. a countable intersection of residual sets is residual. In a Baire space X. if X = U ::".. IC" where the c" are closed sets. then · int( Cn ) ., 0 for some n. For if all int( Cn ) = 0 then 0" = X \ Cn are open. dense. and () jO" == X\ U ::"_ICn - 0 contradicting the definition of Baire space. In other words. Baire spaces are of second Cale1{orV. The following is often useful.
A.4 Propoaltlon. LeI X be a locally Baire space; that is. ('(Ich point x E X has a neighborhood U such Ihal d( U) is a Raire spal'/'. TII('II X is a Baire space.
524
APPENDIX A: THE AXIOM OF CHOICE AND BAIRE SPACES
Proof. Let A c X be residual: Yo
A= nO" II-I
where c1( 0,,) ""' X. Then
A n c1( U ) =
n'"
(0" n d( U )).
II-I
Now On ncl(U) is dense in c1(U) for if II e c1(U) and II e 0. where 0 is an open set. then on U • (lJ and nun 0" • (lJ. Hence cl( U ) c cl( A) and so c1(A)'"' X. •
°
The most important examples of Baire spaces are given by the following theorem. A.S 88lre Category Theorem. pact spaces are Boire spaces.
Complete pseudometric and local(l' com-
Proof. Let X be a complete pseudometric space. Let U c X I}e open and A ... n JOIl be residual. We must show UnA .. (lJ. Now as c1( 0,,) = X. U n 0" .. (lJ and so we can choose a disk of diameter less than one, say VI' such that c1( VI ) C U () 01. Proceed inductively to obtain cl( V,,) c U () 0" n V,,_ I_ where v" has diameter < lin. Let x" e c1(V,,). Clearly (.t ll ) is a Cauchy
sequence. and by completeness has a convergent subsequence with limit point x. Then X E
n I
cI( v"
) and so U n no,,'" (21 : "t I
i.e.. A is dense in X. If X is a locally compact space the same proof works with the following modifications: V" are chosen to be relatively compact open sets. and {XII} has a convergent subsequence since it lies in the compact set cl( VI ). • To get a feeling for this theorem. let us prove that the set of rationals Q cannot be written as a countable intersection of open sets. For suppose Q - () jOlt. Then necessarily each 0 ,~dense in R since Q is and sO CII = R\O" is closed and nowhere .1.' ,,', Since R = Q u U ::'-IC" is a com-
APPENDIX A: THE AXIOM OF CHOICE AND BAIRE SPACES
525
plete metric space (as well as a locally compact space). it is of second category. so some C" should have nonempty interior (hecause intQ = 0,. But this is a contradiction. The notion of category also imposes restrictions on a set. For e"ample in a nondiscrete Hausdorff space. any countable set is first category since, the one-point set is closed and nowhere dense. Hence in such a space every second calegory .fel i.t uncounlahie. In particular. nonfinite complete p~udo metric and loCally compact space... are uncountable.
APPENDIX
B
The Three Pillars of Linear Analysis
We give here the classical proofs of the three fundamental theorems of linear analysis in the setting of Banach spaces. See Chapter 2 for a discussion and applications. 8.1
Hahn-Benach Theorem. ut E he a real or complex vector space. -0 R a seminorm and F § E a subspac·e. If I E F* satisfies I/(")! ~ lIell for all e E F. then there exists a linear map i: E -0 R (or C) suc·h Ihal il F ... I and lj(e)1 ~ lie II for all e E E.
11·11: E
Proof. Real Case. First we show that I E F* can be extended with the given property to FCDspan (eo}. for a given eo E F. For e,.e2 E F we have
I( e. ) + I( e2) - I( e.
+ e2) " lie. + e211 " lie, + eo II + lI e2- eoll.
so that and hence
Let a E R be any number between the sup and inf in the preceding expression and define FEB span (e.} -0 R hy j( e + leo) = I(e)+ tc/. It is
i:
526
APPENDIX B: THE THREE PILLARS OF LINEAR ANAL YSIS
clear that i is linear and that il F = /. To show that note that by the definition of a,
1ft f' + lefl)j "
527
lie + leo II.
/( e2 )-lIe 2 - e"II" a" lIe , + e"l1- /(e , ). so that multiplying the second inequality by t ~ 0 and the first by t < 0 we get the desired result. Second, one verifies that the set :;={(G.g)IFcGcE,G is a subspace of E. If E G*, IfIF = /. and Ig(e)I" lIell for all e E (i) is inductively ordered with respect to the ordering (G"lf,)" (Gz.lfz)
iff G , cG2 .lf2IG,=If,.
Thus by Zorn's lemma (Appendix A) there exists a maximal element (f;,. I,,) of:;. ; Third. using the first step and the maximality of (F". /n)' one concludes that £., = E. Complex Case. Let / = Re / + ilm / and note that complex linearity' implies (1m 1)( e) = - (Re /)( ie) for all e E F. By the real case. Re / eXJends to a real linear continuous map (Ref) -: E ..... R. such that I(Ref) - (e)I" lIell for all e E E. Definei: E ..... C by fie) = (Ref) - (e)- i(Ref) - (ie) and note that / is complex linear and il.F = /. To show that Ifte)I" lIell for all e E E. write fie) = lfie)lexp(i(l). so complex linearity of implies ft e' exp( - i(l» E R and hence
i I.il e)1 = it. e·exp( -
i(l» =.(Re f) - (e·exp( - i8»
.;;lIe·exp( - i(l)1I = lIeli . • The foliowing is the form of the theorem that
was u)o~'d
in Section 2.2.
B.2 Corollary. Lei (E.II·IO be a normed space. F c E a subspace and / E F* (/he topological dual). Then there exists
i
E E* such lhat
ilF =
/ and
lIill""' II/II· We can assume
/-0. Then lIIelil-II/liliell is a norm on
E and the pr~ing theorem we 'get a linear map i: E ..... R (or C) such that II F""' / and lIe e) I .;; IIlelll for all e E E. This says that II ill.;;1I1I1 and since i extends /. it follows that 11111 " II II; i.e.• II ill = II/II and E E*. • Proof.
lI(e)I" 1I/1I'lIell-lIIelll for all e E F.
i
Appl~ing
i
B.3 Open Mapping Theorem of Banach and Schauder. Let E and F he Banach spaces and Suppo.fe A E L ( E, F) is onto. Then A is an open mapping.
528
APPENDIX B: THE THREE PILLARS OF LINEAR ANAL YSIS
Proof. To show that A is an open mapping. it suffices to prove that A(c1( D1(0))) contains a disk centered at zero in F. Let r> O. Since E = U:'_IDM(O). it follows that F= U'::'_I(A(D",(O» and hence U :'_Icl( A( D",(O») - F. Completeness of F implies that at least one of the c1( A( D",(O») has nonempty interior by the Baire category theorem (A.5). The mapping , E E ... n, E E being a homeomorphism. this says that cI(A(D,(O» contains some open set V c F. We shall prove that in fact the origin or F is in int(cl(A( D,(O»». Continuity or ('1"2) E E x E""I -'2 E E assures the existence of an open set U c E such that U - U = {'I '21'1.e2 E U} c D,(O). Thus eI( A( DI(O))):::> cI( A(U)- A(U»:::> cI(A(U»cI(A(U»:::> V-V. But V - V = U r'" ,( V - ,,) is open and clearly contains o E F. It follows that there exists a disk D, (0) c F such that D, (0) c eI( A ( D,(O)). Now let E" = 1/2'" I. II - 0.1.2 ..... so that 1= 1:;:'_of". By the forego. ing result there exists an 11" > 0 such that D~..
no -
:c
L
1/2",1-1
,,-n
and E is complete. Let
x
e=
Thus
,,-II
e"EE.
x
L
Ae=
,,-() i.e.. v E
L
D~,,(O)
x
Ae,,=v.
and
implies v = Ae. lIell
lIell~
L
x
Ilellll~
11=0
~
1: that is.
L
1/2,,'1=1:
II=()
D~"
c
A(eI( DI(O))).
•
•.4 Uniform Bounded.,... Principle of Banach and Steinhaus. Let E and F be normed vector spaces. E complete. and {A,},e I C L(E. F). Iffor each e E E the set <1114 ,elD, e I ;s hound"d ;n F. then {II A /II}, '" , ;s a hOllnded s"t of real numbers. Proof.
Let cp(e) = sup{IIA,elll ; E I} and nole Ihal .),,- \ ... ~EI'I'(t')~II}=
n
{"EEIIIA,"II~I/}
,e I
is elosed and U ::"_IS" = E. Since E is a complete metric space. Ihe Baire
APPENDIX B: THE THREE PILLARS OF LINEAR ANAL YSIS
529
category theQrem (A.S) tells us that some S., has nnncmpty interior; i.c .• there exist r > O. eo E E such that q>( e) ~ M. for all , E \."I( D,( en))' where M > 0 is some constant. . For each i E I. and lIell = I. we have IIA,(re + en~1 ~ q:>(re + eo)~ M. so that
~ (M+q>(en))/r.
i.e..
IIA,II~(M+q:>('n»)jr
for all i E I. •
APPENDIX
c
Unbounded and Self-adjoint Operatorst
In many applications involving differential equations. the operators one meets are not defined on the whole Banach space E and are not continuous. Thus we are led to consider a linear transformation A: D,. c E ~ I:.' where DA is a linear subspace of E (the domain of A). If D,. is dense in E. we say A is densely defined. We speak of A as an opera/or and this shall mean linear operator unless otherwise specified. Even though A is not usually continuous. it might have the important property of being closed. We say A is closed if its graph rA
rA -
{(x. Ax) E EX Elxe DA }
is a closed subset of E x E. This is equivalent to (x"EDA.x,,-XEE
and
implies (xEDA
Ax,,-YEE)
and
Ax=)').
An operator A (with domain D,. ) is called c/o.fable if d( fA I. the closure. of the graph of A. is the graph of an operator. say. A-: We call the closure of A. It is easy to see that A is closable irr (Ix. ED•. xn ~ 0 and Ax. -0)')
..r
tThis appendix was written in collaboration with P. Chernorr.
530
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERA TORS
531
implies y == O}. Clearly A- is a closed operator that .is an extension of A; i.e.• D.i"~ D,4 and A-- A on D,4. One writes this as A-~ A. The closed graph theorem asserts that an everywhere defined closed operator is bounded. (See Section 2.2.) However. if an operator is only densely' defined. "closed" is weaker than" bounded." If A is a closed operator. the map x ..... (x. Ax) is an isomorphism between D,4 and the closed subspace r,4' Hence if we set
V,4 becomes a Banach space. We call the norm III ilion VA the graph norm. Let A be an operator on a real or complex Hilbert space H with dense domain D,4' The adjoint of A is the operator A* with domain D,4o derined as follows:
D,4o={yeHlthereisazeH
such that (Ax.J')=(x.z) for all x e I~~ }
and A*:D,40---H.
...
J......
From the fact that DA is dense we see thaI A* is indeed well defined (there is at most one such z for any J' e H). It is easy to ~ee that if A ::'l B then B*::> A*. If A is everywhere defined and bounded, it f(lll()w~ from the Riesl representation theorem (Box 2.2A) that A* is everywhere ddined: moreover it is not hard to see that, in this case, IIA*II = IIAII. An operator A is symmetri(' (Hermitian in the complex case) if A* ~ A: i.e., (Ax. y) = (x. Ay) for all x. y e D,4. If A* = A (this includes the , condition D,4o = D,4)' then A is -called self-adjoint. An everywhere defined symmetric operator is bounded (from the closed graph theorem) and so is self-adjoint. It is also easy to see that a self-adjoint operator is closed. . One must be aware that. for technical reasons, it is the notiQn of self-adjoint rather than symmetric. which is important in applications. Correspondingly. verifying self-adjointness is often difficult while verifying symmetry is usually trivial. ' . Sometimes it is useful to have another concept at hand. that of essential self-adjointness. First. it is easy to check that any symmetric operator A is dosahle. The closure A- is easily seen to be symmetric. One says that A is euentia[(r self-adjoint when its closure A-is self-adjoint. A related concept is this: Let A be a self-adjoint operator. A dense subspace C cHis said to be a core of A. if C c D... and the closure of A
632
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERA TORS
restricted to C is again A. Thus if C is a core of A one can recover A just by knowing A on C. We shall now give a number of propositions concerning the foregoing concepts. which are useful in applications. Most of this is classical work of von Neumann. We begin with the fo))owing.
C.1 PropoeitIon. Let A be a closed symmetric operator 0/ a complex
Hilbert space H. 1/ A Is self-adjoint then A + "AI is surjective lor every complex number A with 1m A ... 0 (I is the identity operator). Conversely, i/ A is symmetric and A - il and A + il are both surjective then A is self-adjoint.
Proof. Let A be self-adjoint and A= a + iP, P '* O. For x E
DA we have
II(A + A)x1l 2= II(A + a)xll2 + ip(x, Ax) - iP(Ax. x) + P21ixll 2
-II(A + a)xll2 + P211xll2 ~ P211xll2, where A
+ A means A + AI. Thus we have the inequality II(A
+ A)xll ~ (ImA)lIxll.
(I)
Since A is closed, it fonows from (I) that the range of A + A is a closed set for 1m A .. O. Indeed, let y" - (A + A)x" ... y. By the inequality (I), IIx" - x ... 11 < II y" - Y... II/IIm All so x" converges to, say, x. Also Ax" converges to Y - Ax; thus x E DA and y - Ax = Ax as A is closed. Now suppose y is orthogonal to the range of A + AI. Thus (Ax+Ax,y)==O
fora))
xEDA ,
or (Ax, y) == - (x, Ay).
By definition, y E DAo and A· y ... - Xy. As A = A·, y E DA and Ay - - ~, or (A + XI)y - O. By inequality (I) it fonows that y = O. Thus the range of A + Al is all of H. Conversely, suppose A + i and A - i are onto. Let y E DA •• Thus for all xEDA •
«A + ;)x. y) = (x.(A· - ;)y) = (x,(A - i)z) for some
zE
DA since A - i is onto. Thus
«A + i)x. y)
=
«A + i)x. ~
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERA TORS
and it follows that, = z. This proves that DA• C DA and so DA result follows. •
=
533
DA•. The
If ..4 is self-adjoint then for 1m>. .. O,AI -..4 is onto and from (I) is one-to-one. Thus (>. I - ..4) - I: H .... H exists, is bounded, and we have ,
II(AI- A)-III C; 1/IIm>'l.
(2)
This operator (>'1-..4)-1 is called the resolvent of A. Notice that even though..4 is an unbounded operator, the resolvent is bounded. The sam~ argument shows the following.
C.2 Propo8lUon. A symmetric operator A is essentially Jell-adjoint iff the ranges of..4 +'iI and A - il are dense. If ..4 is a (closed) symmetric operator then the ranges of ..4 + il and ..4 - il are (closed) subspaces. The dimensions of their orthogonal complements are called the deficiency indices of ..4. Thus C. t and C.2 can be . restated as: a closed symmetric operator (resp., a symmetric operator) is self-adjoint (resp., essentially self-adjoint) iff it has deficiency indices (0,0). If ..4 is a closed symmetric operator then from (I), A + il is one-to-one and we can consider the inverse (..4 + i/) - I, defined on the range of ..4 + il. One calls (..4 - iI )(..4 + il) - I the C;ayley transform of ..4. It is always isometric, as is easy to check. Thus..4 is se/f-a4joint iff its Cayley transform is unitary. Let us return to the graph of an operator..4 for a moment. The adjoint can be described entirely in terms of its graph and this is often convenient. Define an isometry J: HeN .... HeR by J(x,,) "" (- y, .1'); note that
J2 = -I. C.3 Propo8lUon. Let A be densely defined. Then (fA) 1.
r
= J( A.) and - fA'''' J(fA ).I.. In particular, A* is closed, and if..4 is c1osed,then
NeH
=0
fAeJ(rA .),
where NeN carries the usual inner product: , « .1'1 • .1'2 ),( 'I' JI2}) ,Proof, have
>=
(XI' 'I) + (.1'2.12)·
Let (z.y)eJ(fA.), so yeDA• and ::=-..4*y. Let xeDA. We «.1' • ..4.1').( - A*y.
and so J(fA .) c
fl.
,»
=
(.1'. - ..4*,) + (Ax. y) = O.
534
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
Conversely if (z. y) E
rl.
then
(x.z)+(Ax. y)-O for all xED... Thus by definition. y ED... and z - - A- y. This proves the opposite inclusion. • Thus if A is a closed operator, the statement HeH = fAeJ( fA') means that given '. I E H. the equations {
=,
X -
A-),
Ax
+Y= I
have exactly one solution (x. y). If A is densely defined and symmetric. then Ac A- since A- is closed. There are other important consequences of C.3 as well.
C.4 Corollary. For A densely defined and closable. we have
.o4=A--.
(i)
and
(ii)
.».1.
I'mDf. (i) Note that r A.... - J{(rA')~}'" -(J(rA isometry. But
- (J( rA
.»
.I.
= -
(ii) follows since rj -
(J 2 rl
since J is an
) .I. = ri .I. = rA =' r.i.
ri. •
Suppose A: DA c H - H is one-to-one. Then we get an operator A - I defined on the range of A. In terms of graphs:
rA
•
= K(fA ),
where K(x, y) - (y. x); note that K2 = I. K is an isometry and KJ = It follows for example that if A is self-a4ioint, so i.f A I. si nce r(A .• ) .... -
-
JK.
Jri. = - JKri = KJri = KrA • = rAe •.
Next we consider possible self-adjoint extensions of a symmetric operator.
c.s
PropoeItIon. Let A be a symmetric densely defined operator on. H. The following are equivalent: (i)
A is essentially self-adjOint.
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
535
(ii) A* is self-adjoint. (iii)
A**:::> A*.
(iv) A has exactly one self-adjoint extension. (v)
A-. A*.
Proof. By definition, (i) means (X)* = A-: But we know (X)* == A* and A-= AU by C.4. Thus (i), (ii), (v) are equivalent. These imply (iii). Also (iii) implies (ii) since A c A-c A* c A** = A- and so A* = AU. To prove (iv) is implied let Y be any self-adjoint extension of A. Since Y is closed, Y:::> A-: But A = A* so Yextends the self-adjoint operator A*; i.e. Y:::> A. Taking adjoints, A* = A:::> y* = Y so Y= A.
The proof that (iv) implies the others is more complicated. We shall in fact give a more general result in C.7 below. First we need some notation. Let D + = range( A + il) .1 C II and D _ == range( A - ;/ )
II
.1 C
called the ptAfi'it~ and 1tt'.fllJtlIV .k/n', ,~"''''c''~. l Ish'~ easy to check that D~ =
Ih,' .\I~'"lk·1\1 tIIl',l II I~
{x E DAoIA*x = ix}
and D -' x
C.I Lemma. sum
c
II". A- II
-
"
.
V,fing the graph norm on VAo, we have
'''1' orthowmal (IIrc'c't
D,.o = D,.-eD+eD_.
Since D +, D _ are closed in H they are closed In D~o, Also D,i"C D,.o is closed since A* is an extension of A and hence of A. It IS easy to see that the indicated spaces are orthogonal. For example let x E D,.- and y ED. Then using the inner product
Proof.
«x, y» = (x. y) + (A*x. A* y) «x, y» = (x, y) + (A*x. - iy) = (x. y) - i(A*x, y),
gives Since x
E
D,.-- D,.o by (v). we get «x, y»
= (x.
y) - i(x. A* y)
= (x, y) - (x, y) = O.
536
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
To see that D,.o "" D,.-$O +$0 _ it suffices to show that the orthogonal complement of D,.-$D +$D _ is zero. Let" e (D,.-$D+$O_).J., so
«", x» ... «", y»
=
«",:» ... 0
for all xeDA-.yeO+,:ED_. From «",x»""O we get (",x)+ (.4.", A·x) -0 or A·"e DAo and A·A·" = -". It follows that (/ - iA·)" eD •. But from «",y»=O we have «(I-iA·)".y)=O and so (1jA.)" - O. Hence" e D _. Taking: ... " gives" = O. •
C.7 Propoeltlon. The self-adjoint extensions of a .rymmetric densely defined operator A (if any) are obtained as follows. Let T: D, ..... D be an isometry mapping D + onto D _ and let r reD + $ D _ be it.f Kraph. Then the restriction of A· to DA-$ r T is a self-adjoint extension of A. Thus A has self-adjoint extensions iff its defect indices (dim D., dim D _) are equal and these extensions are in one-to-one correspondence with all isometries of D + onto D_. Assuming this result for a moment, we give the following. Completion of Proof of C.S. If there is only one self-adjoint extension it follows from C.7 that D .. = D _ = {O} so by (",2. A is essentially self-adjoint.
•
Proof of C. 7. Let B be a self-adjoint extension of A--: Then (A)* = A· :J B so B is the restriction of A* to some subspace containing DA-. We want to show that these subspaces are of the form DA-$ r T as stated. Suppose first that T: D + ..... D _ is an isometry onto and let A I be the restriction of A· to DA-$ r T' First of all, one proves that A I is symmetric: i.e.• for II. x e DA-and 0, y ED. that (Ax + A*Y + A·Ty,,, + v + Tv) = (x + Y + Ty, A" + A·v + A*Tv).
This is a straightforward computation using the definitions. To show that AI is self-adjoint, we show that DAr c DA,. If this does not hold there exists a nonzero: e DAr such that either Ar: ... i: or Ar: "" - i:. This follows from Lemma C.6 applied to the operator AI' (Observe that Al is a closed operator-this easily follows). Now A I :J A so A*:J Ar. Thus : e D. or: e D_. Suppose Z E D+. Then: + Tz E D,., ~\' as «DA = 0 « denotes the inner product relative to A I)'
,.:»
(.»
0= «: + Tz.z» = «z.z»+«Tz.z» = 2(z.z). since T: ED _. Hence: = O. In a similar way one sees that if : e D _ then z - O. Hence A I is self-adjoint.
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
537
We will leave the details of the converse to the reader (they are similar to the foregoIng). The idea is this: if A 1 is restriction of A* to a subspace D,.-$V for V c 0+$0_ and 01 is symmetric. then Vis the graph of a map T: WcO+-O_ and (Tu.Tv) ... {u.v). for a subspace WCO+. Then , self-adjoint ness of A 1 implies that in fact W = 0+ and T is onto. • A convenient test for establishing the equality of the deficiency indices is to show that T commutes with a conjugation l/; i.e .. an antilinear isometry U: H - H satisfying V 2 = I (antilinear means V( ax) = iiVx for complex scalars a and V(x + y) "" Vx + Vy for .1'. y E " ) . In fact it is easy to see that V is the isometry required from D. t(l n (use 0 + = range (A
+ il).I.). .
As a corollary, we obtain an important classical result· of von Neumann: Let H be· L2 of a measure space and let A be a (closed) symmetric operator that is real in the sense that it commutes with complex conjugation. Then A admits self-adjoint extensions. (Another sufficient condition of a 'different nature, due to Friedrichs, is given below.) This result applies to many quantum mechanical operators. However. one is also interested in essential self-adjointness, so that the self-adjoint extension will be unique. Methods for proving this for specific operators in quantum mechanics are given in Kato (1966) and Reed and Simon [1974). For corresponding questions in elasticity, see Marsden and Hughes (1983). We now give some additional results that illustrate methods 'for handling self-adjoint operators,
C.B Propos'Don_ LeI A he a self-adjoint and B a bounded self-adjoint operator. Then A + B (with domain D,.) is self-adjoint, If A is essentially self-adjOint on DA then so is A + B.
Proof. A + B is certainly symmetric on D,., Let y
E DCA
I
II,' so that for all
XED,.,
«A + B)x, y) -= (.I', (A + B)*y). The left side is
(Ax. y) + (Bx, y)
=
(Ax. y) + (.I'. By)
since B is everywhere defined, Thus
(Ax, y) = (.I'. (A
+ B)*y - By).
HenceyE DA • = D,. and Ay= A*y= (A DA •
+ B)*y - By.
Hence y E
DA
+.
=
I.e • •
_-I. -._.ISh.-.-w_ .
*'
I.- . - ..- - .
~l'Il.
FIr _
~,...
el);. " ' _ a MJ
.., ...... -b.. n.:. ••
• ..ai;::a. ).- ....-- a-
•
' .... ED. tad' lliI£ .... E~
-L
In genera!. Ihe sum of two self-adjoinl operators need not be selfadjoint. (See Nelson (1959) and Chernoff (1974) for this and related examples.)
C.t Propoeltlon. Let.AI be a symmetric operator. If the range of.AI is all of H then.AI is self-adjoint.
Proof. We first observe that A is one-ta-one. Indeed let Ax = O. Then for any yeD,f.O-(Ax.y) = (x. Ay). But A is onto and so x=O. Thus A admits an everywhere defined inverse A - I. which is therefore self-adjoint. Hence A is self-adjoint (we proved earlier that the inverse of a self-adjoint operator is seJr-adjoint). _ We shall use these results to prove a theorem that typifies the kind of techniques one uses.
C.10· PropoeHIon. Let A be a symmetric operator on H and suppose A < 0; that is (Ax. x) < 0 for x e D,f. Suppose 1- A ha.~ dense ran~e. Then A is essentially self-adjoint.
Proof. Note that
«1- A)u.u) .. (u.u) -(Au.u);;. nun
2
and so by the Schwarz inequality we have
n(l- A)un ~ nun· It follows that (I - A) = 1 - A-has closed range. which by hypothesis must be all of H. By C .9. 1 - A is self-adjoint and so by e.s. Xis self-adjoint. •
C.11 Corolla". If.Al;s .~elf-adjo;nt and A EO O. then for any A > O. A- A is onto, (A - .AI)-I exists and
II(A-A)-III<~. Proof.
As before. we have
(3)
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERATORS
53/J
As If is closed. this implies that the range of'\ - If is dosed. If we can show it is dense, the result will follow. Suppose y is orthogonal to the range:
«A-A)",y)-O foraH "eDA • This means that (A - A)* Y == 0, or since A is self-adjoint. ye D... Taking A)y, y) ~ All yll2 so y == o. •
,,== y gives () = «A -
Note that an operator A has dense range iff A* is one-ta-one; i.e.•
A*", = 0 implies", == O.
For a given symmetric operator A, we considered the general problem of self-adjoint extensions of A and classified these in terms of the defect spaces. Now, under different hypothesis, we constru(.'t a special self-adjoint extension (even though A need not be essentially self-adjoint). This result is useful in many applications. including quantum mechanics. A symmetric operator A on H is called lower semi-bounded if there is a constantc e R such that (Ax, x) ~ cllxll 2 for all x e 1>.4' Upper semi-bounded is dermed similarly. If A is either upper or lower semi-bounded then A is called semi-bounded. Observe that if A is positive or negative then A is semi-bounded. As an example. let A ... - V 2 + V where V 2 is the Laplacian and let V , be a real valued continuous funclion and bounded below. say Vex) ~ cr. Let H - L 2(R".C) and DA the COO functions with compact support. Then - V 2 is positive so
(AI. J) =
(-
v 2/. J) + (VI. J) ~ cr( I, J),
and thus· A is semi-bounded. , We already know that this operator is real so has self-adjoint extensions by von Neumann's theorem. However. the self-adjoint extension constructed below (called the Friedrichs extension) is "natura,l." Thus the actual construction is as important as the statement:
C.12 Theorem. A semi-bounded symmetric (densely defined) operator admits a self-adjoint extension.
Proof. After multiplying by - I if necessary and replacing A by A + (I cr)1 we can suppose (Ax. x) ~ IIx1l2. Consider the inner product on DA given by y» - (AX, y). (Using symmetry of A and the preceding inequality one easily checks that "this is an inner product.)
«x,
540
APPENDIX C: UNBOUNDED AND SELF-ADJOINT OPERA TORS
Let H 1 be the completion of D;4 in this inner product. Since the H 1 norm is stronger than the H norm, we have Hie H (i.e.. the injection °D;4 c H elltends uniquely to the completion). Now let H- 1 be the dual of H I. We have an injection of H into H' 1 defined as follows: if y is fixed and x .... (x, y) is a linear functional on H, it is also continuous on HI since
I(x. y) lor;; II xliII yllor;; III xIII II )'11, where 111·111 is the norm on HI. Thus HI c H C H I . Now the inner product on H I defines an isomorphism B: HI I Let C be the operator with domain Dc = (x E H II B( x) E " ). and C .. = Bx for x E Dc. Thus C is an elltension of A. This will be the extension we sought. We shall prove that C is self-adjoint. By definition, C is surjective; in fact C: Dc - H is a linear isomorphism. Thus by C.9, it suffices to show that C is symmetric. Indeed for x. y E Dc we have. by definition.
--+"
(Cx, y) ... «x, y»
= « y. x» == (CY. x) = (x.Cy) . •
The self-adjoint extension C can be alternatively described as follows: Let H I be as before and let C be the restriction of A* to D;40 n H I. We leave the verification as an ellercise.
APPENDIX
D
Stone's Theoremt
Here we give a self-contained proof of Stone's theorem for unhounded self-adjoint operators on a complex Hilbert space H. This guarantees that the one-parameter group e"..4 of unitary opetators exists. In fact. there is a one-ta-one correspondence between self-adjoint operators and continuous one-parameter unitary groups. A 'continuous one-parameter unitary group is a homomorphism t ...... U, from R to the group of unitary operators on H. such that for each x E H the map t ...... u,x is continuous. The infinitesinwl generator A of U, is defined by
I
.
Uh(x)-x iAx = -d u.x = hm ....!!.":"""";'--dt ' ,-0 h ... O h its domain D consisting of those x for which the indicated limit exists. We insert the factor i for convenience; iA is often called the generator.
Theorem (Stone 119320. bl). Let U, be a conlinrlOu., one-parameter unitary group. Then the generator A of U, is self-ad.Joint. (In particular. hy Appendix C, it is closed and dense(v defined.) Conversely. let A be a gh'en self-a4joint operator. Then there exists a unique one-parameter unitary group U, whose generator is A. D.1
t
This appendix was written in collaboration with P. Chernoff. 541
542
APPENDIX D: STONE'S THEOREM
Before we begin the proof. let us note that if A is a bounded self-adjoint operator then one can form the series
U, = ei,A = I + itA +
;!
(itA)2 +
;!
(itA)' + ....
which converges in the operator norm. It is straightforward to verify that U, is a continuous one-parameter unitary group and that A is its generator. Because of this. one often writes e".4 for the unitary group whose generator is A even if A is unbounded. (In the context of the so-called "operational calculus" for self-adjoint operators. one can show that e i ,.4 really is the result of applying the function e ill " to A; however, we shall not go into these matters here.) Proof of S'OM's ,/womn ( lirst IuIIf). Let U, be a given continuous unitary group. In a series of lemmas, we shall show that the generator A of U, is self-adjoint. D.2 ......ma. The domain D of A is invariant under each A U,x - U,Ax for each xeD. PIvof.
u,. and moreover
Suppose xeD. Then
*(U"U,x - U,x) u,( *< U"x - x»). =
which converges to U,(iAx) = iU,Ax as h .... O. The lemma follows by the definition of A. •
D.3 Corollary. A is clo.fed. Proof.
If xeD then. by 0.2
~ U,x =
iA U,x == iU,Ax.
(I)
Hence
Now suppose that x" e D. X,,"" x. and Ax" .... y. Then we have, by (I).
U,x = lim U,x"
"-or:
= lim {x" II-')C
+ il'U.Ax"dT}. 0
APPENDIX D: STONE'S THEOREM
543
(2)
Thus
(Here we have taken the limit under the integral sign because the convergence is unirorm; indeed IIU.Ax" - U.YII = IIAx" - "II --.0 independent or T E [0. I).) Then. by (2). d dt
u,xl
=
iy.
1-0
Hence xED and y - Ax. Thus A is closed. •
D.4 Lemma. A is densely defined. Let x E H. and let (fI be a Coo runction with compact support on R. Derine x. -.J~oo(fl(t)U,xdl. We shall show that each x.,. is in D. and that x = lim" _ ""x•• ror a suitable sequence {(fin}' To take the latter point rirst. let (fI,,(I) be nonnegative. zero outside the interval [0.1/11). and such that f~ oo(fl,,( I) dl - I. By continuity. ir , > 0 is given one can rind N so large that lIU,x - xII < , ir III < 1/ N. Suppose that n> N. Then Proof.
II x •• - xII = 1I{0<Xi(fI,,( 1)( U,x - x) cllil
1o
1/"
=11
L
(fI,,(I)(U,X-X) dIll
illi
.,.; 0
(fI,,(I)IIU,x-xlidl
1
"';E 0
1/"
(fI,,(t)dl=,.
Finally. we show that x. E D; moreover. we shall sh()w that iAx. = - x.'. Indeed.
. . -fO
- GO
da'(fI'(a)·1'u.+fl xdT 0
544
APPENDIX D: STONE'S THEOREM
Integrating by parts and using the fact that cp has compact support. we get
-1'o U.x",d.,
=
JOO (UO+IX - U.,X )cp( 0) do -00
=
(U,_l)jOO U.,xcp(o) do. -00
That is.
-l'o U.x",d.,
=
U,x" - x".
rrom which the assertion follows. • Thus far we have made no significant use of the fact that U, is unitary. We shall now do so. 0.5 Lemme. A is symmetric. Proof.
Take x. y
E
(Ax.y)=
D, Then we have
~ddt(u,x.Y)1 = !~(x·u,*J')1 , ,_ 0 ,dt , _ ()
I -(x.U d = -:-
, dt
= -
,y)
~, (x. iAy) =
II_
0
= -
I -(x.u,y) d -:,
dl
I
1-0
(x. Ay). •
We can now complete the proof that A is self-adjoint. Suppose that y E D*. Pick any xED. Then by (I). (0.1) and (0.5). (U,y.x) = (y.U_lx) -(y.X)+( y.i!'U• AXd .,) =
(y. x) - i1o-/( y.U.Ax)d.,
=
(y. x)- il- /( y. AU.x)d., o
=
(y. x) + i 1o/(UTA* J'. x) d.,
= (
y + ;{UTA*J'dT. x).
APPENDIX D: STONE'S THEOREM
545
Because D is dense. it follows that
u,J'= y+ il'U A*ydT. T
(I
Hence. differentiating. we see that J' E D and A* y = AJ'. Thus A = 1'4*. Proof of SlOW's I_rem (uCOlld Iudf). We are now given a self-adjoint operator A. We shall construct a continuous unitary group U, whose senerator is A.
If>.. > O. then' + >..1'4 2 : DA , -+ H;s hijectil'e. (I + >"1'4 2 ).1: DA , is bounded by I. and DA ,. the domain of 1'4 2 • i.f detl.~('.
D•• Lemma. H
-+
Proof. If A is self-adjoint. so is .fA A. It is therefore enough to estahlish the lemma for>.. '- l. First we establish surjectivity. By C.3 and D.3. if : E H is given there exists a unique solution (x. ,,) to the equations
x- Ay=O Ax+y=:.
2"
From the first equation. x = Ay. The second equation then yields A + Y = ::. so, + A2 is surjective. For xEDA!. note that «(I+A2)x.x)~lIxIl2. so 1I(1+A2)xll~lIxli. Thus' + 1'4 2 is one-to-one and IK' + A2r 11I:s I. Now suppose that" is orthogonal to DA!' We can find a v such that II =< V + A 2v. Then 0= (II. v) = (v + A 2v. v) = IIvll2 + II A vII 2 • whence v'" 0 and therefore
II
=
O. Consequently DA , is dense in H. •
For>.. > O. define an operator A" by A" = A(I + >"A2)-I. Note that A" is defined on all of H because if xE.1! :,cn (1+>"Al)-IXEDA,CD. so A(I + >..A 2 )-IX makes sense. D.7 Lemma. A" is a bounded sel/-adjoint operator. Also. A" and A" commute for all >...fA > O. Proof.
Pick x E H. Then by D.6.
>"111'4 "xII 2 =
(>..A{I + >"A 2
= (>"1'4 2 ( ,
r
IX.
1'4(1 + >..1'4 2 ) -IX)
+ >"A2) -I X .(1 + >..1'4 2) -IX)
«(I + >..A 2 )(1 + >..A 2) -lx.(I + >..1'4 2 ) -Ix)
11(' + >"A 2 )-l xIl 2
so 111'4,,11
546
APPENDIX D: STONE'S THEOREM
We now show that A~ is self-adjoint. First we shall show that if xED. then
'
Indeed. if xED we have AAx E D,f2 by D.6 and so (I + >'A2)AAX - (I + >.A 2 )A{I + >.A 2 ) 'x = A(I + >.A 2 )(I + >'A 2 )-'x =
Ax.
Now suppose xED and)' is arhitrary. Then (AAX.)')=«(I+>'A2) 'Ax,y) =((I+>.A2) 'Ax,{I+>'A 2 )(/+>'A 2 )
I)')
... (Ax. (I + >'A2)-I)') =(X.AA}')· Because D is dense and A ~ bounded, this relation must hold for all x E H. Hence AA is seU-adjoint. The proof that AAA,. .. A"AA is a calculation that we leave to the reader. • Since AA is bounded, we can form the continuous one-parameter unitary groups U/ = ei/,f,. >. > 0 using power series or the results of Section 4.1. Since AA and A,. commute. it follows that A and u,,. commute for every sand t.
u.
0.8 L....m•• IlxEDthenlimA_oAAx=Ax. Proof.
If xED we have AAX - Ax= (I + >.A 2 ) 'Ax- Ax = -
>.A 2 ( 1+ >.A 2 ) "Ax.
It is therefore enough to show that for every y E H. >.A 2 (1 + >'A2)' I)' --+ O. From the inequality IKI + >'A2)YII2 ~ II>'A 2yII 2, valid for >. ~ O. we see that II>.A 2(1 + >'A 2 )-'II';; I. Thus it is even enough to show the preceding equality for all y in some dense subspace of H.
APPENDIX D: STONE'S THEOREM
547
Suppose Y E D.. J, which is dense by D.6. Then IIAA2(I + AA2) -\ yll = AII(J E;
+ AA2)
I A2YII
AIIA2YII.
which indeed goes to zero with A.••
D.I Lemma. For each x E H. limA _ ou,Ax exists. If we ('(II/ the limit u,x. then {u,} is a continuous one-pararneter unitary group. Proof.
We have
'U,AX-U"X= , ,
1,d 0
-{u.Au" dT T '-T }XdT
=ifo'lf.,Au,"-T{AAX- A"x)dT. whence IIu,AX
-
U,"xll
E;
(3)
Itl'IIAAX - A"xll·
Now suppose that xED. Thcn, by D.8, AAx - Ax, so that IIAAxA"xll- 0 as A, po - O. Because of (3) it follows that {U,Axh > 0 is uniformly Cauchy as A - 0 on every compact t-interval. It follows that limA _ p,Ax ... U,x exists and is a continuous function of t. Moreover. since D is dense and all the u,A have norm I. an easy approximation argument shows that the preceding conclusion holds even if x ff: D. It is obvious that each U, is a linear operator. Furthermore.
(U,x.u,y)
=
lim (u,AX.u,Ay) A-O
... lim (x, y) ... (x, y) A-O so U, is isometric. Trivially. Uo = I. Finally.
(V,u,x. y) .. lim (u,Au,X. y) A-O
=<
lim (U,x.U~.y)
A-O
lim (U,Ai.U~.y) = lim (V,~
A-O
so
=
A-O
,x. y)
u.u, = u,.,. Thus. U, has an inverse. namely U
f •
and so U. is unitary.
•
548
APPENDIX D: STONE'S THEOREM
D.10 Lemma. If x EO, Ihen Ux-x lim' - ;Ax.
,-0
I
I'rfIo/. We have U,A x- x=,'l'VA 0 .AAX d T.
Now
(4)
U.AAAx - U.Ax- U. A( AAx - Ax)+ U.AAx- U.Ax- 0
unirormly ror T E [0, IJ as A ..... O. Thus letting A --+ 0 in (4). we get (5)
U,x - x = ;l'U.AxdT II
for all x EO. The lemma follows directly from (5).
D.11
•
Lemma. If . U,x - x I1m
,-0
I
,
=,w
exisls, Ihen x E O. Proof. It suffices to show that x E 0*. the domain of 14*. since 0 = 0*. Let y E D*. Then by 0.10.
(x. ;Ay) _
lim
(x.
, -{l
= -
. (u,x -x. J' )
hm 1-
= -
So (x. Ay)'" (w, y). Thus x
E
_U.....!...!,.l:...,·---=:...J' , -I /
n
1
(i .... y).
0* and so as A is self·adjoint, xED.
•
Let us finally prove uniqueness. Let c( I) be a differentiable curve in H such that C(I) EO and C'(I) - iA(c(I». We claim that C(I) '"" U,c(O). Indeed consider. II( I) = U _ ,cC I). Then
1111(1 + 7)- 11(1)11 = IIU -1-.c(1 + T)- U -I_P.C( 1)11 =
1Ic(1 + T)-U.c(t)1I
= 1I(c(t + 7)-c(I»- (U.c(I)- c(I»II.
APPENDIX D: STONE'S THEOREM
549
Hence lI(t+1")-II(t) -0 1"
as
T -
0, so N, is constant. But 11(1) -11(0) means e(1) = l/,c(O).
•
From the proof of Stone's theorem, one can deduce the following Laplace transform expression for ttie resolvent, which we give for the sake , of completeness.
D.12 Corollary.
Let
Re>. > O. Then for all x
E H.
~'U,xdt.
(>'-iA) ·' x =lr. e II
Proof. The foregoing is formally an identity if one thinh of U, as e"'·. Indeed. if A is bounded then it follows just by manipulation of the power series: One hase"~le'I;l == e-· ,IA - ;;1,. as one can see by expanding both sides; next.
l
Re-I(A-i;l'xdt
= (>' - iA) -I[X -
e- RI~- i;l,X]
II
(integrate the series term by term.) Letting R - 00, one has the result. . Now for arbitrary A we know that u,x == lim" _. ,Pt"x. uniformly on bounded intervals. It follows that
l
oe
(I
('
~IU.xdt = lim ,
1°Ce- ~1U."xJI
,,~II II
= lim
"
(>. -
iA,,)
IX.
I"· II
It remains to show that this limit is (>' - iA)-I X • Now
But (>' - ;A)-IX ED, (see Proposition C.I) and so by D.lt (>'-iA,,)(>.-iA)-IX-(>.-iA)(>.-iA)-'x=x
Because IK>' - iA,,)-11i ~ IRe>'I-1 it follows that
II( >. -
iA) - I X -
(>. -
iA"
r xll- O. I
•
as 1'-0.
550
APPENDI>' D: STONE'S THEOREM
In closing. we mention that many of the results proved have generalizations to continuous one-parameter groups or semi-groups of linear operators in Banach spaces (or on locally convex spaces). The central result. due to Hille and Yosida. characterizes generators of semi-groups. Our proof of Stone's theorem is based on methods that can be used in the more general context. Expositions or this more general context are found in. for example. Kato (1966) and Marsden and Hughes [1983. ch. 6).
APPENDIX
E
The Sard and Smale Theorems
This appendix is devoted to the classical Sard theorem and its infinitedimensional generalil.ation due to Smale. The exposition is inspired hy Ahraham and Rohhin (1967). Recall that a suhset A c R'" is said to have measure zero if, for every f> 0, there exists a countahle covering of A by closed cuhes K, '(with edges parallel to the coordinate axes) such that the sum of the volumes of K, is less than E.. Clearly a countable union of sets of measure zero has measure zero. Before we proceed to the local version of Sard's theorem it will be useful to have at hand two facts concerning sets of measure zero in R"'. E.1 Lemma. Let U c R m be open and A cUbe of mct/I'ure zero. If f: U -+R'" is a C l map, then f(A) has measure zero. Proof. First write A as a countable union of relatively compact sets Cn • If ' we show that A n CIt has measure zero. then A has measure zero since it will be a countable union of sets of measure zero. But Cn is relatively compact and thus there exists M> 0 such that IIDf(x~1 ~ M for all x E en' By the mean value theorem. the image of a cube of edge length d is contained in a cube of edge length dim M. •
E.2 Lemma (FItbi,,;). Let A be a countable union (If compact sets in R" and asslime that Ac = A n({c}XR,,-I) has measure zero for all c E R. Then A has measure zero. 551
552
APPENDIX E: THE SARD AND SMALE THEOREMS
Proof. It is enough to work with one element of the union. so we may assume A itself is compact and hence there exists an interval [a. b) su.:h that A c [a. b)XR,,-·I. Since A, is compact and has measure zero. for each c E [a. b) there exists a finite number of closed cubes Ke 1••••• Ke N in R,,-I the sum of whose volumes is less than E and such tilat (c}x' Ke., cover A,.• i-I ..... Ne • Find a closed interval I,. with c in its interior such that le.xKe.,cAcxR,,-I. Thus the family (Je.xK, .. ,li-I ..... N,.cE[a.b)} covers A n(ra. b)XR,,-I) - A. But since (int(/,.)lcE [a.b)} covers [a. b). we can choose a finite subcovering 1,., ..... 1,..,. Now find another covering J,., •... •./,.• such that each J e., is contained in some I" and such that the sum of the lengths of all J e is less than 2(b - a). Consequently {J,. x K, ,I j = 1•...• K. i = I ....• N, } cover A and the sum of their volumes' is les~ than 2(h - a)E.. '
We are now ready to prove the local version of Sard's theorem. It will be stated for CA-maps. k ~ I. but the proof we give is strictly for Coo-mappings and follows Abraham and Robbin (1967) and Milnor (1965). We shall indicate the only troublesome spot in the proof for the C k case and how one circumvents it; see Abraham and Robbin (1967) for the lengthy technical details. First we recall the following notations from chapter 3. Let M and N be C I manifolds and/: M -+ N. a C I map. A point x EM is a regular point of / iff T, / is surjective; otherwise. x is a critical point of /. If C c M is the set of critical points of /. then /( C) c N is the set of critical values of / and N \/( C) is the set of regular values of f. The set of regular values is denoted by (.'R f or (:~ (/). In addition. for A c M we define (:11 ,IA by (:11 ,IA = N \/( Ann. In particular. if U c M is open. (:11 ,I U = (:~ (/1 U).
E.3 Sard', Theorem In R". Let m ~ n. U c R'" be open. and /: U -+ R" be 0/ class CA. where k ~ 1 and k > max(O. m - n). Then the .fet 0/ critical values 0/ / has measure zero in R". Proof. . Denote by C = (x E Ulrank D/(x) < n} the set of critical points of If m = O. then.R'" is one
f. We shall show that/(C) has measure zero in R".
point and the theorem is trivially true. Suppose inductively the theorem holds for m - I. Let C, = (x E UIDJ/(x) = 0 for j = I ..... i}. Then C is the following union of disjoint sets C=(C\C1)U(C1\C2 )U'" U(CA_1\CdU{j. The proof that /(C) has measure zero is divided in thr< I.
2. 3.
/( C~ ) has measure zero. /(C\C 1 ) has measure zero. /( C.\C• • I) has measure zero for 1 ...
of ...
k - I.
1'"
APPENDIX E: THE SARD AND SMALE THEOREMS
553
Proof of Step 1. Since k ~ 1.( k -I)n ~ k -I; i.e.• kn ~ n + k -1. But we also have k ~ m - n + 1. so that n + k - I ~ m; i.e .• m $ kn. Let K c V be a closed cube with edges parallel tn the coordinate axes. We will show that/(C, () K) has measure zero. Since CAcan be covered by countably many such cubes. this will prove that I( CA) has measure zero. By Taylor's theorem, the compactness of K. and the definition of CA' we have
I( Y) = I(x)+ R(x. y)
where (I)
for x E CA() K and y E K. Here M is a constant depending only on DAf and K. Let e be the length of the edge of K. Choose an integer I. subdivide K into In! cubes with edge e / k. and choose any cube K' of this subdivision which intersects CA' For x E C, () K' and y E K'. we have' IIx - yll ~ .[m (e / /). By (I )./( K ') C L where L is the cube of edge Nit - I with center I(x); N = 2M«m)I/2/)A t I. The volume of L is N"I--II(" I). There are at' most I'" such cubes; hence. I( C, () K) is contained in a union of cubes ' whose total volume V satisfies.
, Since m ~ kn we have m - n(k + \) < O. so V - 0 as / ..... 00. and thus I( CA() K) has measure zero. Proolof Step 2. KII _I' where
Write C\C I = (XE VII Kq
=
~
rank D/(x) < fl} = KI U
... U
(xEVlrank D/(x) =q}
and it suffices to show that/( Kq) has measure zerQ for q = I. .... n - I. Since Kq is empty for q > m, we may assume q ~ m. As before it will suffice to show that each point of Kq has a neighborhood V such that I( V () Kq) has measure zero. Choose xE K q • By the Local Representation Theorem 2.5.14 we may assume that x has a neighborhood V"" VI X V2 where VI c R It, - q and V2 c R q are open balls such that for x E VI and t E V2 /(x.t)= (1J(x,t).t).
1J,(x)=1J(x.t)
for
XE,VI .
554
APPENDIX E: THE SARD AND SMALE THEOREMS
Then for I
E
V2
This is because, for (x, I) E
VI X V2' Df(x, t)
Df ( X,I ) = [
DlI,( x) 0
is given by the matrix
;J
Hence rank Df(x, I) = q iff DlI,(x) = o. Now 11, is C· and k ~ m - n = (m - q)-(n - q). Since q ~ I, by induction we find that the critical values of 11, and in particular 1I,({ x EVil DT/,( x) = O}) has measure zero for each IE V2 • By Fubini's lemma,J(K q () V) has measure zero. Since Kq is covered by countably many such V, this shows thatf(K q ) has measure zero.
Prool 01 Step J. To show f(C,\C•• I) has measure zero, it suffices to show that every x E C.\Cs+ I has a neighborhood V such that f( C, () V) has measure zero; then since C.\C.. I is covered by countably many such neighborhoods V, it follows that f( C,\C,. I) has measure zero. Choose Xo E C,\ C•• I' All the partial derivatives of f at Xo of order less than or equal to s are zero. but some partial derivative of order .f + I is not zero. Hence we may assume that Dlw( Xo )
'*' 0 and w( xu) = O.
where DI is the partial derivative with respect to .\" I lind w( x) = D" ... D,.f( x).
Define h: U _R m by
where x = (XI' X2'" •• x",) E U c R"'. Clearly h is C· - 'and Dh(xo) is nonsingular; hence there is an open neighborhood V of Xo and an open -set We R m such that
h: V-W is a C 4 • diffeomorphism. Let A = C. () V, A' = h( A) and g = h - I. We would like to consider the function fog and then arrange things such that we can apply the inductive hypothesis to it. If k = 00, there is no trouble.
APPENDIX E: THE SARD AND SMALE THEOREMS
555
But if k <~. then unfortunately f 1{ is only C* -, and the ind~ctive hypothesis would not apply anymore. However, all we are really interested in is that some function F: W -+ R exists which is CA. F( x) = U g)( x) for all x E A' and DF( x) = 0 for all x E A'. The existence of such a function is guaranteed by two technical theorems, the Whitney extension theorem and the Kneser-Glaeser rough composition theorem for whose proofs we refer to Abraham and Robbin (1967). For k = 00, we take F= log. In any case, define the open set Wo c R m I by 0
If
and
0
Fr.): Wo -+ R'" by
Let S = {(x 2 , .•. ,X nr )E Wo IDFo(x 2 , .•• ,x m ) = O}. By the induction hypothesis, Fo( S) has measure zero. But A' = h( C, V) cOx S since for x E A', DF(x) = 0 and since for x E C, n v.
n
because w is an s th derivative of f. Hence f( C. n V) = F( h (C, n V»)
c
F(O x S) =
Fr., ( S)
Hence f(C. n V) has measure zero. As C,\Cs + I is covered by countably many such V, f( C.\ C. + I) has measure rero (s = I •... , k - 1) and the theorem is proved. • We proceed now to the global version of Sard's theorem on finitedimensional manifolds. First recall that a subset of a. topological space is residual iff it is the countable intersection of open dense sets. The Baire category theorem asserts that a residual subset of a complete metric space is dense. The same holds for locally compact spaces (see A.S). A topological space is called Lindeloj iff every open covering has a countable subcovering. In particular, second countable topological spaces are Lindelof. (See 1.1.6).
E.4· Sard's Theorem for Manifolds. LeI M and N be finite-dimensional Ck'manijolds, dim(M)=m, dim(N)=n and f: M-N a C A mapping. Assume in ~ n, k ~ I, M is Lindelof and k > max(O. m - n). Then ~'~'f is residual and hence dense in N.
556
APPENDIX E: THE SARD AND SMALE THEOREMS
Proof. Denote by C the set of critical points of I. We will show that every x e M has a neighborhood Z such that ~II Z is open and dense. where Z- cl(Z). Then. since M is LindelOf we can find a countable cover {Zi} of X with ctlt/lZ, open and dense. Since ctlt / ... n i~/IZ,. it will follow that ~/is residual. Choose xe M. We want a neighborhood Z of x with ~/IZ open and dense. By taking local charts we may assume that M is an open subset of R m and N = R". Choose an open neighborhood Z of x such that Z is compact. ThenC""{xeMlrankDI(x)
E.I propoeltlon. Let I: M -+ N be a C I mapping 01 manilolds. Then the set 01 regular points is open in M. Consequently the set 01 critical points 01I is closed in M. Proof. It suffices to prove the proposition locally. Thus, if E. F are the models of M and N. respectively. and x E U c E is a regular point of I. then DI(x)E SL(E. F). SinceDI: U -+ L(E. F lis continuous. (Df)-I(SL(E. F» is open in U h" the lemma. •
APPENDIX E: THE SARD AND SMALE THEOREMS
557
E.7 Example. If M and N are Banach manifolds, the Sard theorem is false without further assumptions. The foIlowing counterexample is. as far as we know. due to Bonic. Douady. and Kupka. Let E = {x == (XI' .1'2' ... )1 x/ e R.llxll 2 = Ej.I(XJj)2 < oo), which is a Hilbert space with respect to the usual algebraic operations on components and the inner product (x. y) = Ej.,xj'yj//. Consider the mapf: E ~ R given by f(x) = Ej.I(-2x]+3x})/2 J , which is defined since xEf.' implies Ix,l 0 and thus
i.e.• the series f(x) is majorized by the convergent series c"Ej.,jJ/2 j • We have Df(x)·.,-Ej.,6(-x}+x j )vj /2}; i.e.• f is C I . In fact f is C"". Moreover Df( x) == 0 iff all coefficients of v} are zero. i.e.• iff Xj = 0 or x j ... I. Hence the set of critical points is {x e Elx; = 0 or I} so that the set of critical values is
{J(x)lxj=O or X;=I}={
i
E 21.1'11.1'=0 =
I
I
or Xj=l} =,10',1].
But clearly [0. 1) has measure one, .. Sard's theorem holds. however. if enough restrictions are imposed on f. The generalization we consider is due to Smale [1965). The class of linear, operators allowed are called Fredholm. These operators. by definition. enjoy splitting properties similar to those in (he Fredholm alternative theorem (7.5.6).
E.8 DefInition. Let E, F be Banach spaces and A Fmlholm opef'tllor if:
E
1.( E. F). Then A
i.~
a
(i) A is dOuble splitting; i.e .• both the kernel and the image of A are closed and have closed complement; (ii) ihe kernel of A is finite dimensional; (iii) the range of A has finite codimension.
In this case, if n = dim(ker A) and p = codim(range( A». n - p is the iNkx of A; in symbols, index(A)= n - p. If M and N are C I manifolds and if f: M ..... N is a C I map. then f i,~ a FlWIItoim map if for et'el}' X EM. Txf is a Fredholm operator.
A finite-dimensional subspace is closed and splits: hence (i) may be replaced by the weaker condition that the image of A is closed. A map g
558
APPENDIX E: THE SARD AND SMALE THEOREMS
between topological spaces is called locally closed if every point in the domain of definition of S has an open neighborhood U such that slel U is a closed map (i.e.• maps closed sets to closed sets).
E.I Lemm.. A Fredholm map
i.~ local(~'
dosed.
Proof. By the Local Representation Theorem we may suppose our Fredholm map has the form/: B, X B2 .... A P X E where B, open unit balls and for x E B, and e E B2 •
C
Alt and B2 c E are
I(x.e)= (,,(x.e).e)
Let D, and D2 be open balls so that cI D, c B, and cI D2 c B2 • Let U., D, X D2 so that clU = cI D, xci D 2 • Then/lclU is closed. For if A c clU is closed. we see as follows that I( A) is closed. Choose a sequence {( Yj. ej)} such that (y;.e;) .... (y.e) as i .... 00 and (y;.e;)E/(A). say (y,.e;) = I(x;.e;)
where (x,.ej)E A. Since x; E cI D, and cl D, is compact. we may assume x; - xE D,. Then (x;.e;) .... (x,e). Since A is closed, (x.e) EA. and/(x.e) = (y.e), so (y,e)E I(A). Thus/(A) is closed. •
E.10 Sm...•• Den.lty Theorem. Let M and N be C· manilolds with M LindeliJl and I: M .... N a C l Fredholm map. Suppose that k> max(O, index( T..!» lor every x E M. Then ~~ I is a residual subset 01 N.
ProoJ.
It suffices to show that every Xo EM has a neighborhood Z such is open dense. Choose Xo E M. We shall construct a neighborhood Z of Xo so that ~1l/IZ is open dense. By the Local Representation Theorem we may choose charts (U. a) at Xo and (V. P) at/(x o ) such that a(U) c A' X E. P( V) cAP X E and the local representative la/l = polo a - , of I has the form
that
~1l/1 Z
la/l(x.e) "" (,,(x.e).e)
for (x,e)Ea(U). (Here xEA".eEE. and ,,: a(U) .... AP.) The index of T.J is n - p and so k > max(O. n - p) by hypothesis. We now show that '!it (/I U) is dense in N. Indeed it suffices to show that ~R 1._ ·is dense in RPxE. For eEE. (x,e)Ea(U). define 'I ..(x)='I(x.e). Then for each e.1I.. is a C' map defined on an open set of An. By Sard's theorem. ~1l (".. ) is dense in R n for each e E E. But for (x, e) E a(U) C An X E, we have Dla/l(x.e) = [D1J (X)
O
;]
APPENDIX~:
THE SARD AND SMALE THEOREMS
559
so Dfafj(x, e) is surjective iff D'II.(x) is surjective. Thus for e E E 1ilt('II,)x{e}=I!"R!. n(RP x{e})
.-
Thus lilt!. intersects every plane RP x{e} in a dense set and is. therefore. dense in "' P x E. and ~t (fl U) is dense as claimed. Now by Lemma E.9 we choose an open neighhorhood Z of .to such that Zc U andflZis closed. where Z= c1(Z). By Proposition E.6 the set C of critical points of fis closed in M. Hence,f(ZnC) is closed in N. Hence ~,IZ- N\f(Zn C) is open in N. Since '3\,(fl U) C ~R ,
E.11 Theorem. Let M and N be C k manifolds. M being Lindelof. having a boundary aM. and N being boundaryless. Let f: M -+ N be a C k ' Fredholm map and let af = flaM. If k > max(O.index(T,.f» for every x EM,. then ~It, n ~ ill is residual in N. Proof. By c;iefinition. Tx( af) == T,.(f)1 TxC aM) for x E aM. Thus. if."x is a regular point for then it is also regular for I. Hence yEN is a critical value of both I and a f iff it is a critical value of flint M or af. But lnt M and aM are both boundaryless. so Smale's theorem applies. giving '3I'f and Iilt al residual in N. Consequently '31, n ~ is residual in N. •
al.
a,
Sard's theorem deals with the genericity of the surjectivity of the derivative of a map. We conclude with a hrief consideration of the dual question of genericity of the injectivity of the derivative of a map.
E.12 Lemma.
The set IL(E. F) of linear continutJlIJ ,~plil injective maps is
open in L(E; F). Proof. Let A E lL(E. F). Then A(E) is closed and F = A( E)$G for G a closed subspace of F. The map A: EX G -+ F;Ale.g) == A(e)+ g is clearly linear, bijective. and continuous. so by Banach's isomorphism theorem AEGL(EXG.F). The map P: L(ExG.F) ..... L(E.F);P(B)==BIE is linear. continuous, and onto. so by the open mapping theorem it is also an open mapping. Moreover P(A) = A and P(GL(E X G. F)c IL(E. F) for if BE GL(E X G. F) then F== B(E)$B(G) where both B(E) and B(G) are
560
APPENDIX E: THE SARD AND SMALE THEOREMS
closed in F. Thus A has an open neighborhood P(GL(E x G. F» contained' in IL(E.F). • Note that even in finite dimensions IL(E. F) is empty if dim E > dim F and so E.12 is vacuously true in this case.
E.13 PropcHtlUon. Let f: M -.. N be a Cl-map of manifolds. The set p .. {x E Mlf is an immersion at x} is open in M. Proof. It suffices to prove the proposition locally. Thus if E and F are the models of M and N respectively and if f: U -.. E is immersive at x E U c E. then Df(x) E IL(E. F). By the lemma. (Df) -I(IL(E. F» is open in U since Df: U -.. L(E. F) is continuous. • This result can be improved if M is finite dimensional. This is done in the next theorem whose proof relies ultimately on the existence and uniqueness of integral curves of CI-vector fields.
E.13 Theorem. Let f: M -.. N be a C I injective map of manifolds. dim(M)= m. The set P = {x E Mlf is immersive at x} is open and dense in M. In particular. if dim{N) = n. then m ~ n. Proof. (D. Burghelea). It suffices to work in a local chart V. We shall use induction on k to show that ell!" .. " :::) V. where
U, .. ; = I
,
{XE vi TJ(~) ax'l ..... TJ(~) ax" linearly independent} .
The case k - n gives then the statement of the theorem. Note that by the preceding proposition. U,' ... " is open in V. The statement is obvious for I since if it fails would vanish on an open subset of V and thus f would be constant on V. contradicting the injectivity of f. Assume inductively that the statement for k holds: i.e.. U" ... " is open in V and cI U" ... ;':::) V. Define
k-
U'
'1. -I
=
{
X E
U
'1 "'"
TJ
I ( a) * T
xl ".'4.'
0
(1.\
.
and notice Ihat it is open in l~ , and thus in V. It is (by the case k = I) and hence i~ ~! hy induction. Let
u,~ .. '".
I
= { X E
l{ .ITxf( ~) ..... Txf( ~). I ax" iJx" . I
CU"".;,."
} 01'
,tense in U"."
linearly independent}
APPENDIX E: THE SARD AND SMALE THEOREMS
661
We prove that U:, ... i • • , is dense in 11;:. " which then shows that cll1;""i, .. ::::> V. If this were not the case, there exists an open set We Uf.., such that
'al(x)TJ(~)+ ... +ak(x)TJ(~)+TJ(_a_) ... o ax" ax', ax l ,., for some C l functions a l •. ..• a k nowhere zero on W. Let c: (- t, t) .... W be an integral curve of the vector field
a +--. a-E~XI( W). aI-a. + ... +a k --. ax" ax" ax" . , Then
(f 0 c )'(1) = T..("f(c'(I»'" O. so that foe is constant on (- e. e) contradicting injectivity of f.
•
References
Abraham. R. 1963. Lectures 0/ Smale on Differential Topolo1'J·. Notes. Columbia University. Abraham. R. and J. Marsden. 1978. Foundations 0/ Mechanics. 2nd ed. AddisonWesley. Reading. Mass. Abraham. R. and J. Robbin. 1967. Transversal Mappings and Flows. Addison-Wesley. Reading. Mass. Adams. R. A. 1975. Sobolev Space.~. Academic Press. New York. Amol'd. V. I. 1978. Mathematical Method.~ 0/ Classical Mechanics. Springer Graduate Texts in Mathematics. Vol. 60. Springer-Verlag. New York. Amol'd. V. I. and A. Avez. 1967. Theorie ergm/ique cie.f .~,rsleme.f dJ'namiques. Gauthier-Villars. Paris (English ed .• Addison-Wesley. Reading. Mass .. 19(11). Ball. J. M .• J. E. Marsden. and M. Slemrod. 19112. Controllability for distrihuted bilinear systems. SIAM J. Control and Optim. 20. 575 -597. Batchelor. G. K. 1967. An Introduction to Fluid Dynamics. Camhridge Univ, Press. Cambridge. England. Berger. M, and D. Ebin. 1969. Some decompositions of the 8pac:e of symmetric tensors on a Riemannian manifold. J. Di//. Geom. 3. 379-392. Birkhorr. G. D. 1931. Proof of the ergodic theorem. Prot.'. Nat. A (·ad. Sci. 17. 656-660. Bleecker. D. 1981. Gauge Theory and Variational Principles. Global Analysis: Pure and Applied. Vol. I. Addison-Wesley. Reading. Mass. Bonic. R .• and J. Frampton 1966. Smooth functions on Banach manifolds. J. Math. Mech. 16.877-898. Bonic. R .• and F. Reis. 1966. A characterization of Hilbert space. Acad. Bras. de Cien. 38. 239-241. Bony. J. M. 1969. Principe du maximum. inegalite de Harnack et unicite du probleme de Cauchy pour les operateurs elliptiques degeneres. Ann. Inst. Fourier. Grenoble 19. 277-304.
562
REFERENCES
563
Born, M., and L. Infeld. 1935. On the quantization of the new field theory. Proc. Ro/. Soc. A. 150, 141-162. Bott, R. 1970. On a topological ohstruction to integrability. Pro(·. Symp. Pure Malh. 16. 127-131. Bourbaki, N. 1971. Varie-tes dirrerentielles et analytiques. Fasdmle de re.~lIllau JJ. Hermann. . Bourguignon. J. P. 1975. Une stratification de I'espace des structures riemanniennes. Compo Malh. 30. 1-41. Bourguignon. J. P .• and H. Brezis. 1974. Remarks on the Euler equation. J. Funct. An. 15,341-363. Bowen, R. 1975. Equilibrium Slates and the Ergodic Theon' 0/ Anosov Dil/eonrorphisms. Springer Lecture Notes in Mathematics. 470. Bredon. G. E. 1972. Introduction to Compact Trans/omit/lilli' (iroups. Academic Press. New York. Brezis. H. 1970. On a Characterization of Flow Invariant Sets. C"mmun. Pllre Appl.· Math D. 261-263, Brockett, R. W. 1970. Finite Dimensional Linear Systems. Wiley. New York. Brockett, R. W. 1983. A Geometrical Framework for Nonlillear Control and Eslimation. CBMS Conference Series. SIAM. Buchner, M .• J. Marsden, and S. Schecter. 1982. Applications of the blowing-up construction and algebraic geom~try to bifurcation problems. J. Di/l Eq.r. (to appear). Buchner, M .• J. Marsden, and S. Schecter. 1983. Examples for the infinite dimensional Morse lemma. SIAM J. Math. An. (to appear). Burghelea, D .• A. Albu. and T. Ratiu. 1975. Compact lie grollp aellon.r (in Romanian). Monografii Matematice 5, Universitatea Timisoara. Burke. W. L. 1980. Spacetime. Geometry. Co.rmol01{1'. Uniwrsity Sl'ience Books. Mill Valley. Ca. Cantor, M. 1981. Elliptic operators and the decomposition of tensor fields, BIII/. Am . Math. So('. 5. 235-262. Caratheodory. C. 1909. Untersuchungen nber die Grund1agcn der Tbermodynamik. Math. Ann. 67. 355-386. Caratheodory, C. 1965. Calculus 0/ V'fIrialions and Partial Dil/t'renlial Eq/lations. Holden-Day, San Francisco. ('artan. E. I94S. [.,., ".I'.'t;mr., dif!l'rr",leL, r.f(l~rirur.~ rtlrlfr tlppllnllwn., ghmlelrlqlll'.,. Hermann, Paris. Chernoff. P. 1974. Product Formulas, Nonlinear Semigroups and Addition of Unbounded Operators. Memoirs of Am. Math. Soc. 140. Chernoff. P., and J. Marsden. 1974. Properties o/In/inite Dimensional Hamiltonian Systems. Springer Lecture Notes in Mathematics 415. Chevalley, C. 1946. Theory 0/ Lie groups. Princeton University Press, Princeton, N.J. Choquet. G. 1969.l..ecture.~ on Alla~vsis, 3 vols. Addison-Wesley, Reading. Mass. Choquet-Bruhat. Y .• C. DeWitt-Morette and M. Dillard-Bleick, 1977. Analysis. Manifolds and Ph)·sics. North-Holland, Amsterdam. Chorin, A. and J. Marsden. 1979. A Mathefll/lticallntroduction to Fluid Mechanics. Springer Univeritext.
564
REFERENCES
Cborin, A. J., T. J. R. Hughes, M. F. McCracken, and J. E. Marsden. 19711. Product formulas and numerical algorithms, Commun. Pure Appl. Math. 31, 205·-256. Chow, W. L. 1947. Ober systeme von linearen partiellen differential Gleichungen, MfIlh. Ann. 117,89-105. Chow, S. N. and J. K. Hale, 1982. Methods of Bifurcation Theory. Springer. New York. Chow, S. N., J. Mallet-Paret and J. Yorke. 1978. Finding zeros of maps: homotopy methods that are constructive with probability one. Math. Compo 31, 887-899. Clebsch, A. 1859. Ober die Integration der hydrodynamischer Gleichungen, J. Reine Angew. Math. 56, 1-10. Cook, J. M. 1966. Complex Hilbertian structures on stable linear dynamical systems, J. Math. Mech. 16, 339-349. Craioveanu, M., and T. Ratiu. 1976. Elements of Local Analysis, Vol. I, II (in Romanian). Monografii Matematice 6, 7. Universitatea Timisoara. Crandall, M. O. 1972. A generalization of Peano's existence theorem and now . invariance. Proc. Am. Math. Soc. 36,151-155. deRham, G. 1955. Varie;es differentiable.f. Hermann, Paris. Duff. G .• and D. Spencer. 1952. Harmonic tensors on Riemannian manifolds with' boundary. Ann. Math. 56. 128- 156. Duffing, O. 1918. Erzwungene Schw;ngungen be; veranderlichen E;genlrequenz. Vieweg u. Sohn, Braunschweig. Ebin, D., and J. Marsden. 1970. Groups of diffeomorphisms and the motion of an incompressible nuid, Ann Math. 91, 102-163. Eells, J. 1958. On the geometry of function spaces, in Symposium de Topologia Algebrica. Mexico UNAM, Mexico City, pp. 303-307. FJliasson, H. 1967. Geometry of manifolds of maps, J. Dilf. Geom. I, 169-194. FJworthy, K. D., and A. Tromba. 1970a. Differential structures and Fredholm maps on Banach manifolds, Proc. Symp. Pure Math. 15,45-94. FJworthy, D., and A. Tromba. I970b. Degree theory on Banach manifolds, Proc. Symp. Pure Math. II, 86-94. . Fischer. A. 1970. A theory of superspace, in Relativity, M. Carmelli et aI. (Eds.). Plenum. New York. Fischer, A., and J. Marsden. 1975. Deformations of the scalar curvature. Dulce Math. J. 41, 519-547. Fischer, A.• and J. Marsden. 1979. Topics in the dynamics of general relativity, in Iso/atu Gravitating Systems in General Relativity. J. Ehlers (ed.). Italian Physical Society, North-Holland, Amsterdam, pp. 322-395. Flanders, H. 1963. Differential Forms. Academic Press, New York. Fraenkd, L. E. 1978. Formulae for high derivatives of composite functions, Math. Proc. Camb. Phil. Soc. 13, 159-165. Frampton, J., and A. Tromba. 1972. On the classification of spaces of Holder continuous functions, J. Funct. An. 10.336-345. Fulton, T., F. Rohrlich, and L. Witten. 1962. Conformal invariance in physics. Rev. Mod. Phys. 3<1. 442-457. Gaffney, M. P. 1954. A special Stokes's theorem for complete Riemannian manifolds,Ann. Math. fiG. 140-145.
REFERENCES
565
Glaeser. G. 1958. Etude de quelques algchres Tayloriennes, J. Anal. Math. II, 1-118. Goldstein. H. 1980. Classical Mechanics, 2nd ed. Addison-We~kv. Reading. Mass. Golubitsky. M., and V. Guillemin. 1974. Stable mapping' alld their singularities. Graduate Texts in Mathematics, Vol. 14. Springer-Verlag. New York. Golubitsky, M. and J. Marsden. 1983. The Morse lemma in infinite dimensions via singularity theory, SIAM J. Math. An. (to appear). Graves, l. 1950. Some mapping theorems, Duke Math. J. 17. 111-114. Guckenheimer.1. and P. Holmes. 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Applied Math. Sciences, Vol. 43. Guillemin, V. and A. Pollack. 1974. Differential Topology. Prentice-Hall, Englewood Cliffs, N.J. Guillemin, V. and S. Sternberg. 1977. Geometric A.~ymptotics. American Math~mati cal Society Survey, Vol. 14. Gurtin, M. 1981. An Introduction to Continuum Mechanics. Academic Press, New York. ' Hale, J. K. 1969. Ordinary Differential Equation.v. Wiley-Interseicnce, New York. Halmos, P. R. 1956. Lectures on Ergodic Theory. Chelsea, New York. Hamilton, R. 1982. The Inverse function theorem of Nash and Moser, Bull. Am. Math. Soc. 7,65-222. Hartman, P. 1972, On invariant sets and on a theorem of Wazewski, Proc. Am. Math. Soc. 32, 511-520. Hartman, P. 1973. Ordinary Differential Equations, 2nd ed. Reprinted by Birkbliuser. Boston. Hawking. S.• and G. F. R. Ellis, 1973. The Large Scale Struclllre of Space-Time. Cambridge Univ. Press, Cambridge. England. Hayashi, C. 1964. Nonlinear Oscillations in Physical Syst<'fflS. McGraw-Hili. New York. Hermann. R. 1973. Geometrv. Physics, and Systerm. Marcel Dekker. New York. Hern\ann. R. 1977. Differe~tial Geometry a~d the Calculus of Variations. 2nd ed. Math. Sci. Press, Brookline. Mass. Hermann, R. 1980, Cartanian Geometry. Nonlinear Waves and Control Theory. Part B. Math. Sci. Press. Brookline. Mass. Hermann. R. and A. J. Krener. 1977. Nonlinear controllability and observability. IEEE Trans. on Auto. Control. 21,728-740. . Hildebrandt. T. H .• and L. M. Graves. 1927. Implicit functions and their differentials, in general analysis. Trans. Am. Math. Soc. 29, 127 -53. Hirsch, M. W. 1976. Differential Topology. Graduate Texts in Mathematics. Vol. 33. Springer-Verlag, New York. Hirsch. M. W:. and S, Smale. 1974. Differential Equations. DYllamical Systems and Linear Algebra. Academic Press. New York. Hodge. V. W; D. 1952. Theory and Applications of Harmonic Integrals, 2nd ed. Cambridge University Press. Cambridge. England. Holmann. H., and H. Rummier. 1972. Alternierende Differentialformen. BI-Wissenschaftsverlag. ZUrich. Holmes. P. 1979a. A nonlinear oscillator with a strange attractor. Phil. Trans. Roy.
566
REFERENCES
Soc. A 19l, 419-448. Holmes, P. 1979b. Averaging and chaotic motions in forced oscillations, SIA M J. on Appl. Math. 38, 68-80, and 40, 167-168. Husemoller, D. 1975. Fibre Bundles, 2nd. ed. Graduate Texts in Mathematics, Yol. 20. Springer-Yerlag, New York. Irwin, M. C. 1980. Smooth dynamical systems. Academic Pres~. John. F. 1975. Partial Differential Equations. 2nd ed. Applied Mathematical Sciences. Yol. I. Springer-Yerlag. New York. Karp. L. 19111. On Stokes' theorem for non-compact manifolds. PmC'. Am. Math. Soc. 82. 4117-490. Kato. T. 1%6. Perturbation Theory for I.inear Operators. Springer. (Second edition. 1977). Kelley. J. 1975. General Topology. Yan Nostrand. New York. Klingenberg. W. 1978. Lectures on Closed Geodesics. Grundlehren der math. Wissenschaften 23G, Springer. Knowles, G. 1981. An Introduction to Applied Optimal Control. Academic Press, New York. Kobayashi, S., and K. Nomizu. 1963. Foundations of Differential Geometry. Wiley, New York. Kodair&, K. 1949. Harmonic fields in Riemannian manifolds, Ann. Math. 50, 587-665. Koopman, B. O. 1931. Hamiltonian systems and transformations in Hilbert space, Proc. Nat. Acad. Sci. 17,315-318. Lang. S. 1972. Differential Manifolds. Addison-Wesley, Reading, Mass. Lawson, H. B. 1977. The qualitative theory of foliatiOns. American Mathematical Society CBMS Series, vol. 17. Lax., P. D. 1973. Hyperbolic Systems of Conservative Laws and the Mathematical Theory of Shock Waves. SIAM, CBMS Series, vol. 11. Leonard, E.. and K. Sunderesan. 1973. A note on smooth Banach spaces, J. Math. Anal. Appl. 43. 450-454. Linrlenstrauss, J., and L. Tzafriri. 1971. On the complemented subspace problem, Israel J. Math. 9, 263-269. Loomis. L .• and S. Sternberg. 1968. Advanced Calculus. Addison-Wesley, Reading.
Mass. Luenberger, D. G. 1969. Optimization hy Vector Space Methods. John Wiley. New York. Mackey, G. W. 1962. Point realizations of transformation groups. II/inois J. Math. 6, 327-335. Mackey. G. W. 1963. Mathematical Foundations of Quantum Mechanics. AddisonWesley. Reading, Mass. Marcinkiewicz. J., and A. Zygmund. 1936. On the differentiability of functions and summability of trigonometric series. Fund. Math. 16, 1-43. Marsden. J. E. I 968a. Generalized Hamiltonian mechanics, Arch. Rat, Mech. An. za,326-362. Marsden, J. E. 1968b. Hamiltonian one parameter groups, Arch. Rat. Mech. An. za: 362-396.
REFERENCES
667
Marsden. J. E. 1972. Darboux's theorem fails for weak symplectic forms. Proc. Am. Math. Soc .• 31. 590-592. Marsden. J. E. 1973. A proof of the Calderon extension theorem. Can. Math. Bull. . 16. 133-136. Marsden. 1. E. 1974a. Elementary Classical Analysis. W. H. Freeman. San Francisco. Marsden. J. E. 1974b. Application, of Global Ana(vsis in Mathematical Physics. Publish or Perish. Waltham. Mass. . Marsden. J. E. 1976. Well-posedness of equations of non-homogeneous perfect nuid. Comm. PDE t. 215-230. Mar~den. J. E. 19111. IRcture., 0/1 Geometric Methods in Mathemllliral Phy.vics. CBMS Vol. 37. SIAM. Philadelphia. Marsden. J. E .• and T. J. R. Hughes. 19K3. Mathematical Foundations of Elastici~~·. Prentice-Hall. Redwood City. Calif. Marsden. J. E .• T. Ratiu. and A. Weinstein. 1982. Semi-direct products and reduction in mechanics. Trans. Am. Math. Soc. (to appear). Marsden. J. E. and A. Tromba. 1981. Vector Calculus. W. 11. Freeman. San Francisco. Marsden. 1. E .• and A. Weinstein. 1982. The Hamiltonian structure of the MaxwellVlasov equations. Ph.vsica D. 4. 394-406. Marsden. J. E .• and A. Weinstein. 1983. Coadjoint orbits. vortices and C1ebsch. variables for incompressible nuids. P/r.vsica D. (to appear). Martin. R. H. 1973. Differential equations on closed sub~ets of a Banach spal.'C. Trans. Am. Math. Soc. 179.339-414. Massey. W. S. 1967. Algebraic Topology: An Introduction. Harcourt. Brace. New York. Mayer. A. 11172. Ober unbeschrllnkt integrable Systeme von Iinearen Differentialgleichungen. Math. Ann. 5.448-470. Milnor. 1. 1956. On manifolds homeomorphic to the 7-spherc. Ann. Math. 64. 499-'505. Milnor. 1. 1965. Topology from the Differential Viewpoint. University of Virginia Press. Charlottesville. Va. Misner. C. K.'Thome. and J. A. Wheeler. 1973. (lravitatio". W. H. Freeman. San Francisco. Morrey. C B. '1966. Multiple Integrals in the Calculus of Variatiom. Springer-Verlag. New York. ' Moser. J. 1965. On the volume elements on a manifold. TrailS. Am. Math. S(1c. 1l0. 286-294. Nagumo. M. 1942. Ober die Lage der integralkurven gewl'lhnlicher Dirferentialgleichungen. Proc. Phys.-Math. Soc:. Jap. 14. 551-559. Nelson. E. 1959. Analytic vectors. Ann. Math. 70. 572-615 Nelson. E. 1967. Tensor Ana(vsis. Princeton University Press.• Princeton. N. J. . Nelson. E. 1969. Topics in Dynamics. I: Flows. Princeton University Press.• Princeton. N. J. Nirenberg. L. 1974. Topics ill Nonlinear Analysis. Courant Institute Lecture Notes. Oster. G. F .• and A. S. Perelson. 1973. Systems, circuits and thermodynamics. Israel J. Chem. 11.445-478. and Arch. Rat. Mech. All. 55. 230-274. and 57. 31-98.
568
REFERENCES
Palais. R. 19S4. Definition of the exterior derivative in terms of the Lie derivative. I'roc. Am. Math. Soc. 5.902-908. Palais. R. 1963, Morse theory on Hilbert manifolds. Topology 1. 299-340. Palais. R. 1965a. S~m;rror on the At(vah-Singer Inde.y Theorem. Princeton University Press. Princeton. N.J. Palais. R. I 965b. Uctures on the Differential Topolo!{V of lllfinitl! Dimellsional Mani/olds. Notes by S. Greenfield. Brandeis University. Palais. R. 1968. Foundations of Global Nonlinear Ana(vsis. Addison·Wesley. Reading.
Mass. Penot. J-P. 1970. Sur Ie tbeoreme de Frobenius. Bull. Math. Soc. France. 98. 47-80. pnuger. A. 1957. Theorie der Riemannschen Fliichen. Grundlebren der math. Wissenscharten 89. Springer-Verlag. New York. Povzner. A. 1966. A global existence theorem for a nonlinear system and the defect index of a linear operator. Transl. Am. Math. Soc. 51. 189-199. Rao. M. M. 1972. Notes on characterizing Hilbert space by smoothness and smootbness of Orlicz spaces. J. Math. Anal. Appl. 37. 228-234. Rayleigh. B. 1887. The Thf!Qry of Sound. 2 vols .• (1945 ed.). Dover. New York. Redheffer. R. M. 1972. TIle theorems of Bony and Brezis on now-invariant sets. Am. Math. Month(.' 79.740-747. Reed. M .• and B. Simon. 1974. Methods on Modern Mathematical Physics. Vol. I: Functional Arro(.'sis. Vol. II: Self-adjointness and Fourier Ana(vsis. Academic Press. New York. Restrepo. G. 1964. Differentiable norms in Banach spaces. Bull. Am. Math. Soc. 70. 413-414. Riesz. F. 1944. Surla theorie ergodique. Comm. Math. lIelv. 17.221-239. Robbin. J. 1968. On the existence theorem ror differential equations. I'roc. Am. Math. Soc .• 19. 1005-1006. Royden. H. 1968. Real Ana(vsis. 2nd ed.• Macmillan. New York. Rudin. W. 1976. Principles of Mathematical Ana(vsis. 3rd ed. McGraw-Hili. New York. Russell. D. 1979. Mathematics of Finite Dimensional Control Sy.ftems. Theory alld Design. Marcel Dekker. New York. Schutz. B. 1980. Geometrical Methods of Mathematical Physics. Cambridge University Press. Cambridge. England. Schwartz. J. T. 1967. Nonlinear Fun('(ional Ana(I'sis. Gordon and Breach. New York. Singer. I. and J. Thorpe. 1967. Lecture Note.f on Elemelltary TopoloK\' and Geometry'. Scott. Foresman. Smale. S. 1964. Morse theory and a nonlinear generalization or the Dirichlet problem. Ann. Math. 80. 3112-396. Smale.S. 1965. An infinite-dimensional v~rsion of Sard's theorem. A mer . .I. Math. 87. 861-866. Smale. S. 1967. Differentiable dynamical systems. Bull. Am. Mal". Soc '73. 747-817. Smoller. J. 1982. Mathematical Theory' of ShQ('k Waves and Reaction Diffusioll Equations. Graduate Texts in Mathematics. Springer-Verlag. New York. Sobolev. S. S. 1939. On the theory or hyperbolic partial difrerential equations. Mat. Sh. 5. 71-99.
REFERENCES
569
Sommerfeld, A. 1964. Thermodynamics and Statistical Mechanics. I.ectures on Theoretical Phyiics, Vol. 5. Academic Press, New York. Spivak, M. 1979. Differential Geometry, Vols 1-5. Publish or Perish, Waltham, Mass. Stein, B. 1970•. Singu/ar Integrals and Differentiability Properties of Functions. Prince. ton Univ. Press., Princeton, N.J. Sternberg, S. 1963. Uctures on differential geometry. Prentice· Hall. Englewood Cliffs, N.J. Sternberg, S. 1969. Celestial mechanics\ Vols. 1,2. Addison-Wesley, Reading, Mass. Stoker, J. J. 1950. Nonlinear Vibrations. Wiley, New York. Stone. M. 1932a. Linear transformations in Hilbert space, Am. Math. Soc. Colloq. Publ. 15. Stone, M. 1932b. On one-parameter unitary groups in Hilbert space. Ann. of Maih. 33, 643~648. . Sundaresan. K. 1967. Smooth Banach spaces, Math. Ann. 173, 191-199. Sussmann, H. J. 1975. A generalization of the closed subgroup theorem to quotients of arbitrary manifolds,J. Diff. Geom. 10. 151-166. Sussmann, H. J. 1977. Existence and uniqueness of minimal realizations of nonlinear systems, Math. Systems Theory 10, 263-284. Takens, F. 1974. Singularities of vector fields, Publ. Math. lHES 43,47-100. · Trotter, H. F. 1958. Approximation of semi-groups of operators, Pacific J. Math. 8, · 887-919. · Truesdell, C. 1969. Rational Thermodynamics. Springer-Verlag. New York. Tuan, V. T., and D. D. Ang. 1979. A representation theorem for dif(erentiab1e functions, Proc. Am. Math. Soc. 75, 343-350. Ueda. Y. 1980. Explosion of strange attractors exhibited by Duffings equation. Ann. N. Y. Acad. Sci. 357, 422-434. Varadarajan, V. S. 1974. Lie Groups, Lie Algebras and Their Representations. Prentice-Hall, Englewood Cliffs, N.J. . von Neumann;·J. 1932. Zur Operatorenmethode in der Klassischen Mechanik, Ann. Math. 33. 587-648. 789. von Westenbotz, C. 19SI. Differential Forms in Mathematical Phvsics. North-HoIland, Amsterdam. Warner, F. 1971. Foundations of Differential Manifolds and Lie Groups. Scott, Foresman. Glenview. Illinois. . Weinstein, A. 1969. Symplectic structures on Banach manifolds. Bull. Am. Math. Soc. 75. 1040-1041. Weinstein, A. 1977. Lectures on Symplectic Manifolds. ('OMS. Conference Series Vol. 19. American Mathematical Society. Wells, J. C. 1971. C'-partitions of unity on non-separable Hilbert space. Bull. Am. Math. Soc: 77, 804-S01. Wells, J. C. 1973. Differentiable functions on Banach spaces with Lipschitz derivatives, J. Diff. Geometry 8, 135-152. Wells, R. 1980. Differential Analysis on Complex Manifoldt. Springer-Verlag, New York. Whitney, H. 1943a Differentiability of the remainder term in Taylor's formula, Dulce Math. J. 10, IS3-ISS.
REFERENCES
570
Whitney. H. I 943b. Dirrerentiable even functions. Duke Math. J. 10. 159-160. Wu. F. and C. A. Desoer. 1972. Global inverse function theorem. IEEE Tra/l.~ .• CT-19. 199-201.
Wyatt. J. L.. L. O. Chua. and G. F. Oster. 1978. Nonlinear n-port decomposition via the Laplace operator. IEEE Trans. Circuits Sy.ftems 15. 741-754. Yamamuro. S. 1974. Differential Calculus in Topological Linear Spaces. SprinlW Leetun Notes. 374. Yau. S. T. 1976. Some function theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Math J. 15.659-670. Yorke. J. A. 1967. Invariance for ordinary differential equations. Math. Sy.fI. Theory 1.353-372.
INDEX
A absolutely convergent, 51 accumulation point, 4 action, 477 adjoint operator, 445 admissible local chart, 124 vector bundle chart, 137 . admits partitions of unity. 309 Riemannian metrics. 315 A1exandrorr proposition. 35 a-density. 314 alternation mapping. 324 alternative proof of the Poincare lemma, 370 AmJ*re:s law, 498 analytic. 88 runction, 119 angular variables. 380 . antiderivation. 357. 362 antilinear map. 55 antisymmetric multilinear map, 59 antisymmetry. 552 arc,31 arcwise connected. 31
area. 426 ArzcJa-Ascoli theorem. 29 associated tensor. 276, 288 asymptotically stable, 242. 243 unstable. 243 atlas. 123 attractor. 249. 257 attractive closed orhit. 250. 259 automorphisms or function algebras . 229 autopara)lel. 476 axiom of choice. 522
B backward differential equation. 228 Baire category theorem. 524 space. 523 Baker-CampbeIJ- Hausdorff formula, 238, 518 balance of momentum, 484 Banach isomorphism theorem. 63 manifold. 125 space, 43
571
572
INDEX
Banachable space. 139 base integral curve. 480 spate. 133. 138. 139 basin. 249. 250. 257, 259 basis for a topology, 2 Bernoulli's theorem. 492 blowing up a singu1arity. 237 body forces. 484 Bolzano-Weientrass theorem. 26 bordered Hessians. 179 Bom-Infeld bracket. 508, SIO boundary, 4. 404. 407 conditions. 487 orientation. 409 boundaryless double. 424 bounded. S3 set. II Bolt vanishing theorem. 267 Brouwer's fixed point theorem. 449 buckling column equation. 248 bump function. 21 S bundle exact. 14S
c C'-Gateaux.81 C-Banach space. 215 C'-homotopic.447 C-manifold. 125 Calderon extension theorem, 422 canonical coordinates. 464 involution. 161 map. 466 one-form, 465 projection. 19.20.47 transformation. 462 two-form. 465 Caratheodory's inaccessibility theorem. 512 carrier. 309 Cauchy -Bochner integral. 57 -Green tensor. 303 -Riemann equations. 74. 119 -Schwarz inequality. 40 sequence. II stress tensor. 484 traction vector. 484
Cayley transform. 533 chain, 523 rule. 76. 91 change of variables formula, 3%. 397 chaotic attractor. 257 Chapman-Kolmogorov law. 185 characteristic exponents, 241. 242,244. 25S, 259 multiplien. 251, 254. 255, 259 characteristics, 215 charge density, 432. 498 chart. 123 with boundary. 406 choice function. 522 Chow's theorem. 519 Christoffel symhols. 392. 473 circulation. 429. 493 Clausius - Clapeyron formula. 520 -Duhem inequality. 512 class C. 68, 131 classical Gauss theorem. 430 Stokes theorem. 427 classifying map, 143 clean intersection. 169 Oebsch variables, 497 closable operator. 530 closed equivalence relation, 22 form,369 . graph theorem. 63 map. 16 operator. 530 orbit. 249. 255. 256. 259 set. 2 closure. 4 codifferential. 391 codimension. 48. 130 of a foliation. 265 cohomology group. 371. 441 collar. 423 collation. 133. 164 commutation of vector fields. 225 commutators. 4111 commutes with contractions. 294 compact spa~·c. 24 support. 309
INDEX compactness of the unit sphere. 45 complete . metric space, II vector field, 195 completeness criterion, t 97 complex iDDer product, 39 manifold, 128 projective space, 22 complemented, 48 component. 31. 273 of a manifold. 126 ·composite mapping theorem. 76. 154 conronllally invariant. 502 conjugation. 210 connected space. 31 conservation of energy. 469. 487 of mass. 401. 483 consistent algorithm. 198 constant of the motion. 437. 471 constraint. 513 continuous. 14 continuity equation. 484. 501 contractible loop, 34 contraction. 12. 275 mapping theorem, II, 102 contravariant tensor. 272 control. 517 theory. 516 conventions for wedge product. 325, 334 convergence of flows. 206 convergent sequence. 6 converse to Taylor's theorem. 92 convex set. 49 coordinate system, 123 core, ~31 cotangent bundle. 285 Iift,466 . covariant derivative. 392. 474 tensor. 272 covector field. 286 covering map. 183.403 criterion for orientability. 386 critical element. 257
point, Ifl2. 241. 552 value. Ifl2. 55:! cross-product of vectors, 333 curl. 360 current, 499 density. 498 curvature tensor. 274 curve, 151 cylinder, 139
D Darboux's theorem, 265. 463. de Rham's theorem. 441. 452 defect space. 535 deficiencv index. 533 definiteness. J() defomation lemma. 308 degree. 456 of a map. 451 dense set. 4 densely defined operator. 530 density, 341 derivation. 151.213.217.235 derivation property. 229 derivative. hll derived sel. 4 determinant.3.n diagonal. 19 diameter. II diffeomorphic manifolds. 132 diffeomorphism. 102. 132 differentiable. 68 manifold. 124 structure. 124 differential. 71. 72. 211 ideal. 375 one-form. 28h operator. 209. 294.444 differentiating sequences. 1011 under the integral. 89 dimension of a foliation. 265· direct sum. 44 directional derivative. 79. 211 disconnected spa(.'C. 32 discrete foliation. 266 topology. 2 disk. 10
573
574
INDEX
displacement vector, 303 distance function, 9 distinguished charts, 265 distribution, 260 distributional k-form, 403 divergence, 360. 389, 393 free vector field, 400 double covering. 385 dragging along, 214 driVF,515 dual basis, 271 basis, 271 bundle, 148 space. 53, 271 tangent rhombic, 157 dyadic, 274 dynamical system. 185
E elasticity tensor, 274 electric dipole, 510 498 electromagnetism. 498 elementary critical point, 244 elliptic operator, 446 splitting theorem. 446 embedding, 167 mergy,477 density, 503 vector, 503 entropy. 512 equation in divergence form, 240 of continuity, 401 of state. 491,512 equations of motion of an ideal fluid, 487 equicontinuous, 28 equilibrium point, 241, 255 thermodynamics, 511 equivalence class, 20 relation. 20 equivalent atlases, 124 metrics, 10 norms, 43 volume elements, 339 volume forms, 383
nux,
-nux
ergodic, 437 essentially self-adjoint, 531 Euler characteristic, 246 equations for an ideal nuid, 4115 evaluation map, 94. 95 even permutation, 59. 324 evolution operator. 185.227.229 exact form, 369 sequence of vector bundles, 145 existence and uniqueness of flow boxes. 193 of Poincare maps, 252 theorem. 188 for ordinary differential equations. 260 exponential formulas, 238 exponentiation of an operator, II R extension, 531 exterior algebra. 330 bundle. 351 derivative. 357 differential form, 352 k-form.324 F F-valued k-form, 356 Faraday law, 429, 498, 499 two-form, 499 fiber. 133, 139 derivative, 476 product, 139 fibration theorem, 172 final topology, 24 finite-dimensional manifold, 126 first category set, 523 countable, 3 order deformation, 110 principle of thermodynamics, 511 variation. 110 equations, 190 nag manifold, 128 Floquet normal form. 256. 257 theorem. 257, 259 now, 185, 186, 196 box, 192. 202
INDEX nOWS on function spaces and el'lodicity, 433 nuid mechanics, 481 nux, 427, 429 foliated chart, 265, 266 foliation, 260, 265, 267, 268, 270 force of constraint, 514 FrCchet space, 119 Fredholm alternative, 446 map, 557 operator. 557 Friedrichs extension. 539 Frobenius' theorem. 260. 262. 265. 266 functional dependence. 115 derivative. 497 functions in involution. 470 fundamental group. 34 isomorphism theorem. 64 theorem of algebra. 450 theorem of calculus. 73
G g-a-density. 343. 390 g-orthonormal basis. 342 g-volume. 343. 390 Gateaux differentiable. 81 gauge freedom. 504. 506 Gauss' law. 431. 498 theorem. 411. 415 general linear group. 103. 183 generalized Lagrange multiplier theorem. 120 generated differentiable structure. 124 geodesics. 473. 474 global inverse function theorem. 120 section. 143 stable manifold theorem. 256 graded associative algebra. 330 gradient, 72. 259. 288 now, 205 graph norm, 531 of an equivalence relation. 22
Grassmann algebra. 330 bundle. 140 manifold. 127 Grassmannian. 175 gravitational force rield. 186 Green's theorem. 424.458 Gronwall's inequality. 189.244 group of diffeomorphisms. 132 property. 186 H Hahn-Bana~'h thl'orem. 62. 526 hairy ball theorem. 449 half:space.404 Hamiltonian vector field. 468 Hamilton's equations. 462. 469 harmonic rield.444 form. 441 oscillator. 242 vector field. 492 Hausdorff space. 7 Heine-Borclthcl.rcm.28 Helmholtz' theorem. 495 Hermes' theorem. 519 Hermitian inner product. 39 operator. 531 Hilbert basis, 52 manifold. 126 space. 43 Hodge decomposition theorem. 442 star operator. 333. 345 theorem for manifolds with boundary. 444 HOlder c1a.~s. 204 holonomic constraints, 513 holonomy.520 homeomorphism, 14 homocliDic orbit. 260 homogeneous polynomial, 60 homomorphism of differential algebras. 360 homotopic. 381 maps, 457 hyperbolic closed orhit. 256
575
576
INDEX
critical element. 257 critical point. 244. 246 flow. 244
ideal. 151 ideal fluid. 485 identities involving vector fields and differential forms. 371 immersed submanifold. 166.265 immersion. 165 implicit function theorem. 107. 158. 162.260 incompressible. 389 flow. 400 fluid. 489 index Fredholm. 557 lowering operator. 276. 359 of a symmetric covariant twotensor. 342 raising operator. 276. 288 topological. 246 individual ergodic theorem, 439 induced orientation. 409 inductively ordered. 523 infinitesimal generator, 541 inhomogeneous wave equation, 506 initial conditions. 487 topology. 24 injective immersion, 166 inner product. 38 inset. 257 of a closed orbit, 249, 256 of a critical point. 244 of a fixed point of a diffeomorphism. 256 of a point under iterates of a map. 256 integrable. 260. 261. 262, 266 iDtqra\,57 curve. 187 equation. 14 manifold, 260 of a form, 397 of a function, 398 of a vector field. 196. 426 of an ,,-form. 395 integrating factor. 378
integration by parts. 87 over the fibers. 403 interior, 4. 404. 407 product. 274. 362 intermediate value theorem, 33 internal energy. 490 intersect cleanly, 169 intrinsic Hilbert space. 440 invariant k-form.422 set. 201 inverse mapping theorem, 102. 105 inversion map. 103. 117 involution. 471 involutive. 260. 261. 262. 266 isentropic flow, 490 isolated point. 4 isometry. 349
J Jacobian determinant. 388 matrix. 71 jet bundle. 149 transversalitv theorem. 1111 Jordan . canonical form. 242 curve theorem. 34 K k-annihilator of a subbundle. 374 k-form.352 k-th order contact. 150 k-th order differential operator, 444 Kalman criterion. 521 Kelvin circulation theorem, 494 kernel, 145 Killing vector field, 302 Klein bottle. 21 Korteweg-deVries equation, 472 Kronecker delta. 275. 287 L I f -space, 399 Lagrange multiplier. 176 equations. 478
INDEX Lagrangian, 476 coudition, '477 vector field, 477 system, 473 Landau symbol. 68 Laplace-Beltrami operator. 393 Laplace-deRham operator. 441 Laplace transform, 549 latent heat of vaporization, 520 leaf, 265. 266, 267, 268, 269 Leibniz rule, 78, 89 Liapunov stable, 242 theorem, 244. 251, 255 Lie algebra, 222 bracket, 220, 237 derivation, 222 derivative, 211, 226, 293, 296 formula, 220, 227, 299, 304 of functions, 219 of vector fields, 219 product formula, 200 transform, 264 lifetime, 195 limit . cycle, 250 point. 6 Lindel(')f theorem. 3 topological space. 555 line integral of a one-form. 424 linear extension theorem. 56 now on the torus, 267 linearity of the derivative, 69 linearization, 110,241,244 theorem, 245 for closed orbits, 256 for maps. 256 Liouville now, 434 form, 466 theorem, 470 Lipschitz property. 75 local bundle chart, 137 conjugacy. 252 cross-section. 267 immersion theorem. III injeetivity theorem, 110 operator, 217. 357
577
representation theorem, 113 representative. 131, 141, 187 transversal section, 25 I, 252, 254 section, 143 submersion theorem, 111 surjectivity theorem, 108 vector bundle. 133 isomorphism. 134 map. 1]4 localizable. 292 locally arcwise connected. 31 closed map. 55X compact ~pacr. 24 connected span" 31 controllahle. 517 finite. 309 generated ideal. 375 Hamiltonian vector field, 468 logarithm function of an operator, 119 loop, 34 Lorentz group, 501 inner product. 348 invariance. 502 manifold. 502 lowering indices. 276
M magic formula of Carlan, 363 magnetic field,498 nux, 429. 498 manifold. 124 with boundary. 406 with piecewise ~mooth boundary. 416 mass density. 48] material derivative. 485 matrix,278 of second derivative. 75 maximal atlas, 124 element, 523 integral curve. 196. 266 Maxwell's equations, 4911. 500 as a Hamiltonian system, 507 identity. 519 stress tensor, 503
578
INDEX
Mayer system. 378 mean ergodic theorem. 438 mean value theorem. 81 measure zero. 551 method of characteristics. 233 metric. 10 space. J() tensor. 274 topology. 10. 313 minimality.437 multilinear algebra. 271 Milnor manifold. 183 minimal norm element. 49 Minkowski metric. 288 MObius band. 139. 175 model for the violin string. 250 space. 125 momentum density. 497 multilinear mapping. 58 N
n-manifold. 126 natural atlas. ISS. 285 chart. 284. 286 with respect to diffeomorphisms. 364 mappings. 360 push forward. 222. 296 restrictions. 212. 223. 257. 294 negatively oriented. 340 neighborhood. 3 Newton's law of gravitation. 185 second law. 462 Noether's theorem. 481 nondegenerate. 287. 462 nonholonomic constraints. 514 noninvolutive distribution. 261 nonlinear spring with linear damping. 257 nonregular equivalence relation. 269 non-unique integral curves. 206 norm. 38 convergem:e.54 normal bundle. 322 to a foliation. 266 space. 7 normed space. 38 nowhere dense set. 4. 523
o odd permutation. 59. 324 omega-lemma. 94. 96 one-dimensional manifold. 126 form. 352. 353 parameter group of diffeomorphisms. 196 parameter uni tary group. 541 open equivalence relation. 22 map. 16 mapping theorem. 63. 527 neighborhood. 3 rectangle. 18 set. 2 submanifold. 129 operator. 530 norm. 53 order. 522 ordinary differential equation. 188 orientable. 382 double covering. 385 orientation. 340. 384 preserving. 387 reversing. 387 oriented charts. 408 manifold. 384 vector space. 340 orthogonal. 40 orthonormal hasis. 51. 342 oscillation. 250 outset. 257 of a dosed orhit. 250. 256 of a critical point. 244 of a fixed point of a diffeomorphism. 256 outward unit normal. 426 overlap map. 123 p
paracompact. 310 parallel-transported. 476 parallelizable. 219. 239 parallelogram. 41 partial derivative. 70. 82 differential equation. 214 partition of unity, 230, 309 patching construction, 309 permutation group, 59, 323
INDEX perpendicular fonn. 443 Pfartian system. 377. 513 pha'le portrait. 246. 249. 257 space. 462. 5 II volume. 466 cp-related. 208 plane electromagnetic waves. S09 piecewise smooth map. 416 plane field. 260 Poincare duality. 457 - Hopf index theorem. 246 lemma, 265. 369 map. 25 I. 252. 254. 256 recurrence theorem. 402 section. 255. 256 point transformation. 467 Poisson bracket. 470 commute. 471 polarization. 41. 60 identity. 273 positively oriented. 340. 385. 409 potential flow. 496 Poynting's theorem. 503 pressure. 485 presymplectic. 463 principal part. ISO. 187 product bundle. 147 fonnulas. 1911 fonnula for the Lie bracket. 237 manifold. 130. 157 topology. 18 projection. 134. 139 projective space. 128 proper. 167 mapping. 26. 120. 197 pseudometric. II pseudo-Riemannian a-density. 390 metric. 287 volume. 390 pull-back. 207. 279. 280. 289. 354 bundle. 148 push-forward. 207. 212. 2711. 289
Q quadratic fonn. 273 qualitative theory of dynamical systems. 240
quasi-static adiabatic path. 512 quotient bundle. 144 manifold. 173 toporogy. 20 vector space. 47
A raising indkes. 276 range. 145 rank theorl'l11, II' real line. 2 projectivc span', 22 reflcxivity, 522 regular ' equivalence relation. 173. 267 foliation. 267 point. 552 space. 7 value. 162. 552 related vector fields. 208 relative Poincare lemma, 380 topology. I II relatively compal't 'lIb~ct, 24 invariant k-fllrm. 422 residual set. 523. 555 resolvent. 549 restricted bundle. 144 reverse orientatilln. 340. 3114 Riemannian manifold. 463 metric. 287 Riesz representation theorem. 55 rotation group. 31.181
8 saddle point. 248 type clo.'Ied orbit. 259 type critical element. 257 Sard's theorem, 163,552,555 scalar equation, 214 SchrOdinger equation, 473 second category sct, 523 countable. 3
579
580
INDEX
order equation, 478, 480 principle of thermodynamics,S 12 tangent, 100 section, 143 sell-adjoint operator, S30, S31 self-intersections, 165 semibounded operator, 539 semi-now. 186 seminorm. 38 separability. 35 separable space, 4 separatrix, 249, 257. 259 sequence, 6 sesquilinearity. 39 shock wave, 484 shrinking lemma. 315 shufnes, 326 sign, 324 signature, 342 simple pendulum with linear damping, 247 simply connected space, 34 singular point, 241 singularity, 268 skew symmetric, 324 slice, 267 Smale's density theorem, 558 smooth Banacb space, 215 smoothness of local vector bundle maps, 134 Sobolev class, 204 space, 472 source one-form, SOO spectral radius formula, 243 spbere, 122, 125, 159, 163 split exact sequence, 64, 145 subspaces, 48 square root of an operator, 118 stability index. 244, 246 of a critical point, 242 stable. 242 focus, 247, 248. 249 manifold theorem, 245 standard inner product, 39 inner product on C·, 39 topology, 2 stationary now. 492
steady now, 492 steer, SIS Steiner's Roman surface, 182 step function, 57 stereograpbic projection, 122 Stiefel manifold, 164 Stokes' theorem. 409, 418 theorem for nonorientahle manifolds. 413 theorem for piecewise smooth boundary, 415 Stone's theorem, 519 straightening out theorem. 194. 266 strain tensor, 303 stratification, 268 stratified space, 269 stratum, 268 stream function, 497 streamline. 491 stress, 484 -energy-momentum tensor, 503 tensor, 274, 484 strong convergence, 54 symplectic form, 463 strongly nondegenerate, 262 subbundle, 144 subimmersion, 171 submanifold, 129 property. 129 submersion. 162, 261 summation convention, 272 support, 196. 309 suspending a vector field, 191. 229 sustained oscillation. 25 I symbol. 445 symmetril' . heavv top, 26() multlllll,'ar map. 59 operal(lr. 531 rhombic. 161 tensor, 273 symmetrization operator, 66 symmetry. 10 of second derivative, 85 symplectic form, 463 map, 466 transformation, 462 system of integrals, 204
INDEX
T tangent. 76. 151 bundle. 150. 152 projection. 152 to a foliation. 266 form. '443 map. 68. 153 space. 152 Taylor's theorem. 86 tensor algebra. 292 bundle. 284 derivation. 294 field. 286 product. 272 tensorial construction. 147 tensoriality criterion. 287 thickening, 86 Thorn transversality theorem. 181 three pillars of linear analysis. 62 time average. 438 -dependent flow. 186,227 tensor field, 307 vector field, 227 -ordered products. 240 Toda lattice. 480 topological conjugacy. 245. 246 index, 246 manifold, 23 space. 2 ' torus. 20. 159.261 totally bounded set, 27 traces, 277 traction vector, 484 trajectory, 491 transitivity. 522 transport theorem, 40 I with mass density, 488 transpose, 277 transposition, 324 transversal,65. 167, 169 map to a foliation, 269 transversality theorems, 179 triangle inequality, 10 trivial foliation, 265 topology, 2
581
Trotter product formula. 239 twisted k·form, 414 ribbon, 169 two -dimensional manifold, 127 form, 353
u unbounded operator, 530 uniform boundedness principle. 64. 528 contraction principle, 120 convergence. I~. 16 uniformly nlOtinuous, 16 uniqueness, 192 theorem. 11111 universal hllndk~. 140 unstable fncus. 250 upper bound. 523 Vainberg's theorem, 394 Van der Pol equation. 251 variation of constants formula, 205. 243 variational principle, 394 VB equivalent, 137 vector bundle, 137 atlas, 137 isomorphism. 141 of bundle maps, 148 of multilinear maps, 1411 of tensors. 2115 mapping. 141 structure. 137 vector field,I84 fields on spheres, 449 identity, 360 potential. S04 -valued k-form. 356 velocity field, 4112 viscous fluid, 491 voltage, 429, 499 volum~
element, 339 form, 382 manifold, 382 of a parallelepiped, 341 preserving, 364, 3117
582
INDEX
vortex line. 496 tube. 497 vorticity. 493 cquation. 495 ·-strl!am equation. 495
w wave equation. 472 weak pseudo-Riemannian metric. 2R7 Riemannian structure. 463 weakly convergent. 51 nondegenerate. 462
wedge product. 325 well-ordered. 315, 523 Whitney C'·topology. IRO C-topology. 180 cmbedding theorcm. 130 sum, 148 thcorcm. 101 wind toward. 255. 25/\ wrigglc. 515
z Zermclo's lemma, 315 section. 133, 138, 139
lCro