Lectures on Finsler Geometry

K^FGT')-1-

(L22)

By differentiating (1.22) with respect to x J and j/ J ' respectively, we obtain (<5}-F-2FxJ-2/i)^(2)=0 («5j - F ^ i ^ / ) ^ where z := x + y/F(y)

1

^ (z) = 0,

(1.23) (1.24)

- p. It follows from (1.23) and (1.24) that (^Fx3-FFyJ)lpzi(z)yi

= 0.

(1.25)

Observe that v = (vl) G TzRn is tangent to 9 0 if and only if

ipzi(zy

= o.

Thus ipzi(z)yi =^ 0. It follows from (1.25) that Fxj - FFyJ = 0. This implies (1.22).

Q.E.D.

Chapter 2

Finsler m Spaces

In his recent book [Gr4], M. Gromov has set the foundation of the theory of metric measure spaces and shown some potential applications to various problems, in particular, the Levy concentration problem. A metric measure space is a triple M = (M,d,n), where (M,d) is a metric space with a countable base and fi is a cr-finite Borel measure. Here a measure /J. on a metric space (M,d) always means a Borel measure, i.e., all Borel subsets are measurable. In some cases, the underlying spaces are not manifolds. Here we are concerned with regular metric measure spaces, so that we can employ calculus to study the geometric properties. Finsler spaces with a volume form are viewed as regular metric measure spaces. They are simply called Finsler m spaces. In this chapter, we will discuss some basics on Finsler m spaces.

2.1

M e a s u r e Spaces

Let M be a C°° n-manifold. A volume form d/j, on M is a collection of nondegenerate n-forms d/j,i = ai(x)dxl • • • dxn on coordinate neighborhoods {
(vi) M = U ^ ; (V2) if Uif]Uj =/= 0, then d\Xi = d/j,j on Ui O Uj, namely, if d^n = ai{x)dx1 • • • dxn and d/ij = <jj(u)dul • • • dun, then <Ti(x) = |det [-g^j) 19

°j(u).

20

Finsler m Spaces

By (V2), we have / JUiHUj

fdH=[

f dm,

f € C?(Ui n Uj).

JUiHUj

Take a partition of a unity {ipi} for the covering {f/j}, that is a set of nonnegative C°° functions ipi £ C£°(Ui) with the following properties: (a) for each x G M, there are only finitely many Ui's such that ipi(x) ^ 0; ( b ) Ei^i(a;) = l for all i € M. Every volume form dfi = {dfii} defines a measure fi on M by

For any two volume forms dfj,± and d/^2 on M, there is a positive C°° function ip such that d^

= f

d/j,i.

Thus if we fix a volume form dfi„ on M, volume forms d/j, one-to-one correspond to positive C°° functions ip by dfi =

(2.1)

The Euclidean volume of a bounded open subset CI C R n is given by Vol(n)= f dV= JQ

f Jn

dxl---dxn.

More general, on a Riemannian manifold (M, F), where F(y) = the Riemannian metric F determines a canonical volume form dVg := Wdet (gij(x))

dx1 • • • dxn.

We call dVg the Riemannian volume form of F.

\/gij{x)y'ljy:>,

(2.2)

Volume on a Finsler Space

21

We consider an oriented manifold M equipped with a volume form dfx. We can view dfj, as an n-form on M. Let X a vector field on M. Define an (n - l)-form X\d/j, on M by X\dn{X2r--,Xn):=dn(X,X2,---,Xn). Define d(X\dfi) = div(X)dfi.

(2.3)

We call div(X) the divergence of X. Clearly, div(X) depends only on the volume form dp,. In a local coordinate system (xl), express d/i = a(x)dx1 •••dxn. Then for a vector field I = I ' | r o n M , ,.,.., I d , \ 8Xl X{ da div X) = - — • \aXvl)i = — - H -—.

(2.4)

Applying the Stokes theorem to r\ = X\ dji, we obtain

I div(X)dn

=

j d(X\d/j,)=0,

f dW(X)dfi

=

I d(X\dfi)=

[

JM

JdM

JM

2.2

if dM = 0 (2.5) X\dfi,

if <9M ^ 0. (2.6)

Volume on a Finsler Space

There are two canonical volume forms on a Finsler space. Both reduce to the Riemannian volume form when the Finsler metric becomes Riemannian. Let (M, F) be an n-dimensional Finsler space. Let {bi}™=1 be an arbitrary basis for TXM and {0l}?=1 the dual basis for T*M. The set B£ := {(
< l}

is a bounded open strongly convex open subset in R n . Define dVF:=aF{x)01

A---A6n,

(2.7)

22

Finsler m Spaces

where

The volume form dVp determines a regular measure Volp. H. Busemann [Bui] [Bu2] proved that if F is reversible, then Vol^ is the Hausdorff measure of txie induced metric dp. dVp is called the Busemann-Hausdorff volume form . In general, the Euclidean volume of B™ in (2.8) can not be expressed by F in an explicit form. Below we will discuss some special metrics for which we can compute dVp. Example 2.2.1

Let (V,F) be an n-dimensional Minkowski space and

B:={^eV,

F(y)
denote the unit ball of F. Let (xl) denote the global coordinate system on V determined by a basis {bj}™=1. Let B n :={(!,*) € R n , F ( y i b i ) < l } . Then . , aF{x)

=

Vol(B n ) Vol(B") =

constant

We obtain Vol F (B) = [ dVF=

[

aF(x)dx1

• • • dxn = Vol(B").

Example 2.2.2 Consider a Randers metric F = a + / 3 o n a manifold M, where a = y/'aij{x)yiyi is a Riemannian metric and (3 = bi{x)yl is a 1-form satisfying W\\x :=

sup (3{y) = x /a^'(a;)6 i (x)6 J (a;) < 1, a«(!/)=l

V

where (alj(x)) = (aij-(i)) - 1 . Let dVp and dVa denote the BusemannHausdorff volume form of F and a, respectively. By a simple computation,

Volume on a Finsler Space

23

we obtain dVa = Jdet(aij(x))

dxl • • • dxn.

This is just the Riemannian volume form of a defined in (2.2). To find dVp, we take an orthonormal basis {bj}™=1 for (TXM, ax) such that f3x(y) = H/8IU2/1, where y = yxhi. Then the open subset B™ in (2.7) is a convex body in Rn given by

+ (i -11/3112) ±(yaf < i.

(i - ll/C) V + ^%,J The Euclidean volume of B™ is given by

V ^ - ^ f f i ^ . (1 - P i l l )

(2.9)

2

Thus n+l

dVF = ( l - \\P\\2X)

2

dVa.

(2.10)

This implies Vol F < Vol a .

(2.11)

Assume that M is closed. Then Vol F (M) = /

( l - m\l)~dVa

< Vol a (M)

and equality holds if and only if (3 = 0.

E x a m p l e 2.2.3

u

Let

• =

with JJ2™=i(Bn.)2 < 1- Consider an immersion p = (<^) : M -» (R;m+l ||).

it

Finsler m Spaces

24

Let a := \Jaij{x)yly:i

and (3 := bi{x)yz, where

M * ) := £ r W £ r W .

bi(x) := ^ ^ ^ - ( a ; ) M=I

Then the norm of /? with respect to a is given by

||/3|| =

JaVixMxfcix) <

J2w < i.

where (a ij '(x)) = ( a y ( i ) ) - 1 . Thus F = a + /3 is a Randers metric on M. Let 6 with |6| < 1 and n+1

^ ( u " ) 2 + &u n+1 ,

II" 6

\

u = («i)eRn+1.

M=l

Consider a graph in a Randers space (R ra+1 , || • \\b), un+1 = / ( u ) ,

u = (ul) e l l c R " .

The standard immersion ip : fi, —> R™+1 is given by tp(x\---,xn)

(x1,---,xn,f{x1,---,xnj).

=

We obtain

a

^=^ + Bix)§xtJ{xh

b b

^ B{x)-

This gives detidij) = 1 + \df\7 where

i*i-V© + - + ©) df

The inverse of (a^) is given by a lJ = <513 i7-

df

df

l + \df\2dxiy

>dxi

1

Volume on a Finsler Space

25

This gives l + \df\2

\\P\\ We obtain

=

(^iTw)

2

v^WJW-d*"

Example 2.2.4 Let fi be a strongly convex domain in R™. The Funk metric F is defined by z =x

+ i?r-^ yer s n«R".

(2.12)

F{y) where z £ dfl (cf. Example 1.3.4). Note BJ^^eR",

J F(y

i

b1)
Thus ,

x

Vol(B n )

aF{x) =

Vol(B n )

vop^y = -voW

,„ n ,

= constant

-

(2 13)

-

This implies that the Busemarm-Hausdorff volume of (Q., F) is constant Vol F (fi) = / aF(x)dx1 Jn

• • • dxn = Vol(B n ).

(2.14)

There is another important volume form on Finsler spaces. Let (M, F) be an n-dimensional Finsler space. At a point x G M, fix a basis {bj}™=1 for TXM and its dual basis {#*}£=!forT*M. For a vector y = j / b * G TxM\{0}, put 9ij(y)

:=gy(bi,bj).

Finsler m Spaces

26

Each gij(y) is a C°° function on R n \ {0}. Define dy1 • • • dy11

JBndet(gij(y)) aF(x) :-

Vol(B n )

where B™ is defined in (2.7). The n-form dVF :=aF(x)e1

A---A671

(2.15)

is a well-defined volume form on M. Let w

•=

9ij(y)yJdxl.

UJ is called the Hilbert form. Observe that k i du> = -^y dx ox1

A dxj - gijdxi A dyl.

Thus (du)n = dw A • • • Adu = (-l) : i i Tr i i n!det (gaiy^dx1

• ••dxndy1 • • • dyn.

The Hilbert form ui defines a volume form on TM \ {0} by dV:={-\)ZiSF1—Adu})n. (2.16) n! P. Dazord discussed (dui)n in his Ph.D. thesis [Dal]. He further defined the volume of a compact Finsler space (M, F) by Vol(M) :=— w

f n

dV,

(2.17)

JBM

where 7r : B M —• M denotes the unit ball bundle of M. Dazord actually defined the volume using the tangent sphere bundle SM. But his definition is essentially same as (2.17) due to the homogeneity of F (cf. [Da2]). Observe that for a function / on M, f JBM

7T*f dV = 0Jn J f dVF. JM

Thus the volume form dV on BM gives rise to the volume form dVp on M. R.D. Holmes and A.C. Thompson [Th] took a different approach to study Minkowski geometry and discovered this special volume form dVF.

Hyperplanes in a Minkowski m Space

27

Therefore, 6VF is called the Holmes-Thompson volume form in literatures [Al][AlFe].

2.3

Hyperplanes in a Minkowski m Space

Given a volume form dp on a manifold M, there is no canonical way to define a volume form on hypersurfaces from dp. If M is also equipped with a Finsler metric F, then F determines a (local) normal vector field along any hypersurface. Using the normal vector field, one can define a volume form on hypersurfaces from dp. Let (M, F, dp) be a Finsler m space. Let dVp denote the BusemannHausdorff volume form of F. The volume form dp can written dp = ipdVp, where ip is a positive C°° function on M. Thus it suffices to define the "induced volume form" on a hypersurface for the Busemann-Hausdorff volume form d\rF- Since dVp at a point x £ M \s completely determined by the Minkowski norm Fx on TXM, we shall first study Minkowski spaces. Consider a Minkowski space (V, F ) . Given a hyperplane W C V, we claim that there is a unit vector n € V such that W={^V|gn(n,w)=o}.

(2.18)

To prove the claim, take a vector v $ W and let 4>(w) := F(v— w),

weW.

Clearly, attains its minimum m := mine/) at a unique point w0 £ W. Let V -

n :=

Wo

. m If v = Xv + w, X > 0, is another vector on the same side of W, then 4>{w) := F(v — w) attains its minimum fh = Am at w0 = w — Xw0. This implies that v — w0 (Xv + w) — (w — Xw0) __ v — w0 fh Xm m

Finsler m Spaces

28

Thus n is independent of the vectors on the same side of W as v. We call n a normal vector to W. Fix a vector w € W and let /(*) := •^22(w° + tmw) = -^2F2(v

-w0-tmw)

= ^F2{u + t w).

Differentiating / yields 0 = /'(0) = ~

[F 2 (n + t w)] \s=0 = g„(n, w).

This implies (2.18). For any hyperplane W C V, there are exactly two normal vectors n, n' e V to W. In general, n and n' are not parallel unless F is reversible.

In what follows, we are going to introduce an important function on a Minkowski space. This function will be used to define the volume form on hyperplanes in a Minkowski space. Let (V, F) be a Minkowski space. For a vector y € V \ {0}, we obtain a hyperplane W y ::={weV, =

\gy(y,v)=o}.

Take a basis {bj}™=2 for W y and h\ = y such that {bi}™=1 is a basis for V. Let B n:={(yi)€Rn,

n i=l

F^y%)
Hyperplanes

in a Minkowski m Space

29

n a=2

Both B n and B™-1 depend on the choice of {b0}™=2. Define

._ VOKB") . v ^ r 1 )

(219)

The function £ is independent of the choice of {bi}™=2- Moreover, ( has the following homogeneity property C(Ay) = C(V),

A>0, y^O.

Note that if F is Euclidean, then £(y) = 1. A natural question is whether or not F is Euclidean when ( — 1. This is not clear yet. L e m m a 2.3.1 Let (V,F) be an n-dimensional Minkowski space. Let c(n) := Vol(B rl )/Vol(B n " 1 ) and A := sup y6V \{o} F(y)/F(-y). Then (l + A)-"c„
y^O.

(2.20)

In particular, if F is reversible (X = 1), then 2""c n < C{y) < 2"c n .

(2.21)

Proof. Fix a unit vector y = y'hi G V. For any t G (0,1) and (va) € B™-1, F(ty + vaba) < tF{y) + F(vaha) Thus [0,1] x B™"1 c 2B n and VoKB^" 1 ) < 2 n Vol(B n ). This gives the right hand side of (2.20).

< 2.

Finsler m Spaces

30

n-1

„n-l

Let (t,va) e B™, i.e., F(ty + vaha) < 1. If £ > 0, then it follows from (1.13) that t = F(ty)
+

vaba)
< F(ty + vaba) + F(-ty)

< 1 + A.

If t < 0, then it follows from (1.13) again that -t = F(-ty) F(vaba)

< F(-ty

- vaba) < XF(ty + vaba) < A.

< F(ty + vaba) + F(~ty)

< 1 - t < 1 + A,

Thus B n c ( - A , l ) x ( l + A)B;- 1 . This gives the left hand side of (2.20).

2.4

Q.E.D.

Hypersurfaces in a Finsler m Space

Now we are going to define the volume form on hypersurfaces in a Finsler m space.

Hypersurfaces

in a Finsler m Space

31

Let (M,F,d/j,) be a Finsler m space and i : N —> M an embedding. Our goal is to find a volume dAp on hypersurfaces such that the following equality holds, , Yo\F(B(x,p + x AF[S(x,p)) = lim i V

/

e->0+

e))-VolF(B{x,p)) >i '-.

(2.22)

£

However, (2.22) does not hold for the Busemann-Hausdorff volume form dVp of the induced Finsler metric F :— i*F on N. Recall that for the Minkowski norm Fx = F\TXM> there is an important quantity £x : TXM \ {0} ->• (0, oo) (cf. (2.19)). Thus we obtain a scalar function £ = {(,x}x€M cm TM \ {0}. With this scalar function, we can define the desired volume form on hypersurfaces. Definition 2.4.1 Let N be a, hypersurface in a Finsler space (M, F) and n a normal vector field along N. Let dVp denote the Finsler volume form of the induced Finsler metric F on N. Set dAp := C(n)dVp,

(2.23)

where ( is defined in (2.19). dAF is called the induced volume form of dVp with respect to n. For an arbitrary volume form d\i on M, write dfj, := tpdVp, where ip € C7°°(M) and define dv := tpdAp. dv is called the induced volume form by dp. with respect to n. From the definition of dAp, we see that dAp depends on the choice of normal vectors unless F is reversible. By Lemma 2.3.1, we know that if F is reversible, then £ is bounded by two universal positive numbers ci(n) and C2(n). Thus ci(n)dVF < dAp < c2(n)dVF. There is another way to express dAp on a hypersurface i : N —»• M. Fix a normal vector field n along N. For a point x £ N C M, let {t>i}™=1 be a basis for TXM such that b j = n^ and {ba}™=2 is a basis for TXN. Put dVF = <jp{x)e1 A - - - A 0 7 1 ,

32

Finsler m Spaces

where aF is given in (2.8). Let {0l}™=i denote the dual basis for T*M {0a}2=2 the pull-back of {0 a }™ =2 on N. Express dVF = aF{x)62

and

A---A0n.

From the definition of C( n x), we have aF(x)

=

((nx)aF(x)

and = aF(x)92

dAF

A---A0n.

(2.24)

Let X be a vector field on M. Write X\N where X G TN.

= g„(n,X)n + X,

Then for any set of tangent vectors X2, • • •, Xn G TN

X\dn(X2,

•••,Xn)=

g„(n, * ) d/i(n, JC2, • • -,Xn).

(2.25)

At a point x S TV, let {b;}™=1 and {#'}™=1 be as above. P u t dA* = (r(x)O1 A • • • A 9n. By (2.24), we obtain the induced volume v on Ar, dv =
=di/(X2,---,Xn).

It follows from (2.25) t h a t i*{X\dfj)

= g„(n, X)dv.

(2.26)

Assume t h a t M is a compact oriented manifold with smooth boundary dM. By the Stokes theorem, for any (n - l)-form 77 on M, dr]= JM

i*rj. JdM

Applying the Stokes theorem to r\ = X\ dfi, we obtain the following

Hypersurfaces

in a Finsler m Space

33

Theorem 2.4.2 Let (M,F,d^) be a compact oriented Finsler m space and n be the out-ward pointing normal vector to dM. Then for any vector field X on M, [ div(X)dfi= JM

f

Sa(n,X)dv.

(2.27)

JdM

In particular, if M is a closed oriented manifold, then

I div(X)dM = 0. JM

Theorem 2.4.2 is important in many global geometric analysis problems.

Chapter 3

Co-Area Formula

To study Finsler m spaces, we use a family of level hypersurfaces of a distance function. On each level hypersurface, there is an induced volume form. The co-area formula relates the induced volume on level hypersurfaces and the volume on the manifold.

3.1

Legendre Transformations

First, let us study the relationship between a Minkowski norm and its dual norm. Let V be a finite-dimensional vector space and V* the dual vector space. Given a Minkowski norm F on V. F is a norm in the sense that for any y, v S V and A > 0, F(Xy) = XF(y),

F(y + v) < F{y) + F(v).

Let V* denote the vector space dual to V. Define F*(0:=

sup £(y).

(3.1)

F(v) = l

F* is a norm on V* again, i.e., for any £, n £ V* and A > 0, F* (AO = XF* ( 0 ,

F*{t + r,)
+ F* (r,).

We shall show that F* is a Minkowski norm on V* via the Legendre transformation between V and V*. Lemma 3.1.1

Let F be a Minkowski norm on V and F* the dual norm 35

36

Co-Area

Formula

on V*. For any vector y € V \ {0}, the covector £ = gy(2/, •) G V* satisfies

F(y)=F*(t)

= ~ ^

(3-2)

For any covector £ € V* \ {0}, there exists a unique vector j / g V \ {0} such that£ = gv(y,-). Proof. Note that by (1.15) and (3.1)

r[V>

F(y)

F(y)~

^

On the other hand, inequality (1.14) implies av)=gy(y,v)
F*(0=

sup veV-{o}

b

[v)

^
We conclude that F*(£) = F(y). This gives (3.2). Given an arbitrary covector £ £ V* \ {0}. The level sets

W :=

{v e V, £(i>) = A}

are hyperplanes in V. Since the indicatrix S :== F~l(\) is strongly convex, there are exactly one positive number A0 > 0 and a vector y £ S such that WA° is tangent to S at y. Thus £(u) = 0 ,

u e T y S.

Legendre

Transformations

37

On the other hand, TvS = W r = { » e V , g y ( s / , u ) = o } . Thus $(v) = \oSy(y,v),

vGV.

(3.3)

Letting y = \0y, we obtain the desired vector such that £ = gy(y, •). This gives (3.3). Q.E.D. We define a map £ : V \ {0} -» V* \ {0} by i(y)-.= Sy(y,-),

yeV\{o}.

(3.4)

£ is a C°° map satisfying £(Xy) = \£(y),

A > 0.

Set £(0) = 0. It follows from (3.2) that F(y)=F*(e(y)).

(3.5)

Thus £ is a norm-preserving map. Moreover, by Lemma 3.1.1, we know that £ is an onto map. We call £ the Legendre transformation. The Legendre transformation is first used by R. Miron in Finsler geometry and Lagrange geometry.

Co-Area

38

Formula

V

V*

Take a basis {bj}^ = 1 for V and its dual basis {#l}™=1 for V*. Express f = ZiP = 9v(y)yj

Sl = t(y),

(3.6)

where gij(y) = ^[F2]yiyj (y). The Jacobian off is given by

Thus £ is a diffeomorphism from V \ {0} onto V* \ {0}. From (3.5), we can see that F* is C°° on V* \ {0}. Lemma 3.1.2 For any Minkowski norm F on, V, the dual norm F* on V* is always a Minkowski norm. Proof. It suffices to prove that for any £ = £(y), 9*kl(0-=l[F*\-^)=9kl(vh is positive definite, where (gkl{y)) 2

:— (gij(y))

(3-7)

• Differentiating F2(y) =

l

F* (£(y)) with respect to y yields llF%i(y) = \lF*\.(S)gik(y). This implies

g""m Note that

= \n>(t)

= \9ik{y)[F2]yi{y) = vk-

(3.8)

Legendre

Transformations

39

Differentiating (3.8) with respect to yi gives

9iM = g*kl(Z)9ik(v)9ii{v) + g*H{m~{y)

=

g^iOgMwiy)-

Therefore (
Q.E.D.

Let (V,F) be a Minkowski space and (V*,F*) the dual Minkowski space. We can define the Legendre transformation £* : V* —¥ V** and the Minkowski norm F** on V**. Identifying V** = V, we have t*=r1,

F**=F.

(3.9)

For a covector £ € V* \ {0}, let g*^ denote the induced inner product on V*. It is given by

s*H(,v) ••= 9*kl(t)(km,

C = Ckok, v = Vk0k.

The vector y := t~l(£) is determined by C(i/) = g*€(£,C),

C = (iCeV'.

Thus 1/ is given by !/ = l / £ b , = f f * f c ' ( 0 ^ b J = r 1 ( 0 . "

(3.10)

E x a m p l e 3.1.1 ([HrSh]) Let F = a + f3 be a Randers norm on a vector space V, where a is an Euclidean norm and (3 is a linear functional with ||/?|| := sup Q ( 3 / ) = 1 P{y) < 1. Let {bi}? =1 be a basis and {0*}™^ the dual basis for V. Express a and j3 by a

iv) = \]aijViyj,

P(y) = hy1,

y = ylhi

Then ||/3|| = y/a^hbj, where (a^) := ( a ^ ) - 1 - The dual Randers norm F* on V* is still of Randers type. More precisely, F* = a* + (3*, where a* is an Euclidean norm and j3* is a linear functional on V*. They are expressed by

«*(£) = y/a'VZiZj,

£*(0 = b*%,

C=

^

Co-Area

40

Formula

where (l-||/3||2)a^+6^

,y

(i-WPP)2 bi

b*1

i-\\p\\2'

where bl :— b^a%K Let (a*ij) := (a* l J ') _ 1 . a*ij are given by o«i = (1-||/3|| 2 ) ( a 0 - - W , - ) . The norm ||/3*|| := sup Q , ( ? ) = 1 /3*(£) is given by PI

2

= a, y ft"6*' =

1

( QiJ - fc&^ftV = ||/?|| 2 .

_ |

Thus 0 and /3* have the same length with respect to a and a*, respectively. By (1.6) and (3.6), the Legendre transformation £ : V —> V* is given by £ = *(y) =

9ij(y)y*

P = F(v){^

+

fc}^-

(3.11)

Similarly, by (1.6) and (3.10), the Legendre transformation £* — £~x : V* —> V is given by

y = e-\z) = g*klm bk = F*(C,{^

+ 6 «J b f c .

(3.12)

Let F be a Finsler metric on a manifold M. For a point x £ M, Fx := F\TtcM is a Minkowski norm. The dual norm F* : T*M -> R is defined by i£(0=

sup

£fo).

F*(y) = l

We obtain a family of Minkowski norms F* := {F*}XEM- Let £x : T X M —> T£M denote the Legendre transformation of F x . We obtain a family of Legendre transformations £ := {£x}xeM- According to Section 3.1, the map £ : TM \ { 0 } - 4 T * M \ {0} is a C°° diffeomorphism satisfying F*(£(y))=F(y),

yeTM.

Gradients of Functions

3.2

41

Gradients of Functions

Given a function / on a manifold M, the differential dfx at a point x G M,

dfx =

a^ 1 ^''

is a linear functional on TXM. To convert d/x to a vector V / x G T*M, we need a Minkowski norm on TXM. Let F be a Finsler metric on M. By definition, Fx is a Minkowski norm on TXM. Assume that dfx 7^ 0. There is a unique unit vector nx G SXM := F^" x (l) and a positive number A^ > 0 such that

WA* := {dfx(v) = A,} is tangent to S X M at n x . By (3.3), d/x(«) = A* g n j n , , ^ ) ,

v

e r^M.

(3.13)

where F

x(dfx)

= dfx{nx)

= Xx Snx(nx,nx)

= \x.

Define Vfx := A x n, = Fx*(d/X) n x .

(3.14)

Equation (3.13) can be rewritten as dfx(v) = gVfAVfx,v),

veTxM.

(3.15)

If dfx = 0, we set V / x = 0. We call V / x the gradient of / at x. It follows from (3.2) that F*(Vfx)

= Fx*(d/X) = | ^ j .

(3-16)

Co-Area

42

Formula

dfx=0

V/x

Fx = \

Let £ : TM -^T*M and t : T*M ->• T M denote the Legendre transformations. By the regularity of £, we see that V / = £~l(df) is C°° on the open subset {df ^ 0} and C° at {df = 0}. E x a m p l e 3.2.1 Consider a Randers metric F = a + P on a manifold M. In local coordinates (xl) in M, express a

P{y) = Hx)yl-

(y) = yaijitfyw,

y = y

A,

dxilx'

Then \x =

ya^(x)bi(x)bj(x).

By Example 3.1.1,

F(Vf)=F*(df) =

y/(l-\\/3\\2)\df\2 + {0,df)*-(0,df) 1 - WPP

where \df\x := ]/aiHx)~^(x)-^-(x),

{/3,df)x := a» (x^x)

^-(x).

By (3.12), we obtain V/

F*(df) r (1 - | | / ? | | y J £ + (/?, d/>o% i - P I 2 1 \/(l-||/3|| 2 )M/i 2 + {/3,d/)2

Clearly, V / is not C°° at points where df = 0.

J

J dxl

Gradients of Functions

43

The basic properties of V / are shown in the following lemmas. Lemma 3.2.1 Suppose that f is C°° on an open subset U with df ^ 0. Let g := gv/- Then V / = V/,

(3.17)

where V / denotes the gradient of f with respect to F := \fg.

Further,

F(V/)=F(V/).

(3.18)

Proof. (3.15) gives gv/(V/,w) = d / H = 3 ( V / , W ) = g v / ( V / , u ; ) ,

VweTU.

Thus V / = V / . Taking w = V / in (3.19) yields (3.18).

(3.19) Q.E.D.

Lemma 3.2.2 Let f be a C°° function on an open subset U with df j^ 0. Then n := F(y,-. V/|jv t is orthogonal to Nt := f~l(t) with respect to g n . Proof. Since / is a constant on Nt, df(w) = 0,

Vio 6 rJV t .

It follows from (3.15) that 0 = df(w) = F(Vf)zn(n,w),

\/w€TNt.

By definition, n is orthogonal to TNt with respect to g n .

Q.E.D.

Let d = dp denote the metric induced by F, that is defined by

dF(p,q) = mt f F{c(t))dt, c Jo where the infimum is taken over all piecewise C°° curves c : [0,1] —• M with c(0) = p and c(l) = q. If F is reversible, i.e., F(—y) = F(y), then dp is reversible. Given a compact subset A c M, define p.+ (z) := d(A, x),

p_(x) := -d(a;, A).

For xi, a;2 G M and z e A, d(z, xi) < d(z, X2) + d(x2, Xi).

(3.20)

Co-Area

44

Formula

This implies that d(A, xi) < d(A, x2) +

d(x2,xi).

d(A, x2) < d(A,x\)

d(xi,x2).

and +

Thus -d(x2,xx)

< p+(x2) - p+(xi) < d(x1,x2).

(3.21)

By (1.16), we conclude that p + is locally Lipschitz continuous on M. Similarly, one can show that p- is locally Lipschitz continuous on M. Thus both p+ and p_ are differentiable almost everywhere. Lemma 3.2.3 Let (M, F) be a Finsler space and p+, p~ be the functions defined in (3.20). Suppose that for any points p,q € M, there is a shortest unit speed curve from p to q. Then F(Vp+) = 1,

F(Vp_) = 1

hold almost everywhere. Proof. We shall prove F(Vp_) = 1 only. Suppose that p_ is differentiable at 1 £ M. For a vector v € TXM, let c be a constant speed curve with c(0) = v £ TXM. According to Section 1.3, we have

F{v)=

iim feW). s->0+

S

For s > 0, p - ( c ( a ) ) - p - ( c ( 0 ) ) _ d(x,A)-d(c(s),A) s s

K

~

d(x,c(s)) s

Letting s —> 0 + yields dp-{v) < F(v),

VveTxM.

(3.22)

On the other hand, there is a point p_ £ A such that d(x,p^) = d(x, A). By assumption, there is a unit speed curve c : [0, a] —• M with a = d(x, A) from x = c(0) to p_ = c(a). Thus for any 0 < si < s2 < a, d(c(s1),c(s2))

= s2 - si.

Gradients of Functions

45

For small s > 0, p-(c(s))

- p-(x)

=

s

d(x,c(a)) -d(c(s),c{a)) s

> —

d{x,c(s)) s

=

This implies dp_(c(0)) > 1.

(3.23)

Combining (3.22) and (3.23), one obtains F*(dp_)=

sup dp-(v) = l. F(v) = l

By (3.16), one concludes that F(Vp_) = F*{dpJ) = 1.

Q.E.D.

Remark 3.2.4 The condition in Lemma 3.2.3 is satisfied if the Finsler space is positively complete. See Lemma 11.4.1 below. Based on Lemma 3.2.3, we make the following Definition 3.2.5 A locally Lipschitz function p on a Finsler space (M, F) is called a distance function if F(Vp) = l = F * ( d p )

(3.24)

holds almost everywhere. It follows from Lemma 3.2.1 that if p is a distance function with respect to F, then F(Vp) = F(Vp) = 1,

Co-Area

46

Formula

where F = ^/gvp. Thus p is also a distance function with respect to the Riemannian metric F.

3.3

Co-Area Formula

In calculus, we have several important integral formulas for area and volume. Consider a bounded domain D in the Euclidean plane R". For a number t, let L(t) denote the length of the cross section of the line x = t with D. Then CO

L(t) dt. /

-oo

If D is enclosed by two graphs y = f(x) and y = g(x), a < x < b, where f(x) > g(x), then Area(D) = y

{f(x)-g{x)}dx.

This formula can be generalized to Finsler m spaces. We have the following co-area formula. Theorem 3.3.1 Let (M,F,dfx) be a Finsler m space. Let

[

(f

J-OO

V

fdu)dt.

(3.25)

Jv>- 1 (t)

Proof. It suffices to prove (3.25) in a coordinate neighborhood U. For the sake of simplicity, we assume that ip is C°° with dip ^ 0 on U. Fix a number t0 such that
x-

V

^

F(V<^) 2 '

For a point x G

Co-Area

47

Formula

Thus If O C(t) = t.

The integral curves of X give rise to a coordinate map ip = (x1, • • •, xn) : U ->• (-e, e) x B™~ such that (poi(j~1(x1,xaJ

= x1.

Thus Nt ••=
_a_

ax1

t

Rn_1

Let n:

_ V

By Lemma 3.2.2, n is a normal vector to Nt- Take a local frame {bi}™=1 for TM with bi = n and b a = -^, a = 2, • • • ,n. Let {^}™=1 be the coframe for T*M dual to {b;}? =1 . We^have J1 = A dx1 = A d V)

6a=dxa,

(a = 2,---,n).

Observe that

-«>> = » H ^ H & ™

48

Co-Area

Formula

We obtain 91 = Put d\i = a(x)9x A - A f .

———dx1. F(V^)

Then by (2.24) dv = a(x)62 A---0",

where 6a denotes the pull-back of 6a to Nt, a = 2, • • •, n. We obtain du

=

J*}„ .dx1 A--F(V
Adxn,

dv

=

a(x)dx2 A • • • A dxn.

[ fF(V
=

We obtain 1

f

w>n_Ja(x)dx

---dxn

f_e(jw_Je{x)dx2-.-dxn)dxl

=

= r (f J-e

fdv\dt.

K

JNtnU

'

Q.E.D.

We now consider a Finsler space (M, F) with the Busemann-Hausdorff measure Vol/?. By (3.25), [ fF(V
[ ([ fdAF)dt. J-OO yJ
(3.26)

Let F denote the induced Finsler metric on ip~l(t). By definition, dAF = ((n) dVF. where C is defined in (2.19). By Lemma 2.3.1, ("(n) satisfies (1 + X)-ncndVF

< dAF < 2nc(n)dVF,

(3.27)

cn := Vol(B n )/Vol(B n _ 1 ).

(3.28)

where A := sup F(y)/F(-y), yeSM

By (3.25), we obtain the following co-area inequalities for Finsler spaces.

Co-Area

Formula

49

Theorem 3.3.2 Let (M,F) be a Finsler space. Let

J fF{V
f

( f

J-oo

[ fF(V (1 + \yncn JM

V

fdVp)dt,

(3.29)

J ¥ >~ 1 (*)

r J-oo

( f V

fdVp)dt.

^v-1(t)

(3.30)

'

There are many important applications of Theorem 3.3.2. For example, one can use it to extend Gromov's estimates on the Filling radius of Riemannian spaces to Finsler spaces [Grl]. Below is a simple application of Theorem 3.3.2 Proposition 3.3.3 Let S be the indicatrix in a Minkowski space (V, F). Let F denote the induced Finsler metric on S. Then 2-"nw„_ 1 < Voli?(S) < (1 + A) n nw„_ 1 , where A is given in (3.28) and w„_i = Vol(B n

(3.31)

).

Proof. Let p(x) := F(x) denote the distance function from the origin. By definition, the volume of the unit ball B in (V, F) is equal to Vol(B ). Let dVp denote the volume form of the induced Finsler metric F on S(r). Note that Volp(S(r)) = r"- 1 Vol J? (S). It follows from (3.29) that Vol(B") = VoliKB)

<

2 n c„ /

Volp(S(t))dt

Jo

= =

2ncn f Jo

tn-1Vo\F(S)dt

fL^Volp(S).

This gives Vol^(S) > 2- n nVol(B n )/c n = 2- n nVol(B n

1

).

The proof of the inequality on the right hand side of (3.31) is similar, so is omitted. Q.E.D.

Co-Area

50

Formula

When the Minkowski norm is not reversible, the volume Volj?(S) has no universal upper bound. Example 3.3.1 FE on R 2

For 0 < e < 1, consider the following Minkowski norm Fe(u, v) := v u2 + v2 — eu.

The indicatrix SE = F~l(l)

is an ellipse,

Parametrize Se by 1

u=-

£

„

jCos6>+g, 1 - e2 1 - e2

l

v—

•

n

= smfl, VI - e2 r

so that FE{u,v) = y ' ( I - £ - I ) 2 s i „ 2 ^ + - J - j + This gives

Voi^(s£) = fj J^y^e

^ s i n ,

+ ^dB >v T ^ t

- 4 OO,

05 e —> 1 . Compare [Ma2j. Let (V, i*1) be a Minkowski space. F induces a Riemannian metric g on V - {0} b y g(u,v) :=gy(u,v),

u,v€TyV

= V.

Let (yl) be a global coordinate system in V associated with a basis {k>i}™=1. The Riemannian volume form of g is given by dVa =

1

y/det{gij(y))dy

---dyn,

where gi:j(y) = g y (b;,b.,). Let g denote the induced Riemannian metric on S := F~~x(\). Identifying V \ {0} with (0, oo) x S in a natural way, i.e., y = (F(y),y/F(y)). Since

Co-Area

gv(y,w) form.

Formula

51

= 0 for any w £ TyS C V, g can be decomposed to the following g = dt2® t2g.

(3.32)

Denote by Vol r (B) and Volr(S) the Riemannian volume of (B, g) and (S, g), respectively. By (3.32), f1

1 V o l r ( B ) = / tn~1Vo\r{S)dt=-Volr(S). Jo n

(3.33)

Recall that for the induced Finsler metric F on S, the volume Vol^(S) has a universal lower bound, but no universal upper bound. In addition, if F is reversible, then Vol^-(S) has a universal upper bound. For the induced Riemannian metric g on S, we have the following P r o p o s i t i o n 3.3.4 Let (V, F) be an n-dimensional reversible Minkowski space. Then the Riemannian volume of the indicatrix S satisfies Vol r (B) < Vol(B n _ 1 ),

Volr(S) < Vol(S n _ 1 ).

Equality holds if and only if F is Euclidean . Proof. Let {bi}" = 1 be a basis for V and {0l}™=1 the dual basis for V*. Let B" := {(
< l},

B*" := {(fc), F*(CiOl) < l } .

By Santalo inequality [MePa] which requires the symmetry of B n and B* n . Vol(Bn)Vol(B*n) < Vol(B"). Equality holds if and only if B™ is an ellipsoid. The Legendre transformation I: B n ->• B* n given by yl -^-d has Jacobian det(gij(y)).

/

•=9ij(y)yj

Thus

det(9ij(y))dy = f

d£ = Vol(B*n).

52

Co-Area

Formula

Observe that VoL-(B)

=

/

Jdet(gij(y))dy

<

[J ^ det(9ij (y))dy\

-

[vol(B* n )Vol(B n )l"

<

Vol(B n ).

1/2

Vol(B") 1 / 2

Equality holds if and only if det(gij(y)) = constant and B n is an ellipsoid. This completes the proof. Q.E.D. The idea of the proof is borrowed from [Du], where C. Duran established a sharp upper bound on the volume of the Sasaki metric on the unit tangent sphere bundle. Here we are only concerned with the volume of the induced Riemannian metric on the indicatrix. When F is not reversible, then Volr(S) can be greater than Vol(§ n ~ ). Consider the case when dimV = 2. Let c(t) = u(t)hi + v(t)b2,

a
be an arbitrary parametrization for the indicatrix S = i ? _ 1 ( l ) . For any vector v = Xc(t) + (c(t) G Tc{t)V, .

,

.

x2

u'(t)v"(t)-u"(t)v'(t)

~

.on„.

This gives a formula for the Riemannian perimeter L r (S) with respect to g, rb

MS) = I Ja

E x a m p l e 3.3.2

U'{t)v"(t) u(t)v'(t) -

u"(t)v'(t) dt. u'{t)v(t)

Consider the following Randers norm on R 2 , FE(u, v) := V u2 + v2 — eu,

where 0 < e < 1. Parametrize S£ = F'1^) 1

, x

i - e

2•cos(t) w

by c(t) =

£

+ -i -

(u(t),v(t)),

1

e

2j

',

v=

yr

^sin(t).

Co-Area

Formula

53

By (3.34), we obtain

The Riemannian perimeter L r (S £ ) of S e with respect to gE is r27T

L r (S £ )

=

/ Jo

^dt

\/l + + ££COs(t) A/1

= 2r/2{ 7o >

27T

y

i1

+ , yjl

i /1l — £COs(t)

V

-ecos(t)

* u + £ COs(i)

Vl+£cos(t)J

Chapter 4

Isoperimetric Inequalities

The classical isoperimetric inequality for the standard unit sphere S™ C K. states as follows: for any domain fl C § with regular boundary d£l,

S> Vol(S n ) -

hsn(s) S W

'

s-

V

°K»)

Vol(S n )'

where /is»(s) denotes the ratio Vol(9Bs)/Vol(S™) for a geodesic ball B s C S with Vol(Bs) = s • Vol(S n ). The function /ign is called the isoperimetric profile of § . The isoperimetric profile can be defined for Finsler m spaces. In this chapter, we will discuss the relationship between the isoperimetric profile and the Sobolev constants (first eigenvalue) of a Finsler m space. The classical isoperimetric inequality is used by P. Levy to discuss his concentration theory [Le]. Levy's concentration theory is developed further to Riemannian spaces and more general metric measure spaces [GrMi][Gr4]. In his study on the concentration of metric and measure spaces, M. Gromov introduces many new geometric invariants. Among them are the two basic quantities: the expansion distance and the observable diameter [Gr4]. In this chapter, we will discuss the relationship between the isoperimetric profile and these two quantities. 55

56

4.1

Isoperimetric

Inequalities

Isoperimetric Profiles

Let (M, F, d/j,) be a Finsler m space. We define the isoperimetric profile hM • [0,1] ->• [0, oo) by (4.1)

hM{s):=mi

iWy

where the infimum is taken over all regular domains Q C M such that /i(fi) = s • n(M), and v denotes the induced measure on dfl (cf. (2.23)). Note that hM(s) = hM(l - s) > 0,

Vse(0,l)

and hM(0) = 0 =

hM(l).

(j,(£l) = s • fi(M

Therefore we make the following Definition 4.1.1 A continuous function h : (0,1] isoperimetric function if it satisfies h(s) = h(l-s)

> 0 , V s e (0,1)

and h(0) = 0 = h{l).

R_i_ is called an

Isoperimetric

Profiles

57

Given an arbitrary isoperimetric function h, we will construct an open interval (-L, L) and a function Oh{t) on (-L, L) such that /

crh(u)du = s,

ah(t) = h(s).

(4.2)

First, we try to find a solution s = s(t) to the following O.D.E. ds dt s(0)

= h(s)

(4.3)

2'

s 1I

s = s(t) 1

F

-L

- t *

Let r)(s) :=

h(u)

du.

(4.4)

77 is an increasing C 1 function with a symmetric range (—L, L). Let

sW:=^W Then s(t) is a solution to (4.3). Let

ah(t)~h(S(t))

= h(V-l(t)).

Since the range of s(i) is the open interval (0,1), we have lim a hit) = 0. t—>±L

(4.5)

Isoperimetric

58

Inequalities

Further, s(t) = J

°h(t) = /i( /

ah{u)du,

ah{u)duj

= /i( /

(4.6)

ah{u)duj.

This gives (4.2). We can view the Finsler m space ((—L,L),\ • \,ah(t)dt) dimensional model for h.

4.2

(4.7)

as a one-

Sobolev Constants and First Eigenvalue

There are many important geometric invariants for Finsler m spaces. Let (M, F, dfi) be a closed Finsler m space. For 1 < p, q < oo, define

A Pl ,(Af):=

y

inf.

u6C°°(M) i. n f

- '

—

—

—'

u

A p

^R(^)/M! - l ^):

\p,q(M) is called the (p,q)-Sobolev constant. Another geometric invariant is the the first eigenvalue of defined by

Ai M :=

inf

fM\F*(du)\2dn . /M' \ ;' *

(M,F,dfi),

, .

x

4.8

Clearly, Ai(M) = [A 2 , 2 (M)] 2 . The Sobolev constants XPiq(M) are closely related to the isoperimetric constants (cf. [Be] and [Ga] in the Riemannian case). In what follows, we shall establish some relationships between the Sobolev constants and the isoperimetric constants for Finsler m spaces.

Sobolev Constants

and First

Eigenvalue

59

For an isoperimetric function h and 1 < p, q < oo, define

(/oVM'M*)^)17' XPtq(h) := inf — ^ 0

^-—,

(4.9)

(infA6R/>(s)-A|W)

where the infimum is taken over all bounded piecewise C 1 -functions on [0,1]. Xpq(h) can be viewed as the (p, qr)-Sobolev constant of the Finsler m space ((-L,L),\

•

\,ah(t)dtj.

By symmetrization based on the one-dimensional model, we can prove the following Theorem 4.2.1 ([GeSh]) Let (M,d,fi) be a closed Finsler m space. Suppose that fiM > h for some isoperimetric function h. Then A P ,,(M) > A M (/i).

(4.10)

In particular, Ai(M) > [A2,2(/i)]2. Proof. Under the assumption hM > h, for any regular domain fi c M, fi(M) ~

\n(M)/'

Now we process to prove (4.10). Take an arbitrary function u G C°°(M). Since u can be approximated by Morse functions, we may assume that u is a Morse function on M. Decompose u into the difference of the positive and negative parts, u = u+ — u_. For a > 0, let fl+(a) := {x G M,

u+(x) > a}.

Define t+ : [0,supw + ) —> (—L,L) by

fV"'"""^ Jt+

f"1'

60

Isoperimetric Inequalities

The function t+ is an increasing piecewise Cl function. By assumption h-M > h, we have that for a > 0, M(M)

>

/i

=

/i( / Wt + (a)

-

V V u(M) M(M)

=

)

ah{t)dt) ' CTfc(t+(a)).

(4.12)

The last equality follows from (4.7). Define ip+ : (—L,L) —> R + by setting 0 and
t'+(aY We know that (4.11),

f°° f /

F(VM+)

= F*(du+).

wn—^)du

By the co-area formula (3.25) and

= ^+( fl )) = ^(M) /

**(')*• (4-13)

Differentiating (4.13), one obtains /

*

dv =

tiM)oh(t+{a))t+(a).

(4.14)

The Holder inequality yields

„(«,+ (»)) = j

lF*(du+)]'?(b/

Jan+(a) [F*(du+)] « < ( f

[F*(du+)]^y(

[F*(du+)]-ldv)'~\

f

(4.15)

It follows from (4.12) - (4.15) that / [F*(du+)}qdn JM

=

[ ( f JO \Jdn+(a)

[F*{du+)]q-ldu)da ' ' F

d

1

{hn+(a) *( ^r ^)

1-1/9


Sobolev Constants and First

Eigenvalue

61

(Th{t+{a))

> n(M) f Jo

1-1/9.

L

(^(t + (a))t;(o)) J

fx

1

= ^( M )y o

^

=

/x(M) /

v+(*+(a))

=

n(M)J

(a) ] 9 ^(

f

+( ffl )) f +(°) da (7/,(t + (o))t' + (a)da

\d
That is, (4.16)

||F*(du+)||,>/i(M)«||dV+|

Let A be an arbitrary number. By (4.14) and the co-area formula again, we obtain / |u+-A|"d/i Jn+(o)

=

f°|a-A|"[/ l Jo Jdn+(a)F

=

n{M)

) dv\da idu+) i

\a-\\v<jh(t+{a))t'+{a)da Jo r°° i

=

/x(M) / Jo

ip

\tp+(t+{a)) - X\ crh{t+{a))t'+{a)da ' '

= KM) I

\

Jt+{0)

That is, \W+ - A|| LP(n+ (0)) = KM)*

\\
Similarly, for the negative part w_ and a > 0, let n_(a) := {x £ M, u _ ( i ) > a}. Define t_ : [0,oo) -> ( - L , L ) by

L

ah{t)dt

--mr-

Define R+ by setting 0 and if-(t) = 0 if £ > £_(0). By the same argument as above, we obtain !l«- + A||iP(n+(o)) = M(M)1/p|](p_ + A|| L , ( _ Li t_ ( o)),

62

Isoperimetric

Inequalities

\\F*(du-)\\q>^My/*\\d^\\g.

(4.17)

Note that t~(a) < t+(a), Va > 0. Set

_. It follows from (4.16)-(4.17) that

\\u-x\\p =

f,(M)1/"y-x\\p,

\\F%du)\\q > ^(M^Wd.pW,. Therefore

"^''"-'"iPIrs K*h-

(4 18)

-

Take the substitution s = s(t) and let (s) := ¥>(*)• Note that ds = h(s)dt =
(/fi iv'wi'^wdt)1^

(/0: mswhwds)1'9

( i n f W - L | v ( t ) - \\Voh{t)dtf'P

( i n f ^ R / o 1 | # s ) - A|Pd S ) 1 / P (4.19)

(4.18) and (4.19) imply A Pl ,(M) > A Pi ,(h). Q.E.D. For an isoperimetric function h, define MM)

:= i a f - p n

^

r

,

(4.20)

/I(M(«)/M(M)JM(M)

where the infimum is taken over all regular domains fi in M. lh(M) is called the h-isoperimetric constant of M.

Sobolev Constants

and First

Eigenvalue

63

Take an arbitrary domain Q,
h(s)n(M)

> Ih(M).

This gives /i(M)

>lh(M)h(s).

Thus hM(s) >

Ih(M)h(s).

(4.21)

Consider the following isoperimetric function ha(s) := min(s, 1 — s)i - i

1 < a < oo.

Here we assume that ^ = 0 if a = oo. It can be shown that if — > - — -, a

then \p,q(ha)

a — q

p'

> 0. In particular, A2>2(^Q) > 0 for any 1 < a < oo. Let Ia(M)=Ih„(M).

We call / ^ ( M ) the Cheeger constant. The isoperimetric profile of a Riemannian manifold can be estimated under a lower Ricci curvature and an upper diameter bound. This is done by M. Gromov using Levy's method. Gromov applies his estimates on the isoperimetric profiles to obtain some estimates for the eigenvalues of the Laplacian on a Riemannian manifold. See [Gr4] [Gr5] for details.

64

4.3

Isoperimetric

Inequalities

Concentration of Finsler m Spaces

Let R°° be an infinite-dimensional Hilbert space and S°° the unit sphere in R . Consider a finite-dimensional unit sphere S in R , which is the intersection of S°° with a finite-dimensional subspace in R . Let us take a look at this sphere S. The diameter of S is always equal to 7r and the sectional curvature of S is always equal to one (see Section 6.1 for the definition of sectional curvature). A natural question arises: How can we detect the dimension of S and how to tell the difference between two finitedimensional unit spheres in R ? We have a satisfied answer to this problem using the Levy concentration theory of mm spaces [Le]. The Levy concentration theory of mm spaces has been developed further by M. Gromov [Gr4]. He introduce several geometric quantities for mm spaces. Among them is the so-called expansion distance. Definition 4.3.1 Let (M,d,n) be an mm space with fx{M) < oo. For 0 < s < \ , the expansion distance ExDist(M;e) is defined to be the infimal p such that the following statement holds

If a subset A C X satisfies

H(A) > efi{M),

then

n(up(AJ) > (1 - eMM). Here Up(A) denotes the p-neighborhood of A in M,

UP(A) := lx e A, d{A,x) < p\.

Concentration

of Finsler m Spaces

65

The expansion distance tells us how the mass concentrates, with respect to the underlying metric. We say that a sequence of mm spaces (Mi, di,fH) concentrates to a point if ExDist(Xi; e) -> 0 as i -> oo for any e € (0,1/2). A trivial inequality is that ExDist(M;e) < Diam(M). Thus if the diameter of (M i? du /x») converges to zero, then (M i5 di^i) concentrates to a point. There are sequences of mm spaces which do not collapse to a point, but concentrate to a point. E x a m p l e 4.8.1 Let S* denote the standard unit spheres in R* C E°° with the induced Riemannian metric and Riemannian volume form. We have ExDist(S^e)«^, yji

where C(e) depends only on e £ (0,1/2). Thus the sequence {S*} concentrates to a point as i -+ oo, while, {S*} does not collapse to a point.

dim —» oo

8

66

Isoperimetric

Inequalities

The above concentration phenomenon was first discovered by Paul Levy [Le]. Levy explained this by using the sharp isoperimetric inequality for S . Levy's result has been generalized to Riemannian spaces by M. Gromov and V. D. Milman [GrMi]. They proved that for any closed Riemannian spaces {M,g), ExDist(M;e) < -

C

& -

where \\(M) denotes the first eigenvalue of X (see also [Gr4]. The same method can be carried over to Finsler spaces, due to Lemma 3.2.3. Theorem 4.3.2 0 < e < 1/2,

Let (M,F,d/j.)

be a closed Finsler m space. For any

ExDist(M;£) < 2\ — - 1

,

Proof. Fix p > 0 and 0 < e < 1/2. Let A c M be any closed subset and UP{A) the p-neighborhood of A. Define

u{x). im

- ^

where Vp := ii(Up(A)) and V0 := n(A). Let V = fi(M). By Lemma 3.2.3, we see that f -7T1

x € UJA) - A py

F*(du) = F(Vu) - I 7^va \

0

'

X€AU(M-UP(A)):

This

[ [F*(du)}2dn= [

_ J _ d M = i.

Let A € R be an arbitrary number. Note that / JUP(A)

|« - \\2d(i > J \u- \\2dfj, = JA

X2n(A).

Concentration of Finsler m Spaces

67

Thus / \u-X\2dfi JM

f \u-X\2dfi+ JM~Up P

=

[ Jup

\u-\\2dn

* ( A - V ^ ) 2 ( y - ^ + A ^° (V - VP)V0 2 (VP-V0)(V-VP + V0 -P

> We obtain X (M\ <

/ M ^ W ^ M

(VP-Vo)(V-VP + Vo)-2

<

M(M)

-MX€RfM\u-\\*dp-

{V-Vp)Vo

P

•

In other words ' (Vp - V0)(V - Vp + Vq)

>*f


L

vsm-

(422)

'

For any p < ExDist(M;£), there is a subset A satisfying V0 > eV,

VP<(1-

e)V.

Then by (4.22), we obtain l

P<2\/^"1 2e

y/%{M)

This implies that ExDist(M;e) < 2W — - 1 2e vOi(M) Q.E.D. The expansion distance is also controlled by the isoperimetric profile. Theorem 4.3.3 Let (M, F, d(i) be a closed Finsler m space. Suppose that hM > h, where h is an isoperimetric function. Then for any 0 < e < j , ,i-e

ExDist(M;£) < / Js

1

TT^duh(u)

( 4 - 23 )

68

Isoperimetric

Inequalities

Proof. Let A C M be a closed subset and Vp := /i(fp(^)),

V0 = M ( 4 ) ,

V-

ti(M).

Define ip : [0,pi) ->• (~L,L) by rTTdu-

V(P) == /

where pi := smpxeM d(A, x). Applying (3.25) to the special function
( Atdt, Jo

we obtain the

(4.24)

where At := v{ip~l(t)) denotes the induced measure of ip~1(t). Note that

It follows from HM > h that This implies /•v,/v x / TT~\du

=

'Vo/

V(P) - P>

Vp
(4.25)

Let du. Assume that

(J,(A)

> e/j,(M), i.e.. Vo ^

> e.

(4.26)

We assert that n(UPo{A)) > (1 - e)fi(M), i.e., ^ > l -

£

.

(4.27)

Observable

Diameter

69

Note that if p0 > p\, then

>-£ = !>!-.. Now we assume that p0 < p\. In this case, by (4.25) and (4.26), we have /

Je

TT^du

TT^du

- /

M«)

Jv0/V

=

TT^du-

V(P°) ^P°=

h

W

h

Ku)

This implies (4.27). Therefore, we can conclude that fl-e

r ExDist(M;e) < p0 = Je

i

h{uY -—rdu. Q.E.D.

h[U

By (4.21) and (4.23), we see that for any isoperimetric function h, 1 fl~e 1 ExDist(M;e) < ——— / -r^-du. Ih(Af) Je h(u) In particular, for ha(s) = min(.s,1 — s ) 1 - ^ , 1 < a < oo, E

X

Dist(M;,)<^,

where

C7a = ( ^ [ ( 3 ) 1 / a - e V a ] i

-21n(2e)

ifl
if a = oo

The expansion distance of Finsler m spaces can be estimated from below if they satisfy certain curvature bound. See Theorem 16.4.1 below.

4.4

Observable Diameter

To study the Levy concentration of mm spaces, Gromov [Gr4] also introduce the notion of observable diameter. Let (M,d,p,) be an mm space. Denote by Lip x (M) the set of all Lipschitz functions / : M —» R satisfying -d(xi,x2)

< / ( x i ) - f(x2) < d{x2, xi),

Vxi,x2

€ M.

70

Isoperimetric

Inequalities

By (3.21), we know that for any compact subset A C M, the distance function p+{x) := d(A,x) belongs to Lip x (M), so does p~(x) := — d(x, A). For 0 < e < 1, define Diam(/»/i,e) := inf {Diam(J) : I c R, M / _ 1 0 O ) > (1 -

<0A»(AO}

Th° observable diameter is defined by ObsDiam(M;£) :=

sup

Diam(/»/i,e).

/€LiPl(M)

The expansion distance is equivalent to the observable diameter according to the following lemma. Lemma 4.4.1 Let X = (M,F,dp.) be a compact reversible Finsler m space with p,(M) < oo. Then for any 0 < e < 1/2, ExDist(M;£) < ObsDiam(M;£).

(4.28)

ObsDiam(M; e) < 2 • ExDist(M; e/2).

(4.29)

For any 0 < e < 1,

Proof. First we prove (4.28). Let p < ExDist(M;£). There exists a subset A c M such that H(A) > ep(M),

ii{up(A))

< (1 - £)MM).

Observable Diameter

71

Define J

, . _ f dist{A,x) W - \ p

x£Up(A) xeM-Up(A).

Let I = [a, b] be an arbitrary interval such that M

(/-1(/))>(1-£)M(M).

(4.30)

We assert that a < 0 and b> p. Hence Diam(7) = b — a > p. lfb
Kr\l))

< I*(UP(AJ) < (1 - e)MM),

that contradicts (4.30). If a > 0, then / ^ ( I ) C M - £/ a (A). Thus M / _ 1 ( / ) ) < M(M) - /*(tf 0 (A)) < p(M) - M(A) < (1 -

e)n{M),

that contradicts (4.30). Therefore Diam(Z) = b — a > p. This implies Diam(/*/i;e) = inf Diam(J) > p. Thus ObsDiam(M;e) > p. Letting p -s- ExDist(M;e) yields (4.28). We now prove (4.29). Let p < ObsDiam(M;£:). There is a Lipschitz function / G Lip^X) such that Diam(/,/Lx;£) > p. Let a be the supremum of r such that p{f-l{-^r))<£-p{M). Let b be the infimum of R such that Mr1(JR,oo))<|MM). Then for any r < a and R > b, M/

_ 1

M ) = MM) - n(f-l(-™,r))

~ M r ^ o o ) ) > (1 -

e)p(M).

72

Isoperimetric

Inequalities

Thus Diam([r, R]) > Diam(/*/^;£r) > p. This implies b — a > p. Let A = f-l(-oo,a)

and B = /- 1 (6,oo). Then M (A)

> |M(M),

p(B) >

£

-p{M).

Let

For any x G [7^/2(^4) rr

\

^ r,

an

d 2 £ A with d(,z, x) < p/2, ,/

\

x

P

f(x)
^

<

Cl + b

b —d a

+ - _ = _ _ .

Thus z G A For a; G Up/2{B) and z e B with d(.z,x) < p/2, f[x) > f(z)-d(x,z)

> &- £ > 6 - —

=

— .

By the above argument, we conclude that Up/2(A) c A, Since AnB

Up/2{A) c B.

= <8,

p(yp/2(A))

+PL(UP/2(B))

< v(A) + p(B) = p(AnB)

< p(M).

We may assume that p(up/2(A))

<\p{M)<{l-e)p{M).

This implies ExDist(M;e/2) > p/2. Letting p -> ObsDiam(M;e), we obtain (4.29).

Q.E.D.

Observable

Diameter

73

It follows from (4.28) and (4.29) that for any 0 < e < 1/2, ExDist(M; £ ) < 2 m ExDist(M; £ /2 m ). and for 0 < £ < 1, ObsDiam(M;e) < 2 m ObsDiam(M;£/2 m ).

Corollary 4.4.2 0 < £ < 1,

Let (M, F, dfi) be a closed Finsler m space. For and

ObsDiam(M;e) < 4 J - - 1 e

* Ai(M)'

Chapter 5

Geodesies and Connection

In a metric space, minimizing curves are of special interest to us. Locally minimizing curves in a Finsler space are determined by a system of second ordinary differential equations (geodesic equations). With the geodesic coefficients, we introduce the Chern connection and the notion of covariant derivative.

5.1

Geodesies

In this section, we shall derive the Euler-Lagrange equations for locally minimizing curves in a Finsler space (M,F). Let c : [a, b] 4 M be a constant speed piecewise C°° curve F(c) = A = constant. By definition, there is a partition of [a, 6],

a = t0 < • • • < tk = b,

such that c on each [£i_i,£i] is C°°. Fix the above partition and consider a piecewise C°° map H : (—e,e) x [a, b] —> M such that

(a) H is C° on (—£,e) x [a, 6]; (b) H is C°° on each (—£,£) x [U-i,ti], % = 1, • • •, k; (c) c{t) = H(0,t), a
Geodesies and

76

Connection

cu{b)

The vector field

n * ) = ^ ( t ) ^ |l e W( : = ^ ( 0 , * )

dx ' du is called the variationfieldof H. The length of cu(t) := H{u, t) is given by

L(u):= J F(cu(t))dt = J2J*

F(^{u,t))dt

Observe that

i=l

=

^;{[F2]x>-[F2)xiy>yli*}vkdt

/

+ibF-93^vk i=i r6

(5.1) where g^y) := |[F 2 ] y V (y) and G'(y) := \gU(y){[F2}xW(y)yk

- [F2U(y)}.

(5.2)

Geodesies

77

Let

^••-Fikv^l+2Gi{c)}i-Mt)-

(5 3)

-

n(t) is called the geodesic curvature of c at c(t). We can express (5.1) in index-free form L'(0)

=

g c ( « , ^ ) ^ + A- 1 g ( ; ( 6 ) ( C (6),^(&))-g ( ; ( a )(c(a),l/(a))

-Xj

fc-i

+

A_1

E

{Sc(. r ) (c(*D- ^(*i)) " 8c(t+) ( < ^ + ) , n * i ) ) }, (5-4)

where A = F(c(t)) is a constant by assumption. Assume that c has minimal length. Then 1/(0) = 0 for any piecewise C°° variation H of c fixing endpoints. First, we take an arbitrary piecewise C°° variation H oic with -ff (M, ti) = c(ti) (hence V(U) = 0), i = 0, • • •, k. By (5.4), we obtain L'(0) = -\f

g d ( K , 7 ) d t = 0.

This implies /c(t) = 0 . Now for any 1 < i 0 < fc — 1 and v G T c ( t . j M , we take a piecewise C°° variation H of c, that fixes two endpoints of c with v

(ti„)=v,

H(u,ti)

= c(ti),

i^i0.

By (5.4), we obtain L'(0) = A - ^ g ^ + ^ c ^ t ) ^ ) - g ^ - ^ c f o - J . " ) } = 0 . By Lemma 1.2.4, we conclude that t{t:o) =

c(tf).

That is, c is C 1 at each tj. In local coordinates, K = 0 is equivalent to the following system c* + 2G*(c) = 0.

(5.5)

78

Geodesies and

Connection

(5.5) is a system of second order ordinary differential equations. Thus c must be C°° at each t\. We have proved the following Proposition 5.1.1 Let c be a constant speed piecewise C°° curve in a Finsler space (M, F). If c has minimal length, then c is a C°° curve with vanishing geodesic curvature n = 0. In virtue of Proposition 5.1.1, we make the following Definition 5.1.2 A C°° curve in a Finsler space (M,F) is called a geodesic if it has constant speed and its geodesic curvature K = 0. First Variation Formula: Let c : [a, b] —» M be a unit speed geodesic and a : (—e, e) —* M a C 1 curve with • M with H(u,a) = c(a) and H(u,b) = CT(M). By (5.4), the length function L(u) of the curve cu(t) := H(u,t), a < t < b, satisfies £'(0)=gc(6)(c(&),«H0)).

(5-6)

Equation (5.6) is called the first variation formula.

The local functions Gl in (5.2) can be expressed by G\y) = \9U(y){2^(y)-^(y)}y^.

(5.7)

We call G% the geodesic coefficients. The geodesic coefficients G1 give rise to a globally defined vector field on TM \ {0}

G : = ^ - 2 ^ ) A

(5.8)

Geodesies

79

G is C°° on TM \ {0} and C 1 at zero tangent vectors in TM. We call G the spray induced by F. A curve c is a geodesic if and only if it is the projection of an integral curve of G. See [Sh9] for more detailed discussion on sprays. From the definition, the geodesic coefficients Gl (y) satisfy the following homogeneity condition G\\y)

= X^G'iy),

A > 0.

(5.9)

But, Gl{y) are not quadratic in y G TXM in general. Definition 5.1.3 A Finsler metric is called a Berwald metric if in standard local coordinate systems (xl,yl), the geodesic coefficients Gl{y) are quadratic in y e TXM for all x G M, that is, there are local functions Tljk(x) on M such that

Berwald metrics are special Finsler metrics. We are going to show that Riemannian metrics are special Berwald metrics. Example 5.1.1 Let F(y) = \fgij{x)yly^ manifold M. By (5.7), we have

be a Riemannian metric on a

where (gi:i(x)) := (gijix))"1. Clearly, Gl{y) are quadratic in y e TXM. Hence F is a Berwald metric. tf There are many non- Riemannian Berwald metrics [Sz]. Below is a simple example. Example 5.1.2 Consider a Randers metric F = a + 0 on a manifold M, where a(y) = ^Ja,ij{x)ylyi is a Riemannian metric and f3(y) — bi(x)yl is a 1-form with \\(3\\x =

sup px(y) = JaV^biWbjix) a«(l/) = l

V

< 1.

80

Geodesies and

Connection

According to Example 5.1.1, the geodesic coefficients of a can be expressed in the following form &{y) =

\rjk{x)yiyk,

where Tl-k{x) = TkAx) are local functions of x G M. Define b^j by bi{jdxj := dbi - bjTJikdxk.

(5.10)

Let r

ij : = 2 [bi\j + bJ\i) >

S

iJ := g \biti ~ bJ\ V •

The geodesic coefficients G l of F are given by G* = & + Py{ + Q\ where P(V)-Q\y)

=

^^{niVW

: =

-2a(y)brarvSplyl)

aairsriyl.

Assume that (3 is parallel with respect to a, i.e., biy = 0. Then T{j — U — Sij.

Thus P = 0 and Qi = 0. This implies that G\y) = G\y) are quadratic in y G TXM for all x G M. By definition, F = a + ft is a Berwald metric [Hale]. In this case, the geodesies of F coincide with that of a as point sets. By an elementary algebraic argument, one can show that if F is a Berwald metric, then b^ = 0. See [Ma5] [Hale] [Ki2] [SSAY] for details.

5.2

Chern Connection

In 1943, S.S. Chern studied the equivalence problem for Finsler spaces using the Cartan's exterior differentiation method [Chl][Ch2][BaCh]. He discovered a very simple connection. Later on, H. Rund independently

Chern

Connection

81

introduced this connection in a different setting. Thus, Chern's connection was also called the Rund connection in literatures. In this book, we will introduce Chern's connection using the exterior differential method. Let M be an n-manifold and 7r : TM —» M the natural projection. Denote by ir*TM the pull-back tangent bundle over the manifold TM\{0}. The vectors in ir*TM are denoted by (y,v), where y,v £ TXM. Take a local coordinate system (xl) in M. The local natural frame {^fr} for TM determines a local natural frame {di} for n*TM, d d

'i» : = = (»'^?i*)'

y£T*M-

This gives rise to a linear isomorphism between n*TM\y and TXM for every y € TXM. Given a Finsler metric F on M, F(y) — F(yt-^\x)

is a function of

(j/*) € R n at each point x € M. Let

S«(y):=^ 2 ]yV(!/)>

^(!/):=J[F\Vyi(!,).

(5.11)

We obtain two tensors g and C on n*TM defined respectively by g(u,v)

:= QijiyW'V^

C(U, V,W)

:= CiMirViW",

(5.12) (5.13)

where 17 = Uldi\y, V = V3dj\y and W7 = W/fc9fc|y. We call g the fundamental tensor and C the Cartan tensor of F . The Cartan torsion was actually appeared in P. Finsler's thesis [Fi]. But it was E. Cartan who first gave a geometric interpretation of this quantity [Ca]. Let Sy'-dy'

+ NiMdx*,

(5.14)

where N](y) := w(y). Then U*TM := span|dec*|,

V*TM := span j ^ j

(5.15)

Geodesies and

82

Connection

is a well-defined subbundle of T*(TM \ {0}). We obtain a decomposition forT*(TM\{0}), T*(TM \ {0}) = WTM © V*TM. Let {j~r, A } denote the local frame dual to {dxl,5yz},

that is

N

h^h- ^w

(516)

Then UTM := s p a n { ^ - } ,

VTM := s P an{ A }

are well-defined subbundles of T(TM \ {0}). We obtain a decomposition for T ( T M \ {0}), T(TM \ {0}) = WTM © VTM. Observe that ^2]=2cto(2/)yfc^eV*TM. Thus, d[F2] — 2FdF vanishes on horizontal tangent vectors. We obtain the following Lemma 5.2.1 F is horizontally constant, i.e., for any horizontal vector X = X i j | T eUTM X{F) = dF(X) = 0.

(5.17)

Let {e;}7=1 be an arbitrary local frame for ix*TM and {o/,wra+z}™=1 denote the corresponding local coframe for WTM © V*TM. The correspondence is determined by di o dxi «->
If ei = di, then a/ = dx and w

n+!

(5-18)

= J y \ Set

ylei = {y,y)

(5.19)

and 9ij •= 9(ei,ej),

Cijk:=C(ei,ej:ek).

(5.20)

Chern

Connection

83

We obtain a set of local function yl, gtj and djk on TM \ {0}. Theorem 5.2.2 (Chern) There is a unique set of local 1-forms {w/} on TM \ {0} such that djJ

= CJJ Aco/ k

(5.21) k

dgij

= gkjuji +gkjuji

n+i

= dtf + yiu/.

u

n+k

+ 2Cijkoj

,

(5.22)

(5.23)

Proof. Without loss of generality, we may take a natural local frame {e; = di}™=1 for TT*TM so that the corresponding local coframe for T*(TM\{0}) is {uil = dxl,u)n+l = 5yl}™=l. In this case, 9ij(y) = ^[F2}viyj{y),

Cijk(y) =

-[F2]yiyjyk(y).

Let Lijk

:= ^ g f V - 2CijklGl

- CmN\ - CukN] - CtjlNlk,

(5.24)

where

cijki{y) = -g^(y) = ^ W w O / ) We claim that the following set of 1-forms are the unique set of solutions to (5.21) and (5.22), cu/ := T)k dxk, where

^••=S^-9U^-

(5-25)

Observe that duj' - UJJ A to/ = -dxj A T)kdxk = - ^ { r j f c - rlkj\dxj

A dxk = 0.

Thus (5.21) holds. Rewrite (5.7) as follows

^-\{^-a-BW-

<«•»>

Geodesies and

84

Connection

Differentiating (5.26) with respect to yi yields 9klN

* -2\d^

+

^r--Mfy ~2CjklG

•

(5 27)

-

m

Differentiating (5.27) with respect to y , we obtain 1

rfc

[ d9ji dgmi dgjm -i dCjim k m dxi dxl J dxk V 2\dx —2CjkimG — 2CjkiNm — 2CkimNj — Ljim.

=

9kiLjm

(5.28)

From (5.28), we obtain dgji dxr

gkiT>;m + gkjT*rn + 2C3kiN*m.

(5.29)

This implies to (5.22). The proof of uniqueness is omitted. By the homogeneity of F, we have Cijkv1 = Cijkyj

= Cijkyk = 0

(5.30)

LijkV1 = LijkyJ

- Lijkyk

(5.31)

and - 0.

Thus yi k y

J - dyJdyk

y9L kl

i ~~dy3-i-

Thus 5yl = dyl + y]T)kdxk

= dyl +

This gives (5.23).

y'u/. Q.E.D.

From the above argument, we obtain a new tensor £ on w*TM defined by C(U, V, W) := LwMU'ViW",

(5.32)

where U = 11%^, V = V'dj\y and W = Wkdk\y- We call £ the Landsberg tensor. The Cartan tensor and Landsberg tensor play an important role in Finsler geometry.

Chern

Connection

85

Using the set of local 1-forms {w/} in Theorem 5.2.2, we define a map V : T{TM) x C°°(iv*TM) -> ix*TM by

V^U := [dU\X) + U'ujiX)}

® ei:

or simply

V^ — jd^ + C/^/J^ei. V is a linear connection on n*TM. We call it the Chern connection. For a vector field X = (X1, • • • ,Xn) on an open subset U C R71, the directional derivative D„X in a direction v £ TXR™ = R™ is defined by DVX := (dX'iv),

• • •, d X » ) = < / | | .

We can extend the notion of directional derivative to vector fields on a Finsler space. Let (M, F) be a Finsler space. At each point x £ M, define a map D : TXM x C^iTM)

-> T^M

by DyC/ := {dCT(y) + ^ ' ( z ) ^ ) } ^ ! *

(5-33)

where y 6 T X M and U G C°°{TM). Here A/)(y) are local functions on T M defined in (5.15). We call DyU(x) the covariant derivative of t/ at a; in the direction y. Set Dyll(x) = 0 for y = 0. D has the following properties: (a) D„(17 + V) =D y ?7 + DyVr; (b) Dv(fU) = dfx(y)U + f(x)DyU; (c) DXvU = AD„17, A > 0. The family D := {Dy}y£TM is called the connection of F. connection in a usual sense. If, in addition, D is linear, i.e., (d) Dy+vU = DyU + DVU,

D is not a

86

Geodesies and Connection

then D is called an affine connection on TM (or M). The following proposition is obvious. Proposition 5.2.3 The connection D of a Finsler metric F is affine if and only if F is a Berwald metric. Proof. In a standard local coordinate system (xl, yl) in TM, N;(y)

8G\ = —j(y), dyi

N^yW

=2G\y).

Thus Gl(y) are quadratic in y G TXM if and only if NUy) are linear in y G TXM. This proves the proposition. Q.E.D. For Berwald metrics, the connection D is an affine connection on the tangent bundle TM. We call it the Levi-Civita connection. E x a m p l e 5.2.1 Consider a Riemannian metric F(y) = \Jgij{x)ylyi a manifold M. The geodesic coefficients Gl(y) are given by

G\y) = 9Hx){^(*)-l^)}y>yk-

on

(5.34)

Thus Gl{y) are quadratic in y G TXM for all x G M and D is an affine connection. The Levi-Civita connection D satisfies the torsion-free condition BuV-DvU

= [U,V]

(5.35)

and d[g(U,V)](y)=g(pvU,v)+g(u,Dyv),

(5.36)

where y G TXM and U, V G C°°(TM). It can be easily proved that the Levi-Civita connection is the unique linear connection satisfying (5.35) and (5.36). ' jj Now we discuss t h e relationship between t h e Levi-Civita connection D on TM and the Chern connection V on ir*TM for Berwald metrics. For a Berwald metric, in a local coordinate system (xl) in M , the Christoffel symbols F)k(x) •= a ^ f \ (y) are functions of x only, so that NiAy)=T)k{x)yk.

Chern

Connection

87

The Levi-Civita connection is given by DU={dUi

WT)kdxk}»-^i,

+

where U = Ui-^l e C°°(TM). Take an arbitrary local frame {b;}™^ for TM and let {8l}f=l denote the local dual frame for T*M. Write DU = {dUi +

6ji}®bi,

where U = £/*b;. {#/} are called the Levi-Civita connection forms with respect to {bi}™=1. Let 7T : TM —»• M denote the natural projection. We can lift {bj}™=1 to a local frame {ej}™=1 for n*TM, where ei\v •= {y,hi\x),

y€TxM.

The Chern connection forms {u> •*} with respect to {ej}™=1 are given by

Let yl denote the local functions on TM determined by (y,y) = yzei = The local coframe {v*,un+i}?=1 o/=7T*6>\

(y,ylbi).

for T*(TM \ {0}) are given by ojn+i = dyi + yjTr*eji.

Consider a tensor S = S) b; ® 9j on M. Define Q = Qlei by Q\y):=yjSi(x), Then Q is a well-defined tensor on

y= TT*TM.

y%.

On M we define S*,fc by

dS} + Sk6ki-Sl6k=:S^k6k.

(5.37)

While on TM \ {0}, we define Q\k and Q*fc by dQ i + Qkujki =: Qffcwfc + Q > n + f c .

(5.38)

88

Geodesies and

Connection

Observe that

dQ' + Q V

=

y^^dS^

+ Sldy' +

y^^S^)

=

y V [dS) + Sty]

=

yjn* [dS] + Sk6k* - SiOj"] +

=

yin*(S)lkek)

+ Si [un+k - yiir'O/ Slkun+k

Siujn+k.

+

We obtain

5.3

Q\k

=

VJS)\k

(5.39)

Q\

=

Sl

(5.40)

Covariant Derivatives

Let (M,F) be a Finsler space and c(t), a < t < b, a geodesic. For a vector field V = Vl{t)-^\c{t) along c, define

D^(t):={^(*) + V^(c(*))}^| DiV(t) is called the covariant derivative of V{t) along c. A vector V(t) is said to be parallel along c, if D6V(t) = 0. Let U = ^ W ^ T U W and V = ^(*)gfrU(i) be parallel vector fields along c. They satisfy the following equations dJJm —

dVm = -WN™(c):

—

= -VlNr(c).

(5.41)

Consider the function gc(U,V)=9jlUjVl. By (5.41), we obtain Jt [mUjVl]

= { ^ c

m

- gjmNr

-

9jmNr

-

ACjlmGm}wVl.

Covariant

Derivatives

89

Contracting (5.29) with ym yields Vm^

= QmiNJ1 + gimN?

+ 4CjlmGm.

(5.42)

It follows from (5.42) that

|Mt/,v)

Ik""1

Thus Sc{U, V) = constant.

Let c(t), a < t < b, be a geodesic. Define a map Pc : TC^M

—> Tc(p)M

by Pc(v) := V(b),

v G T c ( o ) M,

where V(t) is the parallel vector field along c with F(a) = v. Pc is called the parallel translation along c. The above argument proves the following Lemma 5.3.1 For any geodesic c(t), a < t < b, in a Finsler space (M,F), the parallel translation Pc preserves the inner products gc along c, gc(6) (-Pc(w), Pc{v)j = g c(o ) (u, vj,

u, v £ T c(a) M.

In general, the parallel translation does not preserve the Minkowski norms. Y. Ichijo [ic] proved that on a Berwald space, the parallel translation along any geodesic preserves the Minkowski norms. Thus Berwald spaces can be viewed as Finsler spaces modeled on a single Minkowski space. Lemma 5.3.2 ([ic]) Let (M,F) be a Berwald space and c(t), a < t < b, a geodesic. Then the parallel translation Pc preserves the Minkowski norm,

Fc(b) (P c (v)) = Fc(a)(v),

v G Tc{a)M.

90

Geodesies and

Proof. Let V = Vt(t)-^\c(t)

Connection

be parallel along c. V satisfies dVl dt

VjN}(c).

(5.43)

Consider the function F(t) := F(V(t)). By assumption, N]{yy

= T)k{x)r?yk

= NJ(v)yi

By Lemma 5.2.1 and (5.43), we obtain b

[t)

~

dt

[C dx*

+

=

dF[^rjk(c)Vk

Thus F(t) = constant

dt dyt

- V^)k{c)ck)^-\

= 0. Q.E.D.

On a Berwald space, the canonical connection D is an affine connection. According to Szabo [Sz], if a Finsler metric F is Berwaldian, then there is a Riemannian metric g whose Levi-Civita connection coincides with the canonical connection. There are many non-Riemannian Berwald metrics. See Example 5.1.2 above. 5.4

Geodesic Flow

Let (M, F) be a Finsler space and

the spray of F. For a vector y £ TM, denote by 4>t{y) the integral curve of G with (fio(y) = V- From the definition, we see that the curve c(t) :— 4>t{y) is a geodesic with c(0) = y. We call <j>t the geodesic flow of F. It follows from Lemma 5.2.1 that d F(My))]=dF(GMy))=o. dt

Geodesic Flow

91

Thus F(
(5.44)

This means that the geodesic flow <j>t preserves the Finsler metric. Recall the Hilbert form on TM \ {0} that is given by (5.45) where gij(y) •= \[F2]yiyi{y). Lemma 5.4.1

The Hilbert form has the following property.

For any t,

| [ ( & ) ' W ] = ± d [(&)*[**]]•

(5-46)

Hence {(j>t)*duj — cLo.

Proof. Let (ipl(t),ipl(t)) be the standard local coordinates of y(t) := 4>t{y) in TM. y(t) is an integral curve of G, hence

d

-§{t) =

(5.47)

-2G\y{t)).

By (5.47), 1, dpi ^ (
^

+

Note that ldxiaX

+

V dvJ dy

J t=o

'

dtldxi**

4

dyi

V

\ t=o

V

We immediately obtain d di L

J u=o

±{[F2]x>yiyk

-2[F2]yiy>Gk(y)}dxi

+^[F2}rdyi

(5.48)

It follows from (5.2) that 2[F\iykGk

= [F2]xkyiyk

- \F2}

(5.49)

Geodesies and

92

Connection

Substituting (5.49) into (5.48) yields dt L" '

J t=o

Since 4>t+s -= (t> 4>t° s, oneobtains obtains(5.46). (5.46).Using Using(5.46), (5.46),weweobtair obtain t° s, one

![<*H=4s(<«*")] = M<«''f2iHd r, ,

%i

, l

,r d t , , ,„

1

M

Thus {4>t)*du) = ((j>0)*du — LJ.

Q.E.D. T h e Finsler metric F induces a Riemannian metric of Sasaki t y p e on TM \ {0}, 9 •= 9ij{y)dxl

® dx3 + gij{y)Syl

Sy3,

where Sy1 = dyl + Nj(y)dyJ. T h e Riemannian metric g determines a nondegenerate 2n-form dVg on TM \ {0} by dVs = det (gij)dx1

• • • dxndy1

• • • dyn.

(5.50)

Observe t h a t du =

', ykdx

A dx1 — gijdx1 A dy3 = —g^dx1 A 6y3.

Thus, we can express dVg as follows n(n + l) 1

dVg = ( - 1 )

2

— dw A • • • A da>.

dVg is just the volume form we have mentioned in (2.16). By Lemma 5.4.1, we obtain the following P r o p o s i t i o n 5.4.2 ([Dal]) The geodesic flow cf>t preserves the volume on TM \ {0}.

Riemannian

Let i : SM -» TM \ {0} be the natural embedding. By (5.44), <j>t : SM -» S M . Let CJ := i*u). It follows from (5.44) and (5.46) t h a t

" W*\

dt

o.

Geodesic Flow

93

In other words, (fit preserves u>. Let g :— i*g denote the induced Riemannian metric on SM. Then the induced Riemannian volume form dVg on SM is given by n(n + l)

dVh = (-1)

2

,

1

-.

rrtj Add) A • • • Add}.

(n-1)! Proposition 5.4.3

([Dal]) The geodesic flow (fit preserves dVg on SM.

It is worth making further investigation on the geodesic flow on a Finsler space. The geodesic flow (fit on a reversible Finsler space (M, F) is of Anosov type if there exists a decomposition of T(SM) into

T(SM)=ES®EU®E°, where E° = R • G and Es ^ 0, Eu ^ 0 are invariant under the flow in the sense that

Further, there are constants A, B > 0 such that ||(&).(A-)|| < A e - s t | | X | | , ||(0_ t ).(X)|| < Ae- B t ||X||,

VX e S s , Vt > 0, VX G B", Vt > 0.

P. Foulon [Fol] proved that the geodesic flow on a closed reversible Finsler space of negative curvature must be of Anosov type. Recently, Foulon [Fo2] has done some work on contact Anosov flows on closed manifolds. In particular, he shows that for a smooth contact Anosov flow on a closed three manifold, the measure of maximal entropy is in the Lebesgue class if and only if the flow is up to finite covers conjugate to the geodesic flow of a metric of constant negative curvature on a closed surface. This shows that the ratio between the measure theoretic entropy and the topological entropy of a contact Anosov flow is strictly smaller than one on any closed three manifold which is not a Seifert bundle.

Chapter 6

Riemann Curvature

Riemann curvature is the central concept in Riemannian geometry. This quantity was first introduced by B. Riemann in 1854 for Riemannian metrics. Since then, Riemannian geometry has become one of the important branches in modern mathematics. In 1926, L. Berwald extended the Riemann curvature to Finsler metrics [Bwl] [Bw2]. In this chapter, we will introduce and discuss this very important quantity in Finsler geometry from various points of view.

6.1

Birth of the Riemann Curvature

The Riemann curvature of a Finsler space is a family of linear transformations on tangent spaces. It can be defined via the variations of geodesies. This approach is different from Riemann's and Berwald's approaches. Let (M,F) be a Finsler space. Consider a geodesic c(t), a < t < b. A C°° map H : (—s,e) x [a, b] —• M is called a geodesic variation of c if H{0,t)

=c{t)

and for each u € (—£,£), the curve cu(t)

:=H(u,t)

is a geodesic. We will show that for any geodesic variation, its variation field J(£) := ^ ( 0 , i ) satisfies a special system of second order ordinary differential equations. 95

96

Riemann

Curvature

Lemma 6.1.1 Let (M, F) be a Finsler space. There is a family of transformations ILy : TXM -» TXM, y £ TXM \ {0}, such that for any geodesic variation H of a geodesic c, the variation vector field J(t) := ^ - ( 0 , £) along c satisfies the following equation D a DeJ + R c ( J ) = 0 .

(6.1)

Proof. By assumption, each cu(t) = H(u,t) is a geodesic. Thus dt2

(6.2)

\-dTJ

For simplicity, let rp

rp%

d dx* '

dH dt '

"-"£"£•

Equation (6.2) becomes dT + 2Gl(T) = 0. dt

(6.3)

Note that dT du

d /dW du \ dt )

=U

dU{ dt\

du )

dt

Differentiating (6.3) with respect to u yields dt2

dxk

dt

dyi

Observe that d_ G\T) du . d dGl (T) dt L dyi

dxk(

!+

2{Zni

rph

dt

V QTk

dG (T) + dt dxkdy:>

rpk " Gl


d2Qi

dyidyk

•(T)

0 ^, f c

dxkdy^{T)~2Gk{T)dJd^{T)- dyidyk By the above identities, one obtains

Birth of the Riemann i ( dG dG* k I dx

-ukR\(T)~,

Curvature

i „,„• d22G G^ dx^dyk

97

„„„• d2Gi idyidyk

dGldG^ d • \ A dyi dyk 1dx*

d

where k[V}

:

"

dxk

V

d&dyk

+

G

dyidyk

dyi dyk'

{

'

For every vector y £ TXM \ {0}, define a linear transformation R^ = R\{y)-^-r®dxk\x

: TXM ->• T X M.

We obtain D T D T J7 + RT(i7) = 0 . Restricting the above equation to c, we obtain an equation for J(t) := U(0,t), DcDcJ + Rc(J) = 0 .

(6.5)

This completes the proof.

Q.E.D.

The geodesic variations give rise to a family of transformations R = {R„ : TXM -> TXM | yeTxM\{0},

x € M}.

We call R the Riemann curvature. It follows from (6.4) that Hy(y) = 0.

(6.6)

Moreover, the R^ is self-adjoint with respect to g v , (u),vj

= gy[u,B.y(v)j,

u,veTxM.

(6.7)

This fact will be proved in Section 8.1 below. By granting (6.7), we obtain gtf(R1/(u),y)=g„(«,Rv(y))=0.

(6.8)

98

Riemann

Curvature

Let P C TXM be a tangent plane. For a vector y G P\ {0}, define gv(Ry(u),u) Zy{y,y)%y{u,u)-zv(y,u)%y(y,uy where u G P such that P = sp&n{y,u}. By (6.6) and (6.8), we can easily show that K(P,y) is independent of u G P with P = span{?/,u}. The number K(P, y) is called the flag curvature of the flag (P, y) in TXM. When n = dim M = 2, K(y) := K(TXM, j/),

y£TxM\

{0},

is a scalar function T M \ {0}. We call K(y) the Gauss curvature. For a vector y £ TXM \ {0}, there are infinitely many tangent planes P C T X M containing y. The flag curvature K(p, y) depends on the tangent planes P containing y. A Finsler metric F on a manifold M is said to be of scalar curvature K(j/) if K(P, j/) = K(y) is independent of the tangent plane P containing y for all y G TXM. This is equivalent to the following Ry(u) = K{y) {g y (y, y) u - gy{y, u)y},

y,u e T . M \ {0}.

By definition, every two-dimensional Finsler metric is of scalar curvature. Define n

Ric(y):=^R\(y).

(6.10)

i=i

Ric is a scalar function on TM \ {0} with the following homogeneity: Ric(Ay) = A 2 Ric(y), We call Ric the Ricci curvature.

A > 0.

For the sake of convenience, let

We call R(y) the Ricci scalar. By the homogeneity of JR, we have

Geodesic Fields

6.2

99

Geodesic Fields

In this section, we will give an alternative formula for the Riemann curvature. This formula is not good for computing the Riemann curvature, but it is very useful in comparison geometry. Lemma 6.2.1 Let Y be a geodesic field on an open subset U in a Finsler space (M,F). Let g := gy denote the Riemannian metric on U. Then F := \fg is Y-related to F in the following sense N](Y) = NJ(Y).

(6.11)

In particular, Y is a geodesic field of F.

Proof. Equation (5.27) gives

Note that

It follows from (6.12) and (6.13) that

-2gil{Y)Cjkl{Y)Gk(Y) =

N)(Y).

This gives (6.11). By (6.11), we obtain 2Gl{Y) = Nlj(Y)Yj

= N}(Y)Yj

= 2Gi{Y).

Thus Y also satisfies • dYi Y3-z-^ + 2G,(Y) = 0. Thus Y is also a geodesic field of F.

Q.E.D.

Let Y be a geodesic field on an open subset U C M and g := gy denote the induced Riemannian metric on U. It follows from Lemma 6.2.1 that the

Riemann

100

Curvature

connection D of F and the Levi-Civita connection of F := \fg are related by DYU = t>YU.

(6.14)

Further, we have the following Proposition 6.2.2 Let Y be a geodesic field on an open subset U in a Finsler space (M,F) and g := gy. Let R and R denote the Riemann curvature of F and F = ^fg respectively. Then for y = Yx s TXM R^ = Ay.

(6.15)

' Observe that k [

dx

dYj

= U^Y

'

dYj

-"We* dxk{

(6-16)

'

and dNt

•(

=

d

[ -

i

-•

-i

Yi-£-[Nl(Y)]+2G\Y)rkl{Y)

= Y^(Y)+ril(Y)Yi— =

dYl

yi

+ 2Gl(Y)rkl(Y)

^ j ( y ) - 2G'(r)n,(^) + 2G'(y)r^(y).

This implies y i

"^

( y )

"

2G

'(y)r^(y) =

YJ

-d^W

~ tG\Y)rkl{Y).

(6.17)

Plugging (6.16) and (6.17) into (6.4) yields R\(Y)

= ^fc(F). Q.E.D.

Projectively Related Finsler

Metrics

101

For a Riemannian metric F(y) = \/g(y, y) on a manifold M, the Riemann curvature Rj, : TXM —> TXM is quadratic in y G TXM. Moreover, R y is self-adjoint with respect to g, g(Ky(u), v) = g(u, R » ) ,

u,veTxM.

(6.18)

The proof of (6.18) is given in standard text books on Riemannian geometry. We will prove it later in Section 8.1. Granting that (6.18) holds for Riemannian metrics, we consider a Finsler space (M,F). For a vector y G TXM \ {0}, extend y to a geodesic field Y in a neighborhood U of x. Let R ^ : TZM -> TZM, w e TZM \ {0}, denote the Riemann curvature of g := gy. By (6.18), g(TLY{u),

u) =g(u, Ry(w)),

u,vGTzM.

Restricting the above equation to x yields gy^Ryiu), vj =gy(u, R j » ) ,

u,v€TxM

(6.19)

Thus if (6.19) holds for Riemannian metrics, then it holds for Finsler metrics. 6.3

Projectively Related Finsler Metrics

Two Finsler metrics on a manifold are said to be pointwise projectively related if they have the same geodesies as point sets. The Riemann curvature is determined by geodesies. A natural question arises: What happens to the Riemann curvature if two Finsler metrics on a manifold have the same geodesies as point sets? In this section, we will discuss this problem. See [Dg][BuKe][Ma4] for related discussion. Given a Finsler space (M, F) and a Finsler metric F on M. We view F as a scalar function on TM. Let {ujl,u>n+l} be a local coframe for T*(TM\{0}) corresponding to a local frame {ei}™=1 for IT*TM. See (5.18) for the correspondence. Define dF = F^UJ1 + F.iWn+\

d.F[t - F b V = FW<J + FWjwn-

Riemann

102

Curvature

In a standard local coordinate system (xl,yl) 6y\ F\k and F\k.i are given by F\k — Fxk — NkFyk,

in TM, wz = dx% and u)n+l =

F\k.t = {F\k)yi.

l

Let G (y) and G(y) denote the geodesic coefficients of F and F in a common standard coordinate system (x1, yl) in TM. They are denned by (5.2). From the geodesic equations, one can easily see that F and F have the same geodesies as point sets if and only if Gi{y)=Gi{y)

+ P{y)y\

where P{y) is a scalar function satisfying P(Xy) = XP(y),

A > 0.

The following important result is due to A. Rapcsak [Ra]. Lemma 6.3.1 (Rapcsak) For two Finsler metrics F and F on a manifold M, the following conditions are equivalent (a) F and F have the same geodesies as point sets; (b) There is a scalar function P{y) on TM such that Gi(y)=Gi(y)+P{y)yi. In this case, P is given by

P(y) = %£. Kyi

2F

(c) F satisfies Flk.iyk - F„ = 0. Proof: By a direct computation, one can verify that & = Gi + Pyi + Qi, where 2F ' Q* =

\Fga{F\k.iVk-F\i}-

(6.20)

Projectively Related Finsler

Metrics

103

Thus F and F are pointwise projectively related if and only if Ql = 0, i.e., (6.20) holds. Q.E.D. Example 6.3.1 ([Hale]) Let (M,F) be a Finsler space. Consider a 1form P(y) = bi(x)y1 on M. Assume that ||/3|| := sup (3{y) < 1. F(y)=l

Then F := F + (3 is again a Finsler metric on M. Observe that

_

fc

%

k

, SA^' fc

Note that «V

y

dykdyl

J

dyl

l

'

Thus a

k

n

{dh

dbk\

k

By Lemma 5.2.1, F|fc = F x * - JV'F,,, = 0. We obtain

Thus F = F + /? is pointwise projective to F if and only if (3 is close.

jj

Given two Finsler metrics F and F on an n-dimensional manifold M. Assume that F and F are pointwise projectively related, hence by Lemma 6.3.1 &{y) = G\y) +

P{y)y\

where P(y) = ^f.

(6.21)

104

Riemann

Curvature

Let Rlk{y) and Rlk(y) denote the coefficients of the Riemann curvature of F and F respectively. By (6.4), we immediately obtain R\(y)

= R\(y)

+ z(v)S{ + rk{y)y\

(6.22)

where E = P2 - P\d

rk = 3(P,fe - PP.k) + S.k.

Equation (6.22) is proved in [MaWe]. In index-free form we can rewrite (6.22) as follows Ry(u) = Hy(u)+E{y)I

+ Ty(u)y,

u&TxM,

where / denotes the identity map and TV{U) := Tk(y)uh,

u =•

uk-^\x.

Consider a Finsler metric F on an open domain Q. C R n . According to Lemma 6.3.1, the following conditions are equivalent (a) The geodesies of F are straight lines in O; (b) The geodesic coefficients G1 of F is in the following form Gi(y)=P(y)yi. In this case, P is given by 2F • (c) F satisfies Fxkyiy

=

Fxi.

Assume that F is pointwise projectively flat. Then the Riemann curvature is in the following form Hy(u)=E(y)u

+ Ty(u)y,

ueTxM.

By (6.8), we obtain 0 = gy(RJ/(u),y) =

E(y)gy(y,u)+Tv(u)F\y).

This gives T

v(u) =

-j^sv{y,u).

Projectively

Related Finsler Metrics

105

Let

By the above formulas, we obtain R„(u) = K(j/){g„(y, j/) u - g„(j/,u) y } , Thus F is of scalar curvature.

y,u € TXM \ {0}.

Chapter 7

Non-Riemannian Curvatures

When one looks at a Finsler space, he not only sees the shape of the space, but also the "color" of the space. In this chapter, we will introduce and discuss the Cartan torsion and the Chern curvature, etc. Roughly speaking, the Cartan torsion describes the "color" of the space at a point and the Chern curvature tensor and its portion (the Landsberg curvature) describe the rate of changes of the color over the space. These quantities all vanish for Riemannian spaces. Thus Finsler spaces are much more "colorful" than Riemannian spaces.

7.1

Cartan Torsion

The Cartan torsion is a non-Euclidean quantity of Minkowski spaces. Since every tangent space on a Finsler space is a Minkowski space, the Cartan torsion is defined for a Finsler space. Let (V,F) be an n-dimensional Minkowski space. Recall that the induced inner product gy in V is independent of y G V \ {0} if and only if F is Euclidean. For a vector y € V \ {0}, define Cy(u,v,w)

: = -^|gy+ttu(u,*;)J| t = 0 ,

u,v,w € V.

We have

C (u t w) =

" ' ''

\~dLdt [p2{y+ru+sv+H

L=t=o-

Thus for each y € V \ {0}, Cy is a symmetric multi-linear form on V. The 107

Non-Riemannian

108

Curvatures

homogeneity of F implies Cy(y,v,w)

=0,

v,weV.

The family C = {CJ/}j/6v\{o} is called the Cartan torsion. Let {bj™ =1 be a basis for V. F(y) = F^h,)

is a function of (yl) £ R™.

Let Cijk(y) = - [ F 2 ] y V y * ( y ) .

9ij(v) = ^\F\^{y), We have gy(u,v) = gij(y)ulvJ,

Cy(u,v. w) := Ci:jk{y)utv:,wk,

(7.1)

where u = Mlbj, v = v-'b,- and w = uikh'KThe Cartan torsion is related to the Cartan tensor in (5.13) by Cy(u,v,w)

=C(U.,V,W),

u,v,wGTxM,

where U = (y, u), V = (y, v), W = (y, w) e ir*TM. Define n

M«) : = £

n

^ y (w) c *( b *» b J. u ) = Y, 9lj(y)Cljk(y)uk,

ij=l

u = u'bi,

r/'=l

(7.2) where (glJ(y)) •= (gij(y)) 1. The family I = {ly}yev\{o} is called the mean Cartan torsion. Note that in dimension two, the mean Cartan torsion completely determines the Cartan torsion. Iy can be also expressed as • d

I

M u ) = u%-Q-i l n Vdetfefc(2/)),

" = w'bj.

(7.3)

Thus I = 0 if and only if det(gij(y)) = constant. Using the maximal principles of the Laplacian on S = F _ 1 ( l ) , Deicke [De] proved the following important result Theorem 7.1.1 if 1 = 0.

(Deike) A Minkowski norm F is Euclidean if and only

Cartan

Torsion

109

See also [Bk] and Chapter 14 in [BCSl] for a proof. We should point out that Theorem 7.1.1 does not hold for singular Minkowski spaces [jiSh]. There are infinitely many singular Minkowski norms with vanishing mean Cartan torsion. Define

sup

^

sup

|C„(tj,i;,i;)| rr^.

(7.4)

L

(7.5)

Proposition 7.1.2 Let F = a + j3 be a Randers norm on V, where a is an Euclidean norm and p is a linear functional with \\0\\:=

sup

|/3(J/)|<1.

a(y) = l

The Cartan torsion is uniformly bounded. More precisely,

< ^^/l-yr^^p

IIIII

yen < ^ y T v r = W -

(7.6)

(7-7)

Proof. Fix a basis {bi}™=1 for V. Let P{y)=biV\

ly(u) = Ii{y)u\

y = y'bi, u = ifbi.

By (1.7) and (7.3), we obtain Ti

(y)

=

5TT ln

\fdet(9jk(y))

n + 1 (Fyi 2 I F 2F By (7.4), the norm of Iy is given by

ayi i a )

Non-Riemannian

110

Curvatures

It follows from (1.8), (7.8) and (7.9) that

M-^fMifl'-®!-

<™»

Since (3{y) < \\0\\a(y), we can write (3(y) = ||/3|| cos# for some 0 < 0 < 2n. Assume that F(y) = 1. Then a(i/) = l - / % ) = l-||/?||a(j/)cos0. This gives w

l + 6cos0

Plugging it into (7.10) yields l|I 11

n + 1 / ||/?Psin 2 fl

<

n + l

r

7 = = f

» - - 2 - V l + ll^llcosfl - ^ T V ! " VI " 11/311 •

This gives (7.6). To prove (7.7), we first reduce the problem to the two-dimensional case. Let y0,v0 with F{y0) = 1 and gya(v0,v0) = 1 such that l|C||

=Cyo(v0,v0,v0).

Let V = span< y0, v0 \ and F := F\v. We have 1 d3 r 1 C yo (v 0 ,w 0 ,v 0 ) = -—-^F 2 (y 0 -|-st; 0 )j| s = o = Cj,0(t;0, u 0 ,v 0 ). Let a := a|y and (3 = (i\V. We have 11,311= sup P(y)< a(y) = l

sup /3(2/) = p | | . a(y) = l

Let I(y 0 ) denote the main scalar of F at y0. It is easy to see that ||C|| = max |I|. Taking a special orthonormal basis {bi,b2} for (V,a) such that J3(y) = ||/3||u for y = ubi + vb^. With respect to this basis, F(y) = \lv? +v2 + \\j3\\ u,

y = ubx

+vb2.

Cartan

Torsion

111

It follows from (7.6) that

m^V1-^

(7.11)

Thus |C|| < ||C|| = max |I| < -y=\H

v/i~rPip<4v/w^ N/2

Q.E.D. In dimension two, the bound (7.7) is just the bound (7.6). The bound for the Cartan torsion in dimension two is suggested by B. Lackey. See Exercise 11.2.6 in [BCSl]. The norm of the (mean) Cartan torsion can be as large as one wishes. For example, the norm of the Cartan torsion of the following Minkowski norms F\ on R n approaches oo as A —» oo.

Fx(y) := \ E x a m p l e 7.1.1

£(^)2 + A E(^ 1 ) 4 i=l i=l

Let F(u,v) := lu4 + 3cu2v2 + v4\ ' ,

where 0 < c < 2. F is a Minkowski norm on R 2 (Example 1.2.3). For any y — (u,v) with u > 0, Iv

2v/3(9c 2 -4)(i> 4 -w 4 ) 3 (2c u 4 + (4 - 3c2) u2v2 + 2c vA

Take y = (w, v) with v = J-^u

3/2 '

> 0 , we obtain

,T , (9c2 - 4) 2 Iv = 1 yl 64c

>• oo.

. a s c -> 0"1".

Non-Riemannian

112

7.2

Curvatures

Chern Curvature

Besides the Cartan curvature, there are other quantities which always vanish on Riemannian spaces. In this section, we are going to discuss two non-Riemannian quantities — the Chern curvature and the Landsberg curvature. Let (M, F) be a Finsler space. In a standard local coordinate system (xl,yz) in TM, the local functions gij(y) = \[F2}yiyj(y) and Cijk(y) := \[F2]yiyjyk(y) on TM define the inner product g y and the Cartan torsion Cy on TXM respectively. See (7.1). Let L^k be the set of local functions defined in (5.24). Lijk{v)

= V1-^-

- 2CijklGl

where Cijki := j[F2]yiyjykyi(y). connection are given by jk

where Lzkl quantity:

- CljkN! - CukN] - CijhNl

(7.12)

The Christoffel symbols of the Chern

dyWyk

'

jk

'

:= g%:>Ljki- Differentiating Tl-k with respect to yl yields a new

P

.

(v)

.

jki\yi-

dT

U

&
dUjk

dyidykdyl

Qyl

dyl '

y

'

For a vector y e TXM \ {0}, define Ly(u,v,w)

: =

Lijk(y)uivJwkJ

Py(u,v,w):

=

Pjikl{y)u^vkwl

(7.14) —\x,

(7.15)

where u = u%-^\x,v = v1-£r\x,w = wk-^\x € TXM. L := {Ly}yeTM\{0} is called the Landsberg curvature [Lal][La2] and P := {Py}yeTM\{o} 1S called the Chern curvature [Chl][Ch2]. The Landsberg curvature is related to the Landsberg tensor in (5.32) by Ly(u,v,w)

= C(U,V,W),

u,v,weTxM,

where U = (y, u), V = (y, v), W = (y, w) e w*TM.

Chern Curvature

113

Let c(t) b e a n arbitrary geodesic in (M,F). Take arbitrary parallel vector fields U(t), V(t),W(t) along c. According to (7.12), Lc(t) (U(t),

V(t), W(tj)

= ~ [Ci(t) (U(t), V(t), W(t))].

(7.16)

T h u s the Landsberg curvature measures the rate of changes of the C a r t a n torsion along geodesies. A Finsler metric is called a Landsberg metric if L = 0. By (7.16), we see t h a t on a Landsberg space, the C a r t a n torsion is constant along geodesies. P r o p o s i t i o n 7.2.1 vature by

The Landsberg curvature

Ly{u,v,w)

is related to the Chern

=

-gy(u,

Py(y,v,w)j

=

g y ( p i / ( w , v , w ) , yj.

cur-

(7.17) (7.18)

Proof. By t h e homogeneity of G \ we obtain „.spi

— ,,«

y ^S u - y

sk _

i

d

-

u

dyi

I

Sri

\y

L

\

ji

sk) -

h

—_ji

ki-

^ u-

Thus 9ijysPs\i

= -9ijL\i

=

-LiM.

This gives (7.17). Differentiating (5.28) with y1, t h e n contacting the resulting identity with l y , we obtain y 9kl

~dy1~

=

y

IhF

+2CiikNl -2Lijm

+

AGi kmG

^

+ 2CikmN?

+ 2CikmNk

dLf1 y —^ dyl

l oLjim y'

^•^ijm

o - i( \lf L'jlmj

Ijijml

Thus ylgklPjkml

-

= ~

l y

g

k l i^

dy

= L ijm-

-^i,

Non-Riemannian

114

Curvatures

This gives (7.18).

Q.E.D.

Berwald metrics can be characterized by the Chern curvature. Proposition 7.2.2 j / P = 0.

A Finsler metric F is a Berwald metric if and only

Proof. Assume that F is Berwaldian. By Definition 5.1.3, Gz(y) are quadratic in y e TXM. Then dNk3 QymQyi

d3G' - 0. dyWykdyl

Differentiating (5.27) with respect to ym and yl, then contracting the resulting equation with yl, we obtain 0 = - 2 i / ' -l ^ + 4CijkmGk dx

+ 2CjkmNk

+ 2CijkN^

+

2CikmNk.

By (7.12), we see that Lijm = 0. Thus jk

dyidyk

and p *

Ih. = —a l 1 dy dyWkdy Thus Gz(y) are quadratic in y e TXM and F is a Berwald metric. Assume that P = 0. Thus L = 0 by (7.17). Equation (7.13) implies 3 kl

dT

)k

3 kl

dyl

Thus T%jk(y) are independent of y £ TXM and G\y) = \N]{y)yi

=

\r)k(x)y3yk

are quadratic in y G TXM. We conclude that F is a Berwald metric Q.E.D. From (7.17) and Proposition 7.2.2, we immediately conclude that every Berwald space is a Landsberg space. It is an open problem in Finsler geometry whether or not there is a Landsberg metric which is not a Berwald metric. So far no example has been found.

Chern

Curvature

115

Example 7.2.1 Consider the Funk metric F on a strongly convex domain S I c R " . F is a nonnegative function on TO. = Q x R™. According to (1.21), F satisfies Fxi=FFyi.

(7.19)

By (7.19), one can easily show that p

..k

p

and the geodesic coefficients G* of F are given by G\y) = \F(y)y\

(7.20)

Hence

^ =^ y

+^--

(7.2i)

We are going to compute the Landsberg curvature using (7.12). By (7.19), we obtain

Using (7.20) and (7.21), we obtain yl _

1

pi

dC

ijk „

CijkiGl = -Frf-g± 2 V dyl

=

_1

2

--FCijk

and 1„ ,„ 1_,„ 1 CijkNi = -Fyiy Cijk + -FdiCijk -FCiik. 2 Plugging them into (7.12) yields ^ijk

—

n

^ijk-

In index-free form, L„(u, v, w) =

F(v) ^Cyiu,

v, w).

(7.22)

Let c(t) be a unit speed geodesic in fi with c(0) = y € TXQ, and C/(t) a parallel vector field along c. From (7.22), we see that the function C(t):=Ct(t){U{t),U(t),U{t))

Non-Riemannian

116

Curvatures

satisfies C'(t) + \c{t) = 0.

(7.23)

The general solution of (7.23) is C(i)=C(0)exp(-i*). The maximal interval of c is (—5, oo), where

F(y) In [1 + F(y)

n-y)

Thus C(£) is a bounded function on (—6,00). It is not clear that if C is bounded on Q, when O is not the unit ball in R n . jj Let {t>i}™=1 b e a n a r b i t r a r y basis for TXM. W e define t h e m e a n of hy by n u

3y( )

•=

^ ^ ij = l

J

(

y

)

L

« (

u

'

b i

'

b

j ) '

where gtj(y) = g 3/ (b i , by). The family J = {Jy}y€TM\{o} is called the mean Landsberg curvature. A Finsler metric is called a weak Landsberg metric if J = 0. In dimension two, J completely determines L. Let (M, F) be a Finsler space and c an arbitrary geodesic. Take an arbitrary parallel vector field V(t) along c. It follows from (7.16) that Ji{t)(v(t))=jt[l6{t)(y(t))].

(7.24)

Thus on a weak Landsberg space, the mean Cartan torsion I is constant along any geodesic. There is an induced Riemannian metric g of Sasaki type on TM \ {0}, 9 = gij(y)dxl ® dx3 + gtjSy1 ® 5yJ. T. Aikou proved that if L = 0, then all the slit tangent spaces TXM \ {0} are totally geodesic in TM \ {0} [Ai]. We can show that if J = 0, then all the slit tangent spaces TXM \ {0} are minimal in TM \ {0}.

S-Curvature

7.3

117

S-Curvature

In this section, we will introduce and discuss an important non-Riemannian curvature for Finsler m spaces. It measures the rate of changes of Minkowski tangent spaces over a Finsler m space. Let (M, F, dfj,) be a Finsler m space. Take an arbitrary basis {bj}™=1 for TXM and its dual basis {^}^ = 1 for T*M. Express dy, = a{x) 61 A • • • A 0n, For a vector y £ TXM \ {0}, we define

Jdet(gij(y)) r ( y ) : = In-¥

-,

(7.25)

a

where gij{y) '.= g y (b{,bj). T is called the distortion of (M,F,djj,). The distortion r has the following homogeneity property r(Xy) = r(y),

X > 0, y G TXM \ {0}.

The vertical derivative of T is nothing but the mean Cartan torsion, namely, ft[r(y

+ tv)]\t=0

= Iy(v),

v£TxM.

(7.26)

First, let us give a proof for (7.26). In local coordinates T(y) = ln h/^Mi L

a

(7.27) J

where gij(y) = gtXgfrU, afrU) and d\i = a(x)dx1 • • • dxn. Observe that ^

[in y/det(gij(y))\

=

l

-g^{y)^{y)

= g^(y)Cijk(y)

=

Ik(y).

This gives (7.26). According to Deike's theorem (Theorem 7.1.1), C = 0 if and only if 1 = 0. Therefore, the following conditions are equivalent (a) (b) (c) (d)

r(y) = constant; I = 0; C = 0; F is Euclidean.

Non-Riemannian

118

Curvatures

To measure the rate of changes of the distortion along geodesies, we define S(y) :=

It

-(^)L'

(7.28)

where c(t) is the geodesic with c(0) = y. S is called the S-curvature [Sh9]. It is also called the mean covariation in [Sh2] and mean tangent curvature in [Sh4]. Let Y be a non-zero geodesic field on an open subset U C M. By (7.28), we have S(Y) =

(7.29)

Y[r(Y)]

The S-curvature S satisfies the following homogeneity condition S(Ay) = XS(y),

A > 0.

In general, S(y) is not linear in y. Differentiating S(y) twice with respect to y gives new quantity. For a vector y e TxM\{0}, define Ej, : TxMxTxM -> Rby Ey(u,v) :--

1 d2 2dsdt

S(y + su + tv)

u, v £ TXM.

(7.30)

We call E = {Ey}veTM\{o} the E-curvature. We will show that the Ecurvature is independent of the volume form, although the S-curvature does. We first derive local formulas for the S-curvature and E-curvature. In local coordinates, r(y) is given by (7.27). Observe that in a standard local coordinate system (xl,yl) in TM, _d_ In d dxk . d llnJdetig^) dykr

*M = 2°"lB 9ijQ

t j fe-

(7.31) (7.32)

lt follows from (5.27) that 29

WV

(7.33)

S- Curvature

119

Let c(t) be the geodesic with c(0) = y. By (7.31)-(7.33), we obtain

S(c): = =

=

|[r(c(t)) |(lnydet(3ij(c)))-|(lna(c))

^

t

?

Nm(c)-

—

^

- 2 S «(c)C« fc (c)G*(c) -

^ ^ r ( c )

— (c)

This gives

s(y) = ^ ( « ) - ^ ^ r W -

a-34)

Thus

We obtain a local formula for the E-curvature 1

E*(«.«)

1

f)3(~im

= 2SwV(„)t,V = a ^ ^ ^ d / ) ^ ,

(7.35)

where u = u ^ U ^ = ^ g ^ U G TXM. From (7.35), we see that the E-curvature is independent of the volume form. It is purely a geometric quantity of Finsler metrics. By (7.30), we see that S(y) is linear in y G TXM if and only if E = 0 on TXM \ {0}. In particular, if F is a Berwald metric, then S(y) is linear in y G TXM for all x [Sh2]. In fact, S = 0 for Berwald metrics if we consider the S-curvature of the Busemann-Hausdorff measure Vo\p. More precisely, we have Proposition 7.3.1 For any Berwald space (M,F), S = 0 with respect to the Busemann-Hausdorff volume form dVp. Proof. Let c : (—e,e) —> M be an arbitrary geodesic. Take an arbitrary parallel frame {b,(t)}™=1 along c. Let {#2(i)}™=1 denote the dual coframe

Non-Riemannian

120

Curvatures

along c. By (2.7), the Busemann-Hausdorff volume form dVp of F is given by dVF\c(t)=aF(t)el{t)A---A9n{t), where aF{t) : Vol{(y')eR" | F(i/
s(c(i)) =

it r (

oW)

)

By Lemmas 5.3.1 and 5.3.2, Sc(t){bi(t),bj(t)j

= g £ ( 0 ) (b i (0),b J -(0)J = constant,

F ^ b ; ^ ) ) = F ^ l b i ( 0 ) ) = constant. Thus det [gc(t)(bi(t),bj(i)jj = det [g y (b*, b., J1, Vol{(^) e Rn | F ( V b ^ ) ) < l } = V o l j ^ ) G R n , | i ^ j / ' b i ) < l } . The last equality implies that <XF(£) = constant. Thus S(c(£)) = 0. Since c is arbitrary, we conclude that S = 0. Q.E.D. Now we use (7.34) to prove the following Example 7.3.1 Let F = a + (3 be a Randers metric on a manifold M, where a(y) = y/aij{x)yiyi a n d / % ) = fei(a;)yi with ||/?j| := sup Q ( j / ) = 1 /3(j/) < 1. WE will find a sufficient and necessary condition on a and /3 for S = 0. In particular, we will show that if /3 is a Killing form of constant length, then S = 0. In a standard local coordinate system {xl,yl) in TM, define b^j by

S- Curvature

121

where 6l := dx1 and 6^ := TJikdxk denote the Levi-Civita connection forms of a. Let r

H : = g V^

+ bj]i

Sij

)'

:=

2 V i U ~~

bjli

)'

%

The geodesic coefficients G of F are related to the coefficients Gl of a by G l = & + Py{ + Q\ where P

(V)-

W^-AnjV'v3-2a(y)brar"sply1} 2F{y) aairsriyl.

=

Ql(y) : = Observe that dQ —— = a

1

aijyjalrsriyl

+ aairsri

= a lSjiyjyl

+ aalrsri

= 0.

Thus Nl = Nl +

(n+±)P.

Put dVp = <j(x)dxl • • • dxn,

dVa =
We have
Nl

By (7.34), we obtain a formula for S,

S(y) = (n + l){P(y) - d[ln v T = W ] (l/)}, where P(V) := 2 ^ y ( r y y V " 2a( 2 /)6 r a'-% ; 2 /).

(7.36)

Non-Riemannian Curvatures

122

Now we are going to find a sufficient and necessary condition on /? under which S = 0. For the sake of simplicity, we choose an orthonormal basis for TXM such t h a t aij — Sij. Let bkb fell

lnyr^w

l-IW

Observe t h a t S = 0 if and only if - 2abpspiyl

rijyW

P)AlVl.

= 2(a +

This is equivalent t o the following equations

(7.37) (7.38)

bpSpi + Ai = 0.

We claim t h a t (7.37) implies (7.38). First, contracting (7.37) with bi and bj yields 2bibi\jbj

bibiijbj = - 2 1 | Thus

biAi = - /

l

!LL = o.

i - Ml2

Using this fact, we obtain kbj\i

=

"ZbiTij - bibi\j

=

2\\f3\\2Aj + {l-\\P\\2)Aj

= {l +

\\l3\\2)Aj.

Finally, we obtain bpSpi

=

-(bpbp\i - bpb^p)

= ^{-(l-il/3|| 2 )^-(l-P| 2 )^} = -^. This gives (7.38). We conclude t h a t S = 0 if and only if (7.37) holds. We can rewrite (7.37) as follows. (1 - ||/3|| 2 ) (biy

+ 6 j H ) + 2bkbklibj

+ 2bkbkijbi

= 0.

(7.39)

Note t h a t if 0 is a Killing form (r^- = 0) with constant length (bibiy = 0), then (7.39) holds. Hence S = 0 and E = 0. (t

S-Curvature

123

Below is a specific family of Randers metrics on S 3 with S = 0. Example 7.3.2 Let {C^C^C 3 } De the canonical left-invariant co-frame on the Lie group Sp(l) = S 3 satisfying dC1=2<2AC3,

dC2 = 2C 3 AC\

dC3 = 2C 1 AC 2 -

(7-40)

Consider F = a + (3, where + A 2 [ C W + A2[C3Q/)]2,

a(y) := \ M C W

P(y) ••= e C\v),

where K > 0, A > 0 and e > 0. As Assume that K

Then F is a special Randers metric. Let 61 := {6l ,92,03}

62 := AC2,

K(1,

03 := AC3.

is an orthonormal co-frame for TS 3 with respect to a. Put

where &i = - ,

6 2 = 0,

63=0.

K

It follows from (7.40) that d0i = P A
*• = $«•.

«.'--£»•.

t-&-%)*•

By definition, 6j|j are defined by dbi-bjOi3'=:6i|j0*.

A direct computation gives j.

°2|3 -

£

-^2 ,

».

»3|2 ~

£

^2

and all other components bi\j — 0. Thus /? is a Killing form of constant length. By Example 7.3.1, we conclude that S = 0, hence E = 0. (t

Non-Riemannian

124

Curvatures

E x a m p l e 7.3.3 Let F denote the Funk metric on a strongly convex domain CI in R™. The geodesic coefficients are given by Gi =

2

i -Fy y .

Observe that Nl = l{Fy%

= \{Fy^

+ FS\) =

^ - F

According to Example 2.2.4, the Busemann-Hausdorff volume form dVp = a(x)dx1 • • • dxn has constant coefficient, CTF{X) =

constant.

By (7.34), we obtain S(y) = ^F(y). The angular form hy(u,v)

(7.41)

:= hij{y)ulv:' on TXM is given by ys

yl

9^Y{y)gitF~(y)'

hij := F Fyiyi = gij -

h = {hy} is called the angular m.etric. By (7.35), we obtain 1„ n+1 Eij — o ^ i / V ~

',

-^j/V •

In index-free form, Ey(u,v)

= ^±^hy(u,v).

(7.42)

Chapter 8

Structure Equations

In previous chapters, we introduced the Riemann curvature and many nonRiemannian quantities. We discovered some relationship among these quantities. In this chapter, we are going to use the exterior differential methods to find more relationships among these quantities. At the end, we compute the Riemann curvature of a special class of Randers metrics.

8.1

Structure Equations of Finsler Spaces

In this section, we first introduce the curvature forms by differentiating the Chern connection forms. Then we derive some identities for the curvature coefficients by differentiating the fundamental tensor. Let (M, F) be a Finsler space. In Section 5.2, we introduced the fundamental tensor g, the Cartan tensor C and the Landsberg tensor £ on TT*TM. See (5.12), (5.13) and (5.32) for definitions. Let {ei}™=1 be an arbitrary local frame for TT*TM and {u/,w n+l }™ =1 the corresponding local coframe for T*(TM \ {0}). See (5.18) for the correspondence. Let (y,y) = yleiand

yl, gij, Cijk and L^k are local functions on TM. By Theorem 5.2.2, there is a set of 1-forms {wJJ}™J=1 on TM \ {0}. The Chern connection V on 125

Structure

126 TT*TM

Equations

is expressed by

VU = {dET + C^'u/Jgiej, where U = Ulei G C°°(Tr*TM). T h e Chern connection is uniquely determined by (5.21) a n d (5.22). For convenience, we rewrite (5.21) and (5.22) again.
=

UJJ ALO/

=

k

(8.1) k

n+k

gkjw +gikuj +2Cijkio

.

(8.2)

Let Q/ := doj/-

ujjk A w / .

(8.3)

l

{ilj }ij=1 is a set of local 2-forms on TM \ {0}. T h u s we can express fi •* in t h e following form ft/

= ±R/klu;k

Acol + P/kluk

A ujn+l + \ Q

3

\ ^

k

A «/*+',

(8.4)

where R/ki

+ R/ik = 0,

Q/ki

+ Qjlik = o.

(8.5)

Differentiating (8.1), we obtain

wj' A nj = o. This gives R3\i

+ -Rfc'y + R-i%jk = 0^i'fci

=

Pkju

(8-6) (8-7)

Equation (8.4) simplifies to ft/ - ^ i ? / ^ f c A w ' + P / ^ A o;"+'.

(8.8)

Structure

Equations of Finsler

Spaces

127

Define R(U,V)W:

=

V(U,V)W:

=

R/kl(y)UkVlWjei: Pjikl(y)UkVlWi

<x,

where U = Ukek,V = VleuW = Wjej 6 ir*TM. We call K and V the Riemannian curvature tensor and the Chern curvature tensor respectively.

Let Qi:=dujn+i-u>n+j

ALJJ.

(8.9)

According to Theorem 5.2.2, wn+l are given by un+i=dyi

+ yiwJi.

(8.10)

Differentiating (8.10) yields

By (8.8), f2l can be expressed by Qi = ±I?klu>kAujl

+

Piklu>kA<jn+l,

where R^r^y'R/u,

P'u-ytPSu.

Put R\

:= ffwl/' = ^ i J / f c , j / ' .

(8.11)

We claim that Rlk are just the coefficients of the Riemann curvature in (6.4) and Pjlkl are just the coefficients of the Chern curvature in (7.13). Hence Plkl = —g^Ljki are just the coefficients of the Landsberg curvature in (7.12). Let (xl,yl) be a standard local coordinate system in TM. Take w* = dx\

Lun+i = Sy\

Structure

128

Equations

T h e n by (5.25), d2G

Plugging t h e m into (8.3), we obtain

j k l

dxk

dxl

dym

l

dym

k

+T™lrmk-T™kYiml P<

=

* * dyl

j kl

(8.12)

* & dyidykdyl

_

|

g L

*

dyl

'

{

(813) '

Equation (8.13) is just (7.13). By the homogeneity of F, we obtain

k

P\i

~~

dxk

=

-L*kl.

V

dxidyk

+

dyidyk

dyi dyk'

(

' (8.15)

Equation (8.11) is just (6.4). T h u s Rlk coefficients of the Riemann curvature.

defined in this section are t h e

Now we go back to the general setting. Define Cijk\i and Cijki by dCijk — Cijku:i

— Cuk^j

— Cijiuik

=: Cijk\i(*> + CijkioJ71 •

In virtue of (5.24), we have Lijk = Cijk\iyl.

(8.16)

Differentiating (8.2) and using (8.9), we obtain ft* * V + 9ipttjP + 2ClJpnp

+ 2 ( C m k i 0 k + Ciji.kun+k)

Aoj n + l = 0. (8.17)

From (8.17), we obtain 9pjRi ki + 9ipRj ki + ZCijpR 9PJPi \i + 9iPP? ki + 2CijpP Ciji.k = Cijk-i-

p kl

= 0,

* + 2Cimk

(8.18) = 0,

(8.19) (8.20)

Structure Equations of Finsler Spaces

We first derive some important identities for Rf Rj k '•— Rj

129

ki-

Let

kiV •

Contracting (8.18) with yi yields VJ9vjRiVkl

= -9iPRPkl-

(8.21)

By (8.21), we obtain yj9jpRpki

-yl9iPRPki-

=

This gives ylgipRpkl

= 0.

(8.22)

Contracting (8.18) with yl yields 9viRi\

+ 9iPRjPk

+ 1CijpR\

= 0.

(8.23)

Contracting (8.21) with yl yields gipR\

= -ytgpjR^.

(8.24)

It follows from (8.5) and (8.6) t h a t

Rpk = RiW = -RkpJ

-Ripikyl = Rk\

-*V

It follows from (8.22) and (8.24) t h a t 9iPRPk

-y39pjRPk

=

yj9Pi(Rpik-Rkp)

= =

-yJ9pjRki

=

9kPRPi.

We obtain 9iPRPk

= gkpR'i.

(8.25)

Now we derive some important identities for -PA;three times t o the combination (9PJPAI

+ 9rPPPkl)

- {9PkPPu

+ gPJPkPu)

+ {9piPkji

Applying (8.19)

+

gpkP^i),

130

Structure

Equations

we obtain 9ipPj kl

~

~~^ijpP kl ~ (-'jkpP it " (->ikpi ji -Ciji\k

- Cjkl\i - Ciki\j-

(8.26)

By (8.26), we can easily obtain the following Liu = -gipPPki,

v3-27)

Uk^yig^P^

(8.28)

and VJ9jpPPkl

8.2

= Likiy1 = 0.

(8.29)

S t r u c t u r e E q u a t i o n s of R i e m a n n i a n M e t r i c s

In this section, we will deal with Riemannian metrics in a traditional way. All t h e quantities are defined on the base manifold. At t h e end, we view Riemannian metrics as special Finsler metrics and deal with t h e m as in t h e previous section. We will give a link between these two different approaches. Let (M, g) be an n-dimensional Riemannian space. In a local coordinate system {xl) in M, g — gij(x)dxl ® dx^ and the Christoffel symbols

?fcW •

dyidy*

(V)

29

[

}

\ dx*

{

'

+

dxk

[

are functions of x only, so t h a t N]{y)=T)k{x)y\

V=

d_ ykk7-k\ dx

T h e Levi-Civita connection is given by

d Dy[/ = {d[/i(2/) + ^r} f c y f c }^!

'

dx>

[

}

)

Structure Equations of Riemannian

Metrics

where U = U{-^ £ C°°{TM) and y = yk-£^\xtion is uniquely determined by DVV-DVU

The Levi-Civita connec-

= [U,V},

W[g(U, V)} = g(DwU, V) + g(U, DWV). where U,V,W

131

(8.30) (8.31)

eC°°(TM).

Let {bj}" = 1 be a local frame for TM and {#l}f=1 the dual coframe for T*M. Express the Levi-Civita connection D by Dbj = 0 / ® b i . The set of 2-forms {#/} are called the Levi-Civita connection forms. Let 9ij '•= ff(bi,bj). Equations (8.30) and (8.31) are equivalent to d6i = 6j A 6/, dgij=gkAk

+ gikOjk.

(8.32) (8.33)

Set 6 / = dfl/ - 6k A ^ = i i ? / w 0 f c A 0',

(8.34)

Rhi + */«* = °-

(8-35)

where

Differentiating (8.32) and (8.33) yields 0j A 0 / = 0, and 9kjQik + gikQk

= 0.

These two identities imply Rjiki + Rkiij + Riijk = 0,

(8-36)

9kjR{ kl + 9ikRj kl — °-

(8.37)

Structure

132

Equations

Define R(u,v)w

:= R/klukvlwj

bi;

where u = ukhk,v = vlhi and w = w^bj. We call R(u,v)w curvature tensor. From the definition, we have R(U, V)W = DuDyW where U,V,W

- DvDuW

-

the Riemann

DlUiV]W,

GC°°.

Equations (8.35), (8.36) and (8.37) can be written R(u,v)w + R{v,u)w = 0,

(8.38)

R(w, v)w + R(v, w)u + R(w, u)v = 0,

(8.39)

g(R(u,v)w,

(8.40)

z) + g(w,R(u,v)z)

=0.

It follows from (8.38) and (8.40) that g\R(u,y)y,uj

= -g[y,R(u,y)u)

= g(y,R(y,u)vj

= g(R(y,

u)u,yj.

This gives g[R(u,y)y,uj

g[R(y,u)u,yj

g(u, u)g(y, y) - g(y, u)g(y, u)

g(u, u)g(y, y) - g(y, u)g(y, u)'

Thus the flag curvature K(P, y) is independent of y e P for any tangent plane P C TXM. In this case, we denote by K(P) := K(P,y). We call K ( P ) the sectional curvature of the section P C TXM. In a local coordinate system (xl) in M, 6-z = Tl,kdxk. into (8.34) yields /BY*

3Y^

H kl

\.'d^k+^lLmk)~\~dx^+jkml)-

* ~

Using Gl = \^)kyjyk, pi

R kiyy

j

we can express i?. 8 kl y j y l as follows

j i_odGl 2

Plugging them

~ w~

y

i

d2Gi

lh^

«™

+ 2G

d2Qi

9GidGj

WW'WW'

{

'

Structure Equations of Riemannian

Metrics

133

Comparing (8.41) with the coefficients i?lfc(j/) of the Riemann curvature in (6.4), we obtain = R/uyty1,

R\{y)

(8.42)

that is, Rj,(u) = R(u, y)y.

y,u<= TXM.

Observe that

d2R\

d2Ri, 1

dyidy

dyidyk

-{Ri\qyq + Rp\ivp - Rkliqyq -

dyi

^ kj "I" nj

— 3Rj

kl ~ nk

R;lkyp]

lj ~ -"-j Ik

kl.

We obtain _l
d2R\ -( dyldy^y

(

'

We view a Riemann metric as a special Finsler metric. A natural question arises: What are the relationship between the Riemannian curvature tensor on the base manifold and that on the slit tangent bundle? We lift {bi}™=1 to a local frame {el}f=1 for ir*TM, where eilj, := (y,bj). l

Let y be defined by {y,y)

=yle{.

Let ujn+i := dy1 + j/V'6)/.

J := TT*0\ Then <

:= n*e/

are the Chern connection forms of F with respect to {ei}™=1. We have n/ : =

=

(ko/ - w / A uj

^(dsj-ofsoi)

Structure

134

=

Equations

**(±R>klVMl) R/klLokAojl

Thus the coefficients of the Riemannian curvature tensor are functions of x e M only and the Chern curvature vanishes. However, if we take an arbitrary local frame {ej}™=1 for ir*TM, then the coefficients of the Riemsnnian metric curvature tensor might depend on y € TXM.

8.3

Riemann Curvature of Randers Metrics

In general, it is much more difficult to compute the Riemann curvature. In what follows, we are going to give a formula for the Riemann curvature of a Randers metric F = a + (3, where )3 is a Killing form with constant length. Let F = a + (3 be a Randers metric on a manifold M. In a standard local coordinate system (xl,yl) in TM, a and 3 are expressed by «(j/) = \Jaij(x)yiy^

p(y) = bi(x)y'.

Define b^j by

dbt - b^i =: biUej, where 9l := dxl and 6/ := Tfkdxk denote the Levi-Civita connection forms. Let r

H '•= j [bi\J

+ b3\i) '

S

iJ '•= 2 ( 6 i | i ~

bj l

\)

'

The geodesic coefficients Gl of F are related to the coefficients Gz of a by Gi = Gi + Pyi + Q\ where P(V)--

=

Ql(y) : =

2F\y){TijyiyJ aairsrlyl.

-2a(y)brarpsply1}

Riemann

Curvature of Randers

Metrics

135

From now on, we assume that f3 is a Killing form (r^ = 0) with constant length {bibi\j = 0), that is, (3 satisfies r

H = 2 (bi\J + bJ\i) = °.

haijbjlk

= 0.

(8.44)

Equations (8.44) implies that brarpspi = brarpbp\i = 0. Thus P = 0 and G"s are simplified to Gi = & + Qi

(8.45)

with Q1 =

aatrbr\iyl.

Note that Q"s define a tensor Q = Qldi on T M \ {0}. Let R%k and i?'fc denote the coefficients of the Riemann curvature of a and F respectively. Plugging (8.45) into (6.4) yields R\

= R\ + {2Q% - y^Q'^y,

- ( Q ' ) ^ ( ^ ' ) y * + 2Q^Q%jyk},

(8.46)

where Q1,- denote the covariant derivatives of Ql with respect to a. See (5.38). By Lemma 5.2.1), we know that ay = 0. According to (5.39), we have Q\j = a(airbAlyl\

=

aairbmjyl.

For the sake of simplicity, we take a local orthonormal frame {b;}™=1 on M with respect to a so that a(y) =

'jr.WY,

P(y) = J2biy{'

v = yihi-

\ The local frame {bj}™=1 determines a local frame {ej = (y, bj)}™=1 for +i •K*TM and the corresponding local coframe {LJ\ w " } ^ = 1 for T*(rM\{0}). The formula (8.46) still holds the coefficients of the Riemann curvature with respect to {ej}f=1, although it is stated originally in a standard local coordinate system in TM. We have Qi = abi]pyp

Structure

136

Ql\k J

Equations

= a*i|p| f c 2/ p l

= y3(abiMjyp)yk

y (Q u)yk

(Q%i = a-1{bilpyp) (Q* W

= a~1(bilpiJyJyp)

yk +

abi]k\pyp

yj + abi{j

= """HfypV") $jk " a-3(bilpyp)

yjyk + a~lbAJ yk + a^b^

yj

Plugging them into (8.46) yields R\

=

Rik +

2abilplkyp-abilklpyp-a-1(biMgypycl)yk

-a2bilmbmlk

+ 3(bilpyp) (bk]qyq) + (bi]mbmlpyp)

yk. (8.47)

We have the following Ricci identity for the covariant derivatives of (3, h\j\k — h\k\j — bmRi^k-

Using the Ricci identity, we obtain bi\v\kyp = (bi\k\P + bmRimpkS)yv bi\P\qypyq

= =

= bmpyp

-

bmRimkpyp,

-bplilqypyq -(bPlq\iypyq

bmRpmiq)ypy(l

+

= —bm,R"iPlugging them into (8.47), we obtain R\

=

Rik+abiWpyp~2abmRirlPyp

+

-a2bi]mbm]k

+ (bilmbmlpyp)

+ 3(bilpyp)(bk]gy'!)

a'lbmRmiyk yk.

(8.48)

Let R and R denote the Ricci scalar of F and a respectively. Since a is a Riemannian metric, 77 — 1

D m _ _ n t _ __ • rl i ip -rLm ip ~ n

-

p

flymyv-

From (8.48), we immediately obtain an equation for the Ricci curvature. R = R + abmRymyPyp

+ ^-^{a2(bmlp)2

+ 2(6 m | p 2 / p ) 2 }.

(8.49)

Chapter 9

Finsler Spaces of Constant Curvature

As we know, every Riemannian metric of constant curvature A is locally isometric to a canonical Riemannian metric of constant curvature A. However, for Finsler metrics, this is no longer true. For each A, there are infinitely many non-isometric Finsler metrics of constant curvature A. In this chapter, we will discuss some basic properties and well-known examples of Finsler metrics with constant curvature. 9.1

Finsler Metrics of Constant Curvature

The purpose of this section is to derive some equations for Finsler metrics of constant curvature A. We will show that Landsberg metrics of non-zero constant curvature must be Riemannian. Let F be a Finsler metric on a manifold M. Differentiating (8.1) yields the following Bianchi identities:

dn/ = - n / A
=

*j km\l ~ Pj lm\k + Pj kt^

Im. ~~ Pj lt^

km'

(9-1)

Contracting (9.1) with yi yields P-m ki — Plki-m. + Lzkm\i — L ;m|fe + L ktL

lm

— LlltL

km.

(9.2)

Contracting (9.2) with yl yields Rl kiV1 = R\.m

~ R\m 137

+ L\Mlyl.

(9.3)

Finsler Spaces of Constant Curvature

138

By (8.6) and (9.3), we obtain R%km = Rm klV ~ Rk mlV pi

—

Jx

km

_ pi _9pi IX mk ^£x km-

This gives R\i = \(R\.i-R\.k)-

(9-4)

Theorem 9.1.1 Let (M, F) be a Finsler space of constant curvature A ^ 0. Suppose that J = 0. Then F is Riemannian. Proof. By assumption, Rik = \(F28i-gkpy^yi).

(9.5)

Plugging (9.5) into (9.4) yields Rlki=*{9ji6k-9jk6i}.

(9.6)

Plugging (9.6) into (9.2), we obtain Rfki

= ^[9jisk ~ 9jkSlj + LljkV

- Ll^k

+i? f c f L t J , - L ^ L V

(9.7)

Using (9.7), we can rewrite (9.1) as follows 2A [Cjlm5k - Cjkm^lJ

=

Pj\m\l~

+L u-mL jk ~~ Llkt.mL

Rj\m\k +R/ktL

lm~ Pjl[tL

km

jl + L ltL jk.m — L ktL jl.m l

+ L JHkm - L''jk\l-m-

(9-8)

Thus, for Finsler metrics of constant curvature A, the Cartan torsion, the Chern curvature and the Landsberg curvature satisfy (9.8). Contracting (9.8) with ysgiS yields XCjimgksys

- XCjkmgisV3 = Ljkmll

- Ljim\k.

(9.9)

Then contracting (9.9) with yl, we obtain Ljkm\iyl

+ XF2Cjkm

= 0.

(9.10)

Finsler Metrics of Constant Curvature

139

Contracting (9.10) with gim yields Jk{lyl + \F2Ik

= 0.

(9.11)

Thus if Jk = 0, then Ik — 0. By Deike's theorem, we conclude that F is Riemannian. Q.E.D. Theorem 9.1.1 improves a result by S. Numata in [Nu] where he that L = 0 instead. Let c(t) be a unit speed geodesic in a Finsler space (M, F). Let V = V(t) be an arbitrary parallel vector field along c. Consider C(t) :=Cm(V(t),V(t),V(t)),

L(t)

I(t):=Ii{t)(V(t)),

:=L6{t)(V(t),V(t),V(t)). J(t):=J6{t)(V(t)).

It follows from (7.16) and (7.24) that L(t) = C'(t),

3(t) = I'(t).

By (9.10) and (9.11), we obtain the following ODE: L'(t) + XC(t) = 0 J'{t) + AI(i) = 0. Thus C(t) and I(t) satisfy C"{t) + AC(i) = 0

(9.12)

I"{t) + Xl(t) = 0.

(9.13)

The general solutions of (9.12) and (9.13) are ;iven by C(t) I(t)

= =

s A (t)L(0)+si(t)C(0), s A (t)J(0) + s'A(t)I(0),

(9.14) (9.15)

where s\(t) is the unique solution of the following equation y"(t) + Xy(t) = 0,

y(0) = 0 ,

y'(0) = 1.

Assume that F is complete, i.e., every geodesic is defined on the whole line (—00,00). Let us take a look at the following cases:

140

Finsler Spaces of Constant

Curvature

Case 1: A = —1. In this case, (9.14) gives C{t) = sinh(*)L(0) + cosh(t)C(0). Assume that C is bounded. Then L(0) = 0 = C(0). Since c is arbitrary, we conclude that C = 0 and F is Riemannian. By a similar argument using (9.15), we can show that if I is bounded, then 1 = 0. Hence F is Riemannian by Deike's theorem. Case 2: A = 0. In this case, (9.14) gives C(i) =*L(0) + C(0). Assume that C is bounded. Then L(0) = 0. Since c is arbitrary, we conclude that L = 0 and F is a Landsberg metric. By a similar argument using (9.15), we can show that if I is bounded, then J = 0. Hence F is a weak Landsberg metric. Theorem 9.1.2 ([AZ]) Let (M, F) be a complete Finsler space of constant curvature K = A. Assume that the (mean) Cartan torsion is bounded. (a) If X = 0, then F is a (weak) Landsberg metric; (b) If A < 0, then F is a Riemannian metric. When M is compact, the Finsler metric is complete with bounded Cartan torsion. Thus the conclusions in Theorem 9.1.2 hold. When (M,F) is compact with vanishing flag curvature. In this case, all geometric quantities are bounded. By further argument, we can actually prove that the Chern curvature vanishes. It is known that every Finsler metric with vanishing Chern curvature and flag curvature must be locally Minkowskian [BCSl]. We conclude that every Finsler metric on a compact manifold with vanishing flag curvature must be locally Minkowskian. See also [BCS2][Pa] for some rigidity results on Finsler surfaces. The geometric and topological structures of Finsler spaces of positive constant curvature will be discussed in Section 18.3 below.

Examples

9.2

141

Examples

In this section, we will discuss several important examples of Finsler metrics of constant curvature. These examples show that the classification problem seems to be not solved within my life time.

Example 9.2.1 Let F be the Funk metric on a strongly convex domain fl C R™. According to Example 7.2.1, the geodesic coefficients Gl of F are given by &{y) = \F{y)y\

(9.16)

Thus geodesies are straight lines in 0 and F is projectively flat. Plugging (9.16) into (6.4) and using (7.19), we obtain Rik =

-\{F2(y)Sik-9ki(y)ylyi}.

In index-free form, Ry(u) = —-|gy(y,y) u-gy(y,u)

yj,

u <E TXM.

Thus for any tangent plane P = span{?/,w} C TXM, the flag curvature K of F satisfies K(P ) 3 /) = - ^ .

Now we take a look at a special case when 0, = B is the unit ball in £". From the definition, the Funk metric F is determined by |

.

2

/

|

T

F(y)\ We obtain F{

) = V\y\2-(\x\2\y\2-^y)2) 1 — \x\

2

+ (x>y)_

(9.17)

Finsler Spaces of Constant

142

Curvature

E x a m p l e 9.2.2 Let F denote the Funk metric on a strongly convex domain ft C R". The Klein metric F on ft is defined by F(y)-=\{F(y) + F(-y)}-

(9-is)

By (7.19), one can easily prove that F satisfies the following equations Fxkyiyk

= Fxi

and the geodesic coefficients Gl of F are given by &{y)=l-{F{y)-F{-y))yi.

(9.19)

Thus geodesies are straight lines in ft and F is projectively flat. Plugging (9.16) into (6.4) and using (9.19), we obtain

Rik =

-{F2(y)Sik-gkl(y)ylyi}.

In index-free form, Rj/(w) = -[iy(y,y)

u-£y(y,u)y},

u&TxM.

Thus for P = span{y, u}, the flag curvature K of F satisfies K(P,y) = -l. By (9.17), we obtain the Klein metric F(y) on the unit ball B

C IR":

p, , = V\y\2 - ( N 2 H 2 - (x,y)2) {y) i-kl2 B E x a m p l e 9.2.3 ([Sh8]) Minkowski spaces are the most trivial Finsler spaces with K = 0. We may construct non-trivial Finsler metrics with K = 0 as follows. Let F denote the Funk metric on a strongly convex domain ft in R™. Assume that a Finsler metric F on an open subset U C ft satisfies the following system Fxk=(FF)yk.

(9.20)

Examples

143

Observe that Fxkyiyk

= {FF)ykyiyk

=

(FF)yi

This implies that Fxkyk IF

_ (FF)ykyk

_ 2FF

IF

2F

F.

By the above identities and (5.2), we obtain that & = F(y) y\

(9.21)

Thus, the geodesies of F are straight lines and F is projectively flat. Plugging (9.21) into (6.4) and using (7.19), we obtain that R\

= 0.

Thus the flag curvature K of F vanishes. When fi = B is the unit ball in K , the Funk metric F can be expressed by elementary functions in (9.17). However, no explicit elementary expression for F has been found in this case. (j Now we give some examples of Finsler metrics on t h e n-sphere with positive constant curvature. E x a m p l e 9.2.4 ([Brl][Br2]) Let V 3 be a three-dimensional real vector space and V 3 ® C its complex vector space. Take a basis {bi,b2,bs} for V 3 and define a quadratic Q on V 3 x C by Q(u,v) = e i a u V + e i / 3 u V + e " i Q u V , where u = u ^ v = v{hi and a, (3 £ R. For X G V 3 \ {0}, let [X] := {tX, t > 0}. Then S2 := {[X], X £ V 3 \ {0}} is diffeomorphic to the standard unit sphere S in the Euclidean space K . For a vector Y e V 3 , denote by [X, Y] € T{X]S2 the tangent vector to the curve c(i) := [X + tY] at t = 0. Define F : TS 2 -> R by

where 7^[] denotes the real part of a complex number. Clearly, F is welldefined. Assume that \/3\ < a < -|. Then F is a Finsler metric on S 2 . Bryant has verified that F has constant curvature K = 1.

Finsler Spaces of Constant

144

Curvature

Now we shall give an explicit expression for a special class of Bryant metrics on S 2 with (3 = a. Take an arbitrary vector y G TXR2, let X:=(X,1)GR3,

y:=(y,0)GR3.

We have Q(X,X)

=

e lQ |x| 2 + e - i a

Q(X,Y)

=

e i Q (x,y)

Q(Y,Y)

=

eia\y\2

Hence Q{X, X)Q{Y, Y) - Q{X, Y)2 = e2a 'flxfly I2 - (x, y} 2 ) + |y| 2 . The real part of -i Q{X, Y)/Q{X, X) is r _ . Q(X,Y)i L l Q(X,X)\

=

sin(2a)(x,y) |x| + 2cos(2a)|x| 2 + f 4

For a complex number z, the real part of y/z is given by

KM - ^ ^ For Z =

Q ( x , x ) Q ( y , y ) - Q ( x , y ) 2 _ e2« *(|x| 2 |y| 2 - (x,y) 2 ) + |y| : Q(x,x) 2 2 2 a i | x | 4 + 2|x| 2 + e - 2 a i '

= 1

'

TZ(7) ^ j

=

V(lx| 2 |y| 2 - (x,y) 2 ) 2 + 2cos(2 Q )(|x| 2 |y| 2 - (x,y) 2 )ly| 2 + |y| 4 |x| 4 + 2cos(2a)|x| 2 + l lxl2ly|2-(x;y)2+cos(2Q)iyl2 |x| 4 + 2cos(2a)|x| 2 + l

2

/ sin(2a)(x,y) N2 4 2 V|x| + 2cos(2a)|x| + 1 / '

We obtain F([X, Y]) = Jn{Z)

+ |Z|

+

sin(2a)(x, y) |x| 4 + 2cos(2a)|x| 2 + l '

Define Fa(y) := F([X,Y]),

y G TXR2,

Randers Metrics of Constant

Curvature

145

where X = (x, 1), Y = (y, 0) € R 3 . We obtain a Finsler metric Fa on R 2 . This is the pull-back of the Finsler metric F on S 2 by

V(x) := f ,

X

, . *

V

Clearly, we can generalize Fa to a Finsler metric defined on R n without any modification. One can show that the generalized metric Fa on R™ also has constant curvature K = 1. Note that FQ is just the spherical Riemannian metric on R n , y/|yl 2 T+V I ( | x | 2 | v l 2 X- ( x , y ) 2 ) rl .+ ,J, |x|a: "

F0(y) = VIJI

( 9 - 23 )

All F a are pointwise projective to the Euclidean metric.

9.3

Randers Metrics of Constant Curvature

The Funk metric F in (9.17) is a Randers metric. In fact the Funk metric on any shifted ellipsoid Q C R™ is a Randers metric. We would like to find more Randers metrics of constant curvature. According to Section 8.3, the Riemann curvature a Randers metric F = a + (3 with a Killing form (3 of constant length takes a relatively simple form (8.47). In this section we will investigate this type of Randers metrics. Let F = a + j3 be a Randers metric on M, where (3 is a Killing form of constant length. At a point, take a local orthonormal frame {t>i}™=1 on M so that <x(y)

. E^1)2'

y = yibi.

Piy) = biy\

By assumption,

hi + hi = °'

'°ihi

= °-

( 9 - 24 )

Suppose that F is of constant curvature A, i.e, the coefficients of Riemann curvature with respect to {bi}" = 1 are in the following form:

R\

=

\F2(6ik-^Fyk)

Finsler Spaces of Constant

146

Curvature

\{{a2+p2)8lk-yiyk-l3bkyi}

=

+\a(2(36ik-bkyi")-\a-1Pyiyk Compare it with (8.47), we obtain R\

=

A{(a 2 + / ? 2 ) * f c - y y - / ? 6 f c i / i } +a2bilmbm]k

bmRmiyk

- 3(bilpyp)(bklqyg)

- (bi]mbm\Pyp)y

(9.25)

a2(biWpyP-2bmRimkpyp)

+

-Xf3yiyk + Xa2

=

(2(36ik-bkyi).

According to (8.42) and (8.43), Rlk and R/ki

R\ = Rhrfy',

j kl Rh

(9.26)

are related by

1 r d2R\ 3 I dy'dy1

d2R\ dyidyk .

}

Differentiating Rlk in (9.25) with respect to yi and yl, we obtain R/ki

=

X\6ikSji - Siidjk) + X\bjbi6ik - bjbkSii) +bi\pbp\k5ji - bi\pbp\i6jk + b^bj^ - bi\kbj^ — 26fc|;6j|j(9.27)

Contracting (9.27) with bi and yl, and using (9.24), we obtain bmR^y1

= X(bkyj - (35jk)

bmRmk = X (a2bk - / ? / ) .

Plugging them into (9.26) yields k\k\p = X (bkSip - biSkpJ. By taking the Hessian of M(3\\2 = \{bi)2,

\M

\k\i

=

bi\kbm + bibi\k\i

=

h\kbi\i + Xbi(bk6u - bidkA

=

b^bm + X ^

- \\(3\\2Skiy

(9.28)

Randers Metrics of Constant

Curvature

147

we obtain 6P|ft6P|/ = A(||/3|| 2 J f c ; -6 f c 6 i ).

(9.29)

(&P|*)2=A(n-l)P||2.

(9.30)

From (9.29), we obtain

Thus A > 0. If A = 0, then (3 is parallel and R\ = 0. Plugging (9.29) into (9.25) yields R\

=

A{ [(1 - ||/3|| 2 )a 2 + /32] Slk + a%bk - (1 - ||/?|| 2 )yV -Why*

+ hyk))

- 3(bilpyP)(bklqyq).

(9.31)

Conversely, one can easily verify that (9.28) and (9.31) imply (9.26). Thus under the assumption (9.24), F is of constant curvature A if and only if (9.28) and (9.31) are satisfied. Proposition 9.3.1 Let F = a + (3 be a Randers metric where (3 is a Killing form of constant length. F is of constant flag curvature A if and only if a and (3 satisfy (9.29) and (9.31). In this case, either A > 0 or A = 0 (hence a is flat and (3 is parallel). H. Yasuda and H. Shimada [YaSh] proved a better result than Proposition 9.3.1. They proved that a Randers metric F = a + (3 is of constant flag curvature A > 0 if and only if (a) P is a Killing form of constant length with respect to a, (b) a and (3 satisfy (9.29) and (9.31). However, their proof is much more complicated, so it is omitted here. See [Ma3] for another proof of Yasuda-Shimada's theorem. Recently, D. Bao and Z. Shen have found a specific family of Randers metrics on S 3 with constant flag curvature K = 1. The metrics on S 3 are described in the following example. Example 9.3.1 ([BaSh2]) We view S 3 as a compact Lie group. C^C^C 3 be the standard right invariant 1-forms on S 3 satisfying dC1 =2C 2 AC 3 ,

dC2 = 2 < 3 A C \

dC3 = 2C1AC2-

Let

148

Finsler Spaces of Constant

Curvature

The metric

«i(y) := V I C W + l C W + l W is the standard Riemannian metric of constant curvature 1. For k > 1, define

ak(y) := xA 2 [CW + M<2(»)]2 + M C W and

Then f* := «fc + &

(9.32)

is a Finsler metric on S 3 . Bao-Shen show that Fk is of constant curvature K = 1 for any k > 1. We give a brief argument below which is different from that in [BaSh2]. First, we show that (3k is a Killing form of constant length. Take an orthonormal coframe for T*S 3 , ^:=k(\

62:=y/k(2,

03 =

^(3,

so that

ak(y) 0k(y)

= V[01(y)? + [02(y)]2 + {93(y)?, = b1el(y) + b262(y) + b3e3(y),

where &i =

,

&2 = 0,

b3 = Q.

Clearly, (3k has constant length with respect to ak, that is,

\\pk\U = ^PThe Levi-Civita connection forms 6 -l are given by 6 -1 + 6^ = 0 and

e2l=e\

e3l = -e2,

e2

(H*-

Randers Metrics of Constant

Curvature

149

By definition, b^j are defined by dbi - bjOf =: bi{j9j. A direct computation gives &i|i=0,

62|i=0,

6i|2=0,

6212=0,

&i|3 = 0,

62|3 =

,/£2 — k ,

i/£ 2 — k &3|1 = 0,

&3|2 =

7

,

&3|3 = 0.

Thus h\j + bj\i = 0. We conclude that (3k is a Killing form with respect to akBy definition, b^k are defined by dbi\j - bk\j9i

- bi\k0j

=: bi\j\k9

•

A direct computation gives 0l|2|fc =

T

G

2k,

Ox|3|fc =

d3k,

02|3|fc = U.

Other components are determined by bt|j|fc + ^j|t|fc = 0- O n e c a n easily verify that ak and /3fc satisfy (9.29) with A = 1. Now we compute the Riemann curvature of ak- By definition, Rf kl are defined by

deZ-efAe^^R/^Ae1, where R/ki

+ R/ik = 0. A direct computation gives R2 12

=

1)

R2 13

^ 3 13

—

1>

- ^ 3 12 — 0 '

^ 3 23 — .

—

3,

R3

=

13

0,

— 0,

i t 2 13 = 0,

-^3 23

R3

—

12

0'

— 0.

150

Finsler Spaces of Constant

Other components of R? Rjki

are determined by

kl

+ R/ki

Curvature

=

0,

Rjki

+ Rkij

+ R-iljk = °-

One can easily verify t h a t ak and (3k satisfy (9.31) with A = 1. By Proposition 9.3.1, we conclude t h a t Fk = ak + (3k has constant flag curvature K = 1. According t o Example 7.3.2, Fk satisfies S = 0,

E = 0.

T h e volume form of ak is given by dVQk = k2(l

A C2 A C3 =

k2dVai.

By (2.10), t h e volume forms of Fk and ak are related by

dVFk = (l - H & H ^ ) 2 ^ = (l - (l - l))2k2dVai

= dVai.

This gives Vol Ffe (S 3 ) = V o l a i ( S 3 ) = Vol(S 3 ). T h a t is, the volume of (S3,Fk) is a constant. This implies t h a t for any point p e S 3 , there is a unique point q € S 3 with d(p,q) = n such t h a t all geodesies c(t), 0 < t < oo, with c(0) = p is minimizing on [0,7r] with c(t) = q. T h e details will be given in Section 18.3 later. Let (x, y, z, u, v, w) be t h e standard coordinate system in T R 3 = R 3 x R 3 . Let ip± : R 3 - * S3^. denote t h e diffeomorphisms defined by rpe{x, y, z) :=

( x , y, z, e), +y +z +1 >

1

2

yjx

2

(x, y, z) G R 3 ,

v

where e = ± 1 . We can express FK on R 3 . Pulling back £' by ipe onto R 3 , we obtain —edx — zdy + ydz

x

~

x2 + y2 + z2 + 1 zdx — edy — xdz

=

x2 + y2 + z2 + 1 —ydx + xdy — edz x2 + y2 + z2 + 1

2 C

^3 ~

Randers Metrics of Constant Curvature

151

Plugging them into aK and (3K, we obtain ah

=

\Jk2{su + zv — yw)2 + k(zu — ev — xw)2 + k(yu — xv + ew)2 1 + x2 + y2 + z2 1 + xz + yz + z2

One can extend above construction to odd-dimensional spheres S 2 n + 1 , because of the special fibration structure S 2 n + 1 -4 C P " = S ^ + V S 1 . fl

A Finsler metric F on an n-dimensional manifold M is of constant Ricci curvature A if Ric(y) = (n-1)A J F , 2 ( 2 ;),

y G TM.

Finsler metrics of constant Ricci curvature are also called Einstein

metrics.

Let us take a close look at Randers metrics of constant Ricci curvature A. Let F...= a + (3 be a Randers metric. We still assume that /? is a Killing form of constant length, so that the Ricci curvature of F is in a relatively simple form (8.49). Take a local orthonormal frame {bi}™=1 for TM so that

X^)2-

a(y) = \

P(y) = biyi,

y=

yibieTxM.

i=i

By Assumption, bi\j + bj\i = 0,

bibi\j = 0,

bibj\i = 0.

According to (8.49), the Ricci scalar R and R of F and a are related by R = R + abmRymyvVv

+ ^j{a2(bmlp)2

+ 2(6 m | p 2 /") 2 }.

(9.33)

Assume that F is of constant Ricci curvature A. Then \{a + 0)2 =R + abmRymyPyP

+ _ L - | a 2 ( 6 m | p ) 2 + 2(6 m | p 2 / p ) 2 }.

Finsler Spaces of Constant

152

Curvature

We obtain bmRymyPyp

= 2X0,

R+ ^j{a2(bmlp)2

(9.34) + 2(bmlpyP)2}

=X(a2+p2).

(9.35)

^ V + — [ { M 6 m | P ) 2 + bm\ibm\j} = 2\(6ij + bibj).

(9.36)

Equation (9.35) is equivalent to the following

Contracting (9.36) with bi and bj and using (9.34), we obtain (bm]p)2 = (n-l)\\\(3\\2.

(9.37)

From (9.37), we see that A > 0. Plugging (9.37) into (9.35) or (9.36), we obtain R = \{(1-

||/3|| 2 )a 2 + P2} - ^(bm\Pypf-

(9-38)

or equivalently, Ryiyj = 2A{(1 - ||/3|i 2 )^ + bibj} - ^jbmlibmlj.

(9.39)

Note that if A = 0, then bm]p = 0 by (9.37) and R = 0 by (9.38). Thus a is Ricci-flat and /? is parallel. Conversely, assume that the Ricci scalar R of a satisfies (9.38), one can easily show that (9.34) and (9.35) hold. We have proved the following Proposition 9.3.2 Let F = a + j3 be a Randers metric where P is a Killing form of constant length. F is of constant Ricci curvature A if and only if the Ricci scalar R of a satisfies (9.38). In this case, either A > 0 or A = 0 (hence a is Ricci-flat and (3 is parallel).

Chapter 10

Second Variation Formula

In previous chapters, we have introduced several important curvatures for Finsler metrics. We have also discussed some relationships among these quantities. In this chapter, we are going to introduce another nonRiemannian quantity — the T-curvature. This quantity has a close relationship with the distance function.

10.1

T-Curvature

Let (M,F) be a Finsler space. Given a vector y 6 TXM \ {0}. Extend y to a geodesic field Y in a neighborhood U of x. Let D denote the Levi-Civita connection of the induced Riemannian metric g = gy on U. Define T„(«) := g„(D„V,y) -gx(i>„V,y),

v G TXM,

where V is a vector field with Vx = v. T = {Ty}yeTM\{o} T-curvature. T is also called the tangent curvature in [Sh4]. From the definition of T we have Tv(Xv) = X2Ty(v),

(10.1) is called the

A> 0

and T v (l/) = 0. Now we are going to derive a formula for Ty(v) 153

in a local coordinate

Second Variation

154

Formula

system. First, contracting (5.28) with yl gives , / „ rk

-

1 d 9 j l

i d9ml

f

d

9jm\

r,r

,„ ,

rk

Observe that dfjij _ dgij

dY

# = igf<m*w)dx

k'

Since Y = Y%-^-z is geodesic, Y satisfies BY1 Y -— + 2Gi(Y)=0. k

(10.3)

We obtain vVidM _ = ±Y*^(Y), k k

(10.4)

Y

(10.5)

dx

dx

= Yk^(Y)-ACi3k(Y)Gk(Y).

"&

Let f} fe denote the Christoffel symbols of g. By (10.2), (10.4) and (10.5), we obtain v^ 9kl rk m > ~ =

1 dg l

( i 2\dx™

.

d9ml

+

\{B%{Y)

dxi

dffj" 1 vi dxi V

+

°ixT{Y)- l&W}*1

+

2C

^n(Y)Gk(Y)

Ylgkl(Y)Tkm(Y)

=

In particular, the following holds ylgkl(x)tkm(x)

= ylgkl{y)Tkjm{y).

(10.6)

Thus gx(pvV,y)

= ylgkl(x){dVk(v) =

+

rkrn(x)^vm}

Vl9ki(v){dVk(v)+r*m(y)v*vm}.

(10.7)

On the other hand, K„(pvV,y)

= ylgki(y){dVk(v)

+ Tkm{v)v^vm).

(10.8)

T-Curvature

155

From (10.7) and (10.8), we obtain T »

:= ylgkl{y){vkjm{v)

- Tkm{y))^vm.

(10.9)

From (10.9), we see that Ty(v) is independent of the extensions of y and v. Proposition 10.1.1

P = 0 if and only i / T = 0.

Proof. Suppose P = 0. By (7.13), we see that Tjm(y) y. (10.9) immediately implies T = 0. Suppose that T = 0. In local coordinates,

are independent of

yl9ki(y){rUv) - r* m (y)}^ m = o.

(10.10)

This implies that Vjk(v) are independent of v. Hence P = 0. The proof is left for the reader. Q.E.D. The T-curvature is defined for Finsler metrics. There is another interesting quantity which depends only on the spray of the Finsler metric. Let F b4 a Finsler metric and

be a spray of F. Let Nl(u,v)

:= vjNUu) - ujNUv) = vj^-^(u)

-

uj^—(v).

For a pair of vectors (u,v) G TXM x TXM, define N(u,„) := One can verify that N : TxMxTxM it is anti-symmetric,

N\u,v)^-\x. -» TXM is a well-defined map. Further,

N(u, v) +N(v,u)

= 0,

and satisfies the following homogeneity N(Xu,v) = XN(u,v), Proposition 10.1.2

N(u,Xv) = AN(u,u),

P = 0 if and only if N = 0.

The proof is left for the reader.

A > 0.

156

Second Variation Formula

10.2

Second Variation of Length

In this section, we will derive the second variation formula for a geodesic. The geometric meanings of the Riemann curvature and the T-curvature lie in this important formula. Let (M,F) be a Finsler space. Take an arbitrary local frame {e*}f=1 for n*TM. Let { w > n + , } ? = 1 denote the local coframe for T*(TM \ {0}), corresponding to {e;}. The correspondence is given by (5.18). For vectors X, Y on TM \ {0} and a section U = Wej of T T T M , define Q{X, Y)U := Ujn/(X,

Y) e*.

Let V denote the Chern connection on n*TM. By (8.3), we have v

x V Y U ~ v f vx&

~ V[x,Y}U = n(X, Y)U,

where U e C°°{TV*TM) and X, Y e C°°(T(TM \ {0})). Let {ei,ei}f =1 be the local frame for T*(TM \ {0}) that is dual to { w ^ w " ^ } ^ . We have n(e fc) e«)ej

=

R/kl

a

(10.11)

n(e fc ,e,)ej

=

P/ f c , *.

(10.12)

For a vector o n T M \ { 0 } , X = Xiei + Yiei, we always denote X =

Xiei.

Let V = Viei + Uiei,

f = Tlei + Slei

be vector fields on TM \ {0} . Fix a vector y e TXM \ {0}. Assume that T% = yl,

S% = 0,

where yl are determined by (y,y) = yzei.

Second Variation of Length

157

By (8.24) and (8.29), we obtain g(n(V,f)V,f)\y

V^Vkyiylg(n{ek,el)ej,el)

=

+V^Ukyiylg(n(ek,el)ej,ei) =

V>Vk ylgipR3pklyl

=

V*Vk y'gipR;,

- V*Uk + VWk

yigtpPjplkyl y'g^P^

k

=

-VJV gjpR\.

We obtain an important identity = -VWk gjpR\.

g(n(V,f)V,f)\y

(10.13)

Let {b,;}™=1 be the basis for TXM determined by ei\y = G/,bi). Let V := V^i € TXM. We rewrite (10.13) as follows g(n(V,f)V,f)\y

= -Sy(Ky(V),v)

(10.14)

Let c : [a, b] —> M be a geodesic with F(c) = A > 0. Consider a piecewise C°° variation of c if : (-£,£) x [a,b] ->• M More precisely, there is a partition a = to < • • • < tk = b such that (a) # is C° on (-£,£); (b) H is C°° each (-£,e) x \ti,ti+i) and (c) # ( 0 , i ) =c(t), a
x

dH

The map if : (-£,£) x [a, 6] - 4 T M

Second Variation

158

Formula

is not C° on (—£,e) x [a, b], but it is C°° on each (—e,e) x [£i,£i+i]. i? is not C° on (e,e) x [a, 6] unless H is C°°. Let -

V:

dW d d2H{ { dt dx dt2 dH* d d2^ „s n d 9a;,* + dsdt

dH dt dH = — 5s =

d dyi d dyl

Then -

dH*

T:

d

-

dW V:=

-dTdi-

=-df "

Let

v(t)--=v^£Mt) = f{0,t). Then f(0, t) = (c(t), c(t)),

V(Q, t) = (c(t),

V{t)).

Let V denote the Chern connection on ir*TM. The torsion-free condition (5.21) is equivalent to that V ^ y - VyX

= {X,Y}.

(10.15)

Without loss of generality, we may assume that T and V can extended to smooth vector fields on TM \ {0}. Then [T, V] is independent of any extensions. Observe

\fv]-(d2Hi

d2Hl

1

dsdt) dxi

'

J

\dtds

\

d

(d3Hl

d3Ri

\dt2ds

dsdt2)

\

d

dyi

By (10.15), VyT=VfV.

Now we consider the following length function

(10.16)

Second Variation of Length

159

We obtain L'(s)

= J

g{f,f)-V2g(Vvf,f)dt

= J g{f,f)-l'2g(vfV,f).

(10.17)

Differentiating (10.17) and using (10.16) yield pb

L"(s)

g(f,f)-1f2{g(V^tV,f)+g(^tV,Vfv)}dt

= J

g{t,f)-*l2g(vtV,t)2dt

- J = J

g{f,f)-ll2g(si(V,f)V,f)dt

+ J g(f,f)~1/2{g(VfVyV,f) 2

g(f,f)-3'2g(yfV,f)

-J

dt +g(vfV,Vfv)}

dt

Now we deal with the above identity along c term by term. Since c is a geodesic, A,

^

9H* ,n , d =

d2H\n

.d

c\t)-^- 2Giv(c(t))^= c{t) —l . v v 'dxl " dyl Sx

Thus T| s = 0 is horizontal. Equation (10.14) gives g(Q(V,f)V,

f)|,=o =

-Sc(t)(^nt){V(t)),V(t)).

Observe that g(vfVyV,f) First, we study VfT.

= f [fl(VvV-,f)] - 5 ( v ^ , V f T ) .

(10.18)

160

Second Variation

Formula

We obtain V t f | . = 0 = {c + 2Gi{c)}di\c = 0. Now we fix t with a < t < b. Let at(s) := H(s,t). Similarly,

We obtain

V^V-|fl=0

=

{al + &iTjk{c)dkt]di\t

=

[a1 + 2Gi(&t)}di\6

+ {Fjk(c) - Fjk(at)}&ia?

d^.

Assume that at has constant speed with F(&t(s)) =: Ct- Then the geodesic curvature Kt of at (s) at s = 0 is given by

- = ^ { ^ ) + ^(0))}£-

o-«(0)-

This gives 5(v^V-,f)|i=o

= (C t ) 2 gc W («t,c(t)) - T fi(t) (<7 t (0)).

Note that the above function is piecewise C°°. Thus

J g(f,fr1/2g(vtVyV,f)\^0dt A" 1 (Cbf

=

g£(6) («6, C(6)J - (C„) 2 g c ( a ) ( « a , c ( a ) J

-A" 1 T c ( f c ) ( ^ ( 0 ) ) - T , ( a ) ( c r a ( 0 ) ) . Observe that

We obtain

V t V-|, =0 = {^

+

™$(c)}ft|

Second Variation of Length

161

This gives

g (ytv, v f v) | s=0 = gc(t) (DC^W, DdK(t)), 5(vtV-

, f ) | s = 0 = gc(t)(DcV(t), c(t)).

Let F x ( t ) := K(i) -

X~2gc{t)(V(t),c(t))c(t).

We have gc(t)(De ( t )^ X (t),D ( ; ( t ) K X (t)) = gc(t)(pnt)V{t),B m V{t))

-

A-2gi(t)(vmV{t),c(t)).

We obtain the following Second Variation Formula: L"(0)

=

A-1/

{gi(D^,D^-L)-gc(Rc(^x),^i-)}^

+A- 1 [F 2 (y(6))g ( i ( 6 ) (K 6 (0), C (6)) -F 2 (V(a)) g ( i ( a ) (rv a (0),c(a))] +A" 1 [T £ ( a ) (V(a)) - T 6(6) (V(6))].

(10.19)

The second variation formula is derived in [AbPa]. But the T-curvature is not clearly defined. The second variation formula for variations with fixed endpoints is derived in [BCSl] in a different way. Let c(t), a < t < b, be a unit speed geodesic in (M, F). Take a special geodesic variation H(s, t) of a geodesic c(t) such that H(s, a) = c(a) and
L"(0)

=

/

{gc(DeVx,D^x)-K(P,c)ge(v±,Vx)}dt

Second Variation

162

Formula

-T£(6)(
(10.20)

where P(t) = span{c(£), V(t)} is a family of tar gent planes along c. Thus the behavior of the geodesies issuing from a point depends on the flag curvature and the T-curvature. 10.3

Synge Theorem

The second variation formula has many applications. In this section, we will apply this formula to prove that any even-dimensional compact Finsler space with K > 0 has simple fundamental group m = {0} or Z 2 (Synge Theorem). First we prove the following Lemma 10.3.1 Let (M,F) be an even-dimensional, oriented Finsler space and c be a closed geodesic. There is at least one parallel vector field W along c, which is not tangent to c. Proof. Let c : [0,r] -> M denote the closed geodesic with c(0) = x = c(r) and Pc denote the parallel translation along c. By Lemma 5.3.1, Pc : TXM —> TXM is a linear isometry with respect to gy, where y — c(0) = c(r), i.e., g„ (P c («), P c (v)) = sy(u, v),

u,ve TXM.

(10.21)

Let

/ : = f e W gy{y,v) = 0.}. It follows from (10-21) that Pc : y1- -> y1- is a linear isometry. Since dimj/ x = n — 1 is odd, there is at least one vector w G y x such that Pc(w) — w. Let W(t) be the parallel vector field along c with W(0) =w = W{r). Then W(0) = Pc(w) = W(r). Thus W must be a smooth parallel vector field along c.

Q.E.D.

Lemma 10.3.1 has important application. Theorem 10.3.2 (Synge, 1926) Let (M, F) be an even-dimensional closed oriented Finsler manifold with K > 0. Then M is simply connected.

Synge

Theorem

163

Proof. Suppose that TTI(M) contains a non-trivial element a. By Proposition 12.3.1 below, there is a closed geodesic c G a with L(c) = \a\. Assume that c(i), 0 < t < \a\, is parametrized by arclength. It follows from Lemma 10.3.1 that there is a parallel vector field W{t) along c. Take a variation H : {-£,£) x [0, |Q|] -> M of c such that

W(t) =

^-(0,t).

Applying the second variation formula to

and using the fact that Wc(0) = Wc(|a|); we obtain

L"(0) = -Ja

gm (Rc(t) (W(tj), W(t))dt < 0.

Thus cs(t) := H(s,t) is shorter than c for small s > 0. Since cs is homotopy equivalent to c, we have \a\ < L{cs) < L(c) = \a\. It is a contradiction, since c is assumed to be the shortest closed curve in a. Q.E.D.

Chapter 11

Geodesies and Exponential Map

Geodesies are the most important objects in a Finsler space. In this chapter, we will define the exponential map and use it to discuss some basic properties of geodesies. We will show that the geodesic completeness is equivalent to the metric completeness (Hopf-Rinow theorem).

11.1

Exponential Map

In this section, we will define the exponential map and discuss its regularity. The exponential map plays a very important role in comparison geometry. Let (M,F) be a Finsler space and ir : TM —> M denote the natural projection. For a vector y e TXM \ {0}, let cy(t) denote the integral curve of the induced spray G in TM \ {0}. By definition, cy(t)

:=irocy(t)

is a geodesic in M with cy(Q) = y. In a standard local coordinate system (xl,yl) in TM, geodesies are characterized by the following equations
(11.1)

By the O.D.E. theory, cy smoothly depends on y € TM \ {0}. The homogeneity of G implies that for any A > 0 and y G TM \ {0}, c\y(t) = cy(\t),

Vf>0.

Let U(M) denote the set of all tangent vectors y € TM such that cy is 165

166

Geodesies and Exponential

Map

defined on [0,1 + e) for some e > 0. Define exp : U(M) ->• M by exp(y) := cy(l). We call exp the exponential map. Since cy smoothly depends on y € TM \ {0}, we conclude that exp is C°° on U{M) \ {0}. For a point x £ M, let expx := exp \UXM • UXM -> M. expx is C°° on UXM - {0}, where UXM := UM n T X M. We call exp x the exponential map at a;.

For any vector y £ UXM, we identify TyUxM with T X M in a natural way. Then the differential of exp x at a vector y G UXM is a linear map ^(expJij, : T^M -> T Z M,

z=

expx(y).

Thus, d(exp)| 0 :TXM

-*TXM

is an endomorphism, provided that exp^ is differentiable at the origin of TXM. The problem is whether or not exp^. is C°° at the origin. This problem was answered by Whitehead. Theorem 11.1.1 (Whitehead [Wh]) Let (M,F) be a Finsler space. The exponential map exp is C1 on the zero sections of •K : TM —> M and for

Jacobi Fields

167

any x G M, d(exp x )| 0 : TXM —> TXM is the identity map at the origin 0 G TXM. The proof is technical, so is omitted. See [BCSl] for details. For any precompact open subset A c M, let Ur(A) denote the rneighborhood Ur(A) of A in TM, Ur{A) := {
A}

C TM.

For a point a; G M, let B r (:r) := Ur({x}), i.e., B r (z) := {y G TXM, Fx{y) < r } C T X M. By Theorem 11.1.1, we immediately obtain the following Corollary 11.1.2 Let (M,F) be a Finsler space. For any precompact open subset A C M, there is a positive number r0 = r0(A) > 0 such that for any z G A, exp z : TZM -> M is a C1 -diffeomorphism on B 2 (r) C TZM onto its image for all 0 < r < r0. In Section 11.3, we will show that expx[Bx(r)] coincides with the metric ball B(x,r). 11.2

Jacobi Fields

The Riemann curvature is defined via geodesic variations. Thus we will use geodesic variations to study the relationship between geodesies and the Riemann curvature. Let (M, F) be a Finsler space. Let c : [a, b] -¥ M be a geodesic and H : (—e,e) x [a,b] —• M a geodesic variation of c, namely, each cs(t) := H(s,t) is a geodesic. Let J(t) := Q?L(Q,t). By Lemma 6.1.1, we know that J(t) satisfies DeDe J + Rc( J) = 0.

(11.2)

Equation (11.2) is called the Jacobi equation of c. Conversely, for every vector field J(t) along c satisfying the Jacobi equation (11.2), there is a geodesic variation H of c whose variation field is equal to J{t). Thus we call a vector field J{t) along c satisfying (11.2) a Jacobi field.

Geodesies and Exponential

168

Lemma 11.2.1 F(c) = A. Then

Map

(Gauss) Let J(t) be a geodesic along a geodesic c with

gc(t) (•/(*). c(t)) = A2(a + bt),

g d ( t ) (p6J(t),

c(t)) = A26.

(11.3)

where Jx(t)

:=J{t)-{a

+ bt)c(t)

is also a Jacobi field along c such that J±(t) to c(t) with respect to g^t)-

and DcJ x (i) are orthogonal

Proof. Observe that |[gc(t)(j«,c(*))]

=

gc W (Dc7(t),c(t))

_[g.(t)(j(i),c(*))]

=

gc(t)(DcDc7(t),c(t))

=

-gi{t)(^c(t)(J(t)),

=

-gc(t)(^(*), Rc(t)(c(<)))=0.

C(t))

Thus (11.3) holds for some constants a,b. Plugging J{t) = J±(t) + {at + b)c(t) into (11.2), we obtain D£DeJx(i)+Re(J±)=0. Thus J x ( t ) is also a Jacobi field along c. By (11.3), gc(t) (jHt),

c(*)) = 0,

ge(t) ( D , ^ ( i ) , c(t)) = 0. Q.E.D.

Fix a unit vector y e TXM and let c(i) := expa.(iy), 0 < £ < a. Consider a special geodesic variation H(s, t) := expx[t(y + sv)}),

0 < t < a, |s| < e.

Jacobi Fields

169

By Lemma 6.1.1, J(t) := ^ ( 0 , t ) = d(expx)\ty(tv)

(11.4)

is a Jacobi field along c. Namely, J(t) satisfies (11.2) on (0, a). Since exp x is C 1 at the origin, J(t) is C° at t = 0. We can extend c to a geodesic c on (-S, a) for some 6 > 0. Take a number r close to 0 with 0 < r < a. There is a unique Jacobi field J along c satisfying J{r) = J ( r ) ,

D £ J » = D, J(r).

By the uniqueness, J(t) = J(i),

0 < * < a.

Thus J(i) can be extended to a Jacobi field J(i) along c. By this observation, we can prove the following L e m m a 11.2.2 The Jacobi field J{t) in (11.4) JS C°° along c{t) = expx(ty), 0 < t < a. It satisfies the following initial conditions: J(0) = 0,

Dd J(0) = v.

Proof. First observe that J(0)=d(expx)|o(0)=0. By the above argument, we may extend J to a Jacobi field J along a geodesic c(t), —6 < t < a. Write j(t) = tW(t),

-6


where W(t) is a C°° vector field along c and W(t) = d{expx)\ty(v),

0
Observe that BrJ(t) = Note that W(t) = d(expx)\ty(v),

W(t)+tBcW(t). t > 0, satisfies

W{0) = lim. d(expx)\ty(y)

= d(expx)l0{v)

= v.

Geodesies and Exponential

170

Map

We obtain Di J(0) = W(0) = v. Q.E.D. Let c : (—£, e) —> M be a unit speed geodesic with c(0) = j / G T p M. For a vector i> € T P M, let J be the Jacobi field along c satisfying J(0) = 0,

D d J(0) = v.

Consider the following function f(t):=Si(t)(J(t),Jt)). Define Rc ( t ) : Tc{t)M -> T c ( t ) M by R* ( t ) (n*)):=Dc( t ) [Rc(t)(V r (t))], where V(£) is a parallel vector field along c. Similarly, we can define Rc(t)By the Jacobi equation (11.2), we obtain /'

=

2g6(D,J,j),

/"

=

2ge(DeJ,Dej) - 2g, ( R , ( J), j ) ,

/(3)

=

-8ge(Rc(J),Dcj)-2ge(Re(J),j),

/(4)

=

-8ge(Rc(DeJ),Dej)+8ge(Rc(J),Ri(J)) -12g£(Ri(J),Dcj) -2ge(R,(J), j ) .

We obtain the following P r o p o s i t i o n 11.2.3

T/ie aftoae Jacobi field J(t) satisfies

gc(t) (-/(*), . 7 ( 0 ) = gy(«, ^

Sd(t)(D6J{t),J(t))

- \sy ( R J M v)tA + o(* 4 ),

=Sy(v,v)t-lgy(Ry{v),vy

+ o(t:i).

(11.5)

(11.6)

At the beginning, we have mentioned that exp is C°° on U(A) \ {0}. Now we discuss the regularity of exp on the set of zero sections in TM.

Minimality

of Geodesies

171

We first consider the case when F is a Berwald metric, i.e., the geodesic coefficients Gl(y) = ^Yljk{x)y3yk are quadratic functions in y G TXM for all x G M. Geodesies are characterized by dV

„; . s dcj dck

n

By the O.D.E. theory, cy(t) smoothly depend on y G TM (for small t). Thus exp is C°° at y — 0 G TM. In particular, exp x is C°° at y = 0 for all x e A. Assume that exp x is C2 at y = 0 for all a;. Let f(y) := exp(y). By definition, for any r > 0 and any precompact open subset A, there is a number e > 0 such that c(t) = f{ty), 0 < £ < l + £ , is a geodesic for every y G TXM, where x € A, with F(j/) < r. Substituting c into the geodesic equation (11-1) and letting t = 0 yields

In other words, Gl(y) are quadratic in y G TXM. By definition, such a Finsler metric is a Berwald metric. We have proved the following Theorem 11.2.4 (Akbar-Zadeh [AZ]) Let (M,F) be a Finsler space. The exponential map exp is C2 at zero sections if and only if F is a Berwald metric.

11.3

Minimality of Geodesies

In Section 5.1, we define a geodesic to be a critical curve of the arc-length function of curves joining two fixed points. In this section, we shall prove that geodesies are locally minimizing curves. We have the following Proposition 11.3.1 Let (M,F) be a Finsler space. Suppose that expx : TXM —*• M is a C1 diffeomorphism on B x (r) C TXM. Then for any y G SXM, the radial geodesic c{t) = expx{ty),

0 < t < r,

Geodesies and Exponential

172

Map

has minimal length r among all piecewise C1 curves in M joining x to z = c(r). Moreover, for any 0 < t < r, the exponential map restricted to Bx(t) C TXM and Sx(t) = dBx(t) are diffeomorphisms onto the metric t-ball B(x,t) C M and S(x,t) = dB(x,t) respectively, exp, : Bx(t) -> B(x, t),

exp x : S x (t) -> S{x, t).

(11.7)

Proof By Corollary 11.1.2, we may assume that for some r and e > 0, exp x is a C1 -difFeomorphism from B x (r + e) C TXM onto its image. Take an arbitrary piecewise C 1 curve a : [0,1] -> M,


= X = C(0),

cr(l) = z = c{r).

First we assume that a is contained in exp^B^fr + e)\. Then a(s) = expx[t(s)y(s)]

y(s) e SXM,

where t = t(s) is a piecewise C°° function with t(0) = 0, t(l) = r and 2/ : (0,1] -»• S X M is a piecewise C°° map with y(l) = y. It is easy to show that lim

y(s)

=
s->0+

Hence, we may set y(0) :=
0 < s < 1, 0 < t < r ,

such that a(s) =

H(t(s),y(s)).

Minimality

of Geodesies

Let

a* "

of Then

a = y— + v. as

For each s € [0,1], Vs(t) := V(t, s) is a Jacobi field along cs{t) := exp x (*i/(s)),

0 < t < v.

V(t) satisfies

V.(o) = o,

D6svs(o) = y(s)^-y(s).

By the Gauss Lemma (Lemma 11.2.1), Vs(t).Lcs(t) with respect to Sc.{t)Thus SY(Y,V)=Q.

(11.8)

Recall (1.14) g„(l/, w) < F(y)F(w)

\/y ^ 0 .

(11.9)

Taking w = a(s) in (11.9) and using (11.8), we obtain F(&) > SY(Y,&)

= gy(Y,Y)^s+Sy(Y,V)

= £.

Hence, L(a)=

f F(d)ds> Jo

f ^-ds = t(l)-t(0)=r Jo as

= L(c).

174

Geodesies and Exponential

Map

This shows that if a is contained in expx[Bx(t+e)}: then its length is greater than or equal to that of c. Now we suppose that a is not completely contained in expx[Bx(r + e)]. Let 0 < s0 < 1 be the smallest number such that a(s0) G expx[Sx(r)]. Then by the above argument, the length of cr|[o,s„] is n ° t less than r. We obtain L(a) = L(a\[0iSo]) + Z-(a|[So,i]) > L(a\[0iSo]) >r = L(c). Thus c has minimal length among all piecewise C 1 curves joining x to z. Now it becomes obvious that (11.7) holds for 0 < t < r. Q.E.D. Corollary 11.3.2 Let (M,F) be a Finsler space and c{t), a < t < b be a geodesic. Then for any t0 (= (a, b), there is a small interval [t0 — e, t0 + e\ C (a, b) on which c has the smallest length among all curves issuing from c(t0 - e) to c{t0 + E). Proof. Take a sufficiently small e > 0 such that exp^. is a diffeomorphism from Bx(r) C TXM onto B(x, r) C M for x — c(t0 - e) and r > 2e. This is guaranteed by Corollary 11.1.2. Note that for y = c(t0 — e), exp, (ty) = c{t0 -e + t),

Vt G [0,2e].

By Proposition 11.3.1, we have d(c{t0 - e), c(t0 -

e

+ tj)=

tF(y),

Vi e [0,2e].

Thus c is minimizing on [t0 — e, t0 + e}.

Q.E.D.

Now we study distance functions in a Finsler space (M,F). the distance function from x £ M, dx(z) := d(x,z),

Consider

z e M.

According to Theorem 11.1.1, exp x is C°° on TXM — {x} and C 1 at y = 0 G TXM. By the local Minimality of geodesies (Proposition 11.3.1), we see that for z nearby x, dx(z)

=Fx(exp-l(z)).

Thus dx is C°° nearby x and C 1 at x. A natural question arises: Is dx always C°° at xl This question is answered in the following

Completeness

of Finsler Spaces

175

Proposition 11.3.3 Let (M,F) be a Finsler space. Suppose that d\ is C2 at x, then Fx is Euclidean at x. Thus dx is C2 for all x G M if and only if F is Riemannian. Proof. Fix an arbitrary basis {b;}™=1 for TXM. Let