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0 and c(ip(t)) = c(t), Vt € [ti-i.tj. A (piecewise) C°° curve C in a manifold M is an equivalence class of regular (piecewise) C°° maps from an interval / into M. For the sake of simplicity, we do not distinguish a regular map c = c(t) and the curve C represented by c. For a C°° curve (represented by) c : / = [a, b] —> M, its reverse c_ : I —> M is defined by c_(£) := c(6 + a - t). The class represented by c_ is different from that represented by c. All C°° curves in this book are oriented. Consider a piecewise C°° curve C from p to q in (M, F). Let C be a piecewise C°° curve represented by c = c(t) with c(a) = p and c(6) = q. The length of C is defined by
Figure 1.4 If C is represented by another map c = c(f) with c(a) = p and c(b) = q, then there is a positive function i = (p(t) such that c(t) = c(i) with a =
c(<) = 5(t)p'(t)-
16
Finsler Metrics
We have
J F(c(t),c(t))dt = j
F(c(t),KW(t))dt
= J F(c(t),Kt))
dF(p,q):=inf£F(C),
where the infimum is taken over all piecewise C°° curves C issuing from p to q. dF is a function o n M x M with the following properties: (a) dF(p,q)>0; (b) dp(p, q) = 0 if and only if p = q; (c) dF{p, q) < dF(p, r) + dF(r, q). The proofs of (a) and (c) are elementary. To prove (b), it suffices to prove the following fact [55]. At every point xo € M, there is a local coordinate system
U cRn such that A- 1 !^)! < F(x,y) < X\(y% — sqxp" R n + 1 . f(x) is an eigenfunction corresponding to the first eigenvalue \i = n. In fact, it satisfies a stronger PDE: (6-12) J,
y = y * ~ € TXU,
where A > 1 is a constant. Choosing a smaller neighborhood U if necessarily, one can easily show that there is a constant C > 1 such that C-1\V(x2) - p ( n ) | < dF(xux2) < C\
(1.25)
where x\,x2 G U. This implies (b). The function dF is called the distance function of F. Let A C M x M denote the diagonal. It can be shown that dF is C°° on U \ A for some neighborhood U of A in M x M. The proof is very technical, so it is omitted here. Conversely, dF determines the Finsler metric F by F(Xyy)
=
hm M^c(e))^ e—>0+
y e r:cMi
£
where c(t) is an arbitrary C°° curve with c(0) = x and c(0) = y.
Length Structure and Volume Form
17
Figure 1.5 Clearly, if F is reversible, then the induced distance function dF is reversible, i.e., dp(p,q) = dF(q,p), for all pairs of points p,q S M and vice versa. Every Finsler metric F = F(x,y) on an n-dimensional manifold M defines a volume form. At a point x e M, let {bj} be a basis for TXM and {0z} be the basis for T*M dual to {b^}. Then the following n-form at x € M is, up to an orientation, well-defined, dVF:=aF{x)O1A---A0n, where , x aF{x) :=
Vol(B"(l)) -; r-. (1.27) Vol(( 2 / i )eR«| J F(2r,^b i )
18
Finsler Metrics
First, let us consider a Riemannian metric a on an n-dimensional manifold M. Let a= ^aij{x)yiy^
y =yi—i\x
eTxM.
Let A be a matrix such that ATA = (a,ij). Then the linear transformation x = Ay : R™ —» R n sends the convex domain Ux := {(yl) 6 R-" I ^aij(x)yiyi < 1} onto the unit ball B n (l). We may assume that the Jacobian matrix has positive determinant det(^4) > 0. We have det(A) = -Jdet(a,ij(x)). Observe that Vol(B n (l))= /
7B"(I)
-
/
Jux
dx1---dxn
det{A)dy1
•••dyn
= v
Jdet{atj(x))Vo\{Ux).
That yields, Vol(», ) =
V
°" B "< 1 » . Wdet (aijix))
Then dVa = aa(x)dx1 • • • dxn is given by "aOO = v/det (aijix)). Consider a Randers metric F = a + f3, where a = y/aij(x)yiyj is a Riemannian metric and j3 = bi{x)y% is a 1-form on an n-dimensional manifold M. Let Q,x := {(y*) e Rn | F(x,y1—— \x) < 1}. By an elementary argument using linear algebra, we obtain Vol B 1
vow = (l ~
II/3,II2Q)
^ ^ ><>
ydet (oy(x)j
where ||/3x|U denotes the norm of /3 at x with respect to ax. Plugging the above formula into (1-27) yields /
n \(«+l)/2
19
Length Structure and Volume Form
Thus /
o\(n+l)/2
dVF = (1 - \\(3x\\l)
dVa.
(1.28)
Note that for any open subset Q C M,
f dVF< [ dVa. Equality holds if and only if F = a. Example 1.3.1 Consider the Randers metric F = a + (3 on the unit ball B n (l) C R", where a = ^/aij{x)yiy^ and /? = bi(x)yl are given by a
' =
V\y\2-(\x\2\y\2-(x,y)2)
T^W
„ ^ {x, y) P ' l-\x\2
'
(a,y) l + (a,x)'
where y e TxRn ^ R™, a € B n (l), | • | and ( , ) denote the standard Euclidean norm and inner product in R™, respectively. When a = 0, F is the Funk metric denned in (1.15). Applying Lemma 1.1.1 to the following matrix
we obtain det(M =
(1_|l
1
|a)n+r
Then dVa = ( l - | x | 2 )
2
dx1---dxn.
By Lemma 1.1.1,
aij = (1 - \x\2){sij - x^. Then the norm of (3 is given by
||/y a = y/aVixM'Mx) = 1 - ( 1 ^ 1 'f ( ) f^ ) l a | 2 ) -
(1-29)
20
Finsler Metrics
Plugging the above formulas into (1.28) yields
dVp
1
t |2
n+l
= L i 7 m l 2 ^ • • • ^nL(l + (a,a:)) 2 J
1.4
Navigation Problem
In this section, we will discuss Randers metrics from a navigation point of view. We shall see that non-Riemannian metrics are not avoidable even though we live in a Riemannian world. Consider an object moving in a metric space, such as Euclidean space, pushed by an interval force and an external force field. The shortest time problem is to determine a curve from one point to another in the space, along which it takes the least time for the object to travel. This problem in some special cases was studied by E. Zermelo [102], hence called the Zermelo navigation problem. Here we shall discuss the navigation problem in the most general case. Suppose that an object on a Finsler manifold (M, $) is pushed by an internal force U with constant length, $(x, Ux) — c, and while it is pushed by an external force field V with $(£, — Vx) < c. The combined force at x is Tx :— Ux + Vx. The condition, $(x, —Vx) < c, guarantees that the object can move forward in any direction.
Figure 1.6 Due to the friction, the object moves on M at a speed proportional to the combined force T. For the sake of simplicity, one may assume that c = 1 and the velocity vector at any point x e M is equal to Tx. Given a pair
21
Navigation Problem
of points p,q £ M, let C be an arbitrary piecewise C°° curve in M. Since $(x, Ux) — 1, we have (1.30)
$>(x,Tx-Vx)=${x,Ux) = l.
On the other hand, for any vector y € TXM\ {0}, there is a unique solution F = F(x, y) > 0 to the following equation (1.31)
*(x,?L-Vx)=l. Observe that for any A > 0,
i-$(x
A
^
rV-fYx
A
^
r)
By the uniqueness,
F{x,Xy) = XF{x,y). One can show that Fx := F\TXM is a Minkowski norm on TXM. Thus JF = F(x,y) is a Finsler metric on M. Comparing (1.30) and (1.31), one can see that the combined force Tx has unit F-length, F(x,Tx) = l.
(1.32)
This observation leads to the following Lemma 1.4.1 Let (M, $) be a Finsler manifold and V be a vector field on M with $(x,-Vx) < 1, Vx e M. Define F : TM -» [0,oo) 6y fi.51;. For any piecewise C°° curve C in M, the F-length of C is equal to the time for which the object travels along it. Proof. Let c : [0, to] —> M be the parametrization of C such that the velocity vector c(t) — Tc(t). Then t0 is the time for which the object travels along C. It follows from (1.32) that F(c(t),c(t))=l. This implies
to = J° F(c(t),c(t))dt = £F(C). Q.E.D.
22
Finsler Metrics
For a pair {$, V} on a manifold M, where $ = $ ( i , y) is a Finsler metric and V is a vector field with $(x, — Vx) < 1, we define a Finsler metric F = F(x, y) by (1.31). The Finsler metric F can also be defined in the following way. First, define $* and V* on T*M by
$*(*,£):= sup J ^ L ,
V(O:=£(f*),
£erx*M.
Then F* := $* 4- V* is a co-Finsler metric on M and i*1 is dual to F*, i.e., F
^^)=
SU
P
J^TY
The proof is left to the reader. Lemma 1.4.2 Let $ = $(x,y) be a Finsler metric on an n-dimensional manifold M and V = V%{x)-g~ be an arbitrary vector field on M with $(x, —Vx) < 1, x e M. Let F = F(x,y) denote the Finsler metric on M defined by (1.31). Then the Finsler volume forms of F and $ are equal, dVF = dV*.
(1.33)
Proof. Fix a basis {b4} for TXM and let Vx := t/bj. Let
W* :={(!,*) €R n | $ ( x , y i b i ) < l } , UF : = {(i/') G R" |F(x, 2 / i b,)
Q.E.D.
The above proposition shows that the volume of an open subset on a Finsler manifold is not disturbed by any vector field. Example 1.4.3
Let
U:=[yeRn\
23
Navigation Problem
Let $(x, y) :=
i€M.
^ R n \ {0}, define 0 = G(x, y) > 0 by
*(af'9(b-V-)=1-
^
Then 0 = 0(x, y) is a Finsler metric on U, which is called the Funk metric on U. The Funk metric 9 = O(x, y) can also be defined by e * • • = * + S7^-T 6(x,y)
aW
-
Figure 1.7 Equation (1.34) can be written as Q(x,y)=<j>(y + e(x,y)xy
(1.35)
Differentiating (1.35) with respect to xk and yk respectively, one obtains (l -<j)wi(w)xljQxk{x,y)
= (j)wk(w)Q{x,y),
(1.36)
(l-
=
(1.37)
where w := y + ©(z,y):r. It follows from (1.36) and (1.37) that Qxk=eeyk.
(1.38)
24
Finsler Metrics
The above argument is given by T. Okada [77]. If (j) = |y| is the standard Euclidean norm on R", U = B n (l) is the unit ball in R". In this case, 6 = F as defined in (1.15). Given a Riemannian metric h = yjhij{x)ylyi and a vector field V = V*(X)^L on a manifold M with /i(x, -V x ) = yfJi~(3?jV^xjW(x) < 1, one can define a Finsler F = F(x,y) by (1.31), i.e.,
K*,f-V-) = f«(£-V<)(£-V0)=l.
(1.39)
Solving (1.39) for F, one obtains F = a + f3, where a = ^aij(x)y'ty^ /? = bi{x)y% are given by
(l-z^n^ + y ^ ^ ^ "
'
(l-hpqVrV«r
and
(
j
It is easy to show that II&IU = ^ M j
= yJhijViVi = h(x,-Vx) < 1.
(1.42)
Thus F is a Randers metric. Conversely, every Randers metric F = a + j3 on a manifold M can be constructed from a Riemannian metric h and a vector field V on M. The construction is given as follows. Let a = x/aifj/yj and /? = biy%. Define
^ : = (1-Il/3*H 2 ){^-^}' F* • =
(L43)
(1 44)
^h
Then F is given by (1.39) for h = ^hij(x)yiy^> and V = V^x)-^. Moreover, (1.42) holds. Thus h(x, -Vx) < 1 for x € M. See [45] andl46] for a similar type of duality between Randers metrics defined as a function on TM and Randers co-metrics defined as a function on T*M. Example 1.4.4 h
'~
Let B n c R n be the standard unit ball and let
VT^W L{, m l + (a,x)\j
2(a,y){Xly) l + {a,x)
(1 - |»|a)
(
'
25
Cartan Torsion V: =
_i_M^>(x +
a)|
(L46)
where y € TxBn = Rn and a £ R n is a constant vector with \a\ < 1. By (1.39), one obtains F=
+
T^NP
I^^
+
TT(^)-
(L47)
Both h — h(x,y) and F — F(x,y) have some special geometric properties. See Examples 3.4.2 and 3.4.6 below. When a = 0, F is the Funk metric on B" defined in (1.15). 1.5
Cartan Torsion
To characterize Euclidean norms among Minkowski norms, E. Cartan introduces a quantity for Minkowski norms [23]. Let F = F(y) be a Minkowski norm on a vector space V. For a vector ye V\{0}, let C
>'U'W)
:=
\dStfr [^
+ SU + tV +
rw)
]s=t_o'
where u, v, w G V. Each Cy is a symmetric trilinear form on V. We call the family C := {Cy \ y € V \ {0}} the Cartan torsion. Let {hi} be a basis for V. Let gij := g y (bj,b ;7 ), C^fc := Cj,(bi,bj,bfc). Then
9ij
— 2 ^
JV'J/ 3 ''
Define the mean value of the Cartan torsion by n
\y{u) := 2^(l/)C v («, b i; b,), i=l
ueV.
We call the family I := {Iy \ y € V\{0}} the meon Cartan torsion. Observe that
A[det(s.fc)]
= det(^fc)5«^ =
2det(^fc)^Ctpg.
26
Finsler Metrics
We have Ii = gJkCijk = ~
[In y/det (ft*)] •
(1.48)
It follows from the homogeneity of F that C I ,(i/,i;,u;)=C I ,(«,i/,«;) = C 1 ,(u > t; > y)=O
(1.49)
Iv(v)=0.
(1.50)
and
Moreover, C Av = A- 1 C w ,
I Ay = A- 1 I w ,
A>0.
(1.51)
From (1.49)-(1.51), one can see that C y and Ij, depend only on the geometry of the indicatrix Sp of F. Intuitively, the indicatrix of F can be viewed as a color pattern on V, then Cy (resp. Iy) is the rate (resp. average rate) of tangential change of the color pattern at y. It is obvious that F is Euclidean if and only if Cj, = 0 for any y € V\{0}. In fact, Euclidean norms can be characterized by the mean Cartan torsion. The following result is due to Deicke [34]. Theorem 1.5.1 ([34]) A Minkowski norm on a vector space V is Euclidean if and only if 1 = 0. The proof does not fit in this book, so it is omitted. One can see [5] for a proof. To characterize Randers norms among Minkowski norms, M. Matsumoto introduces the following quantity [64] [66]. For y = ylhi € V, define Mijk := Ciik - ^rj{lihjk
+ Ijhik + Jfc/iy},
(1.52)
where h^ := FFyiyJ = gtj - -pj9iPypgjgyq. Let My(u,v,w)
:=Mijk{x,y)uiviwk,
(1.53)
where u = ulhi, v — v^hj and w = wkbk- Each M y is a symmetric trilinear form on V. We call the family M := {M y | y £ V \ {0}} the Matsumoto torsion. Clearly, M = 0 for all two-dimensional Minkowski norms.
27
Carton Torsion
Example 1.5.2 ([64]) Let F = a + /? be a Randers norm on a vector space V, where a = y/a^yi and /3 = biyi with \\0\\a < 1. Then gi:i := 2 h[F ]yiyj(y) are given by (1.3) and det (&.,•) is given by (1.4). Note that det(ay) is independent of y. By (1.48), one obtains
Differentiating (1.3) with respect to yk yields Cijfc = ^^jTfl/Afc + 7Afc + 4/iij},
(1.55)
where hij := FFyiyj are given by
q+^/
a
yiyj \
a2 )
\
This implies that Mijk = 0. Minkowski norms with M = 0 are said to be C-reducible. It turns out that every C-reducible Minkowski norm is a Randers norm in dimension n>3. Proposition 1.5.3 ([64], [70]) Let F be a Minkowski norm on a vector space V of dimension n > 3. The Matsumoto torsion M = 0 if and only if F is a Randers norm. The proof does not fit in this book, and so is omitted. See [70] for more details. Given a Minkowski space (V, F), using the family of inner products gy on V, one can define the norm of I, C and M in a natural way.
||I||:=
sup
E®2M,
y,ueV\{0} Vgy(U, U)
||C||:=
sup
ny)|C y (u,,, W )|
=
y,u,v,weV\{0} \/gy(u, U)gy(v, V)gy(w, w)
||M|| : =
sup
F(y)\My(u,v,W)\
y,u,v,w€V\{0} Vg y (u, U)gy(v, v)gy(iV, V))
28
Finsler Metrics
By (1.52), ||M|| is bounded by ||C||. It is easy to construct a family of Minkowski norms Fj on R n with ||Cj|| —> +00 as i —> +00, where Q denotes the Cartan torsion of Fj. The (mean) Cartan torsion of any Randers norm is bounded from above by a number depending only on the dimension. Lemma 1.5.4 ([49]) Let F = a+0 be a Randers norm on an n-dimensional vector space V. Then
Hil = ^^Ji-V^r¥E<
^ -
(i-56)
Proof. Let a = ^/a^yi and 0 = biy\ Then gij := \[F2)yiyi are given by (1.3). By (1.3) and Lemma 1.1.1, one can find the inverse matrix (g^) =
9* = j«lj - I ^ V + *V) + ^%^-yW.
(1-57)
By (1.54) and (1-57), one obtains
Since |/3(y)| < ||/9||Qa(j/), we can write j3(y) = \\/3\\aa(y) cos0, where 0 < 8 < 2?r. Assume that y is a unit vector, i.e., F(y) = a(y) +/?(«/) = 1. Then a(y)=l-f3(y)
= l-\\p\\aa(y)cose.
Thus a{y)
=
l + \\0\\acoS0-
Plugging it into (1.58) yields Ulj9
" I
2 j l+||/?|| a cos0'
PII2 = ^ ^ ( i - v / ^ w ) . This gives the upper bound (1.56) immediately.
Q.E.D.
29
Cartan Torsion
It follows from (1.55) and (1.56) that
I|C|| < ^y/l-y/T^WE
< ^=-
(1-59)
Namely, the Cartan torsion is uniformly bounded by 3/\/2- The bound (1.59) for two-dimensional Randers norms is given in Exercise 11.2.6 in [5] which is suggested by Brad Lackey. Example 1.5.5 Consider the generalized Funk metric F = a + (3 on the unit ball B"(l) c R", V\y\2-(\x\*\y\2-(x,y)i) l-|x|2
(x,y) 1-N2
(a,y) l + {a,x)'
Let ||I||X denote the norm of the mean Cartan torsion at a; € B™(1). By (1.29) and (1.56), one obtains iiTii -
n + 1
/T
V(i-W2)(i-N2)i
Note that at x = —a, Ix = 0, namely, Fx is Euclidean. However, as x -> 9B n (l), \\I\\X -* {n + l)/\/2. The point x = -a can be regarded as the Euclidean center of F.
Chapter 2
Structure Equations
In 1943, the first author introduced a connection for Finsler metrics and gave a solution of the local congruence, i.e., a complete system of local invariants which ensures that two Finsler structures differ by a change of coordinates [28]. This connection is a natural generalization of the Levi-Civita connection in the Riemannian case and seems to be the right analytical basis of the subject. This connection is now called the Chern connection. The aim of this chapter is to give a short derivation of the Chern connection, introduce various notions of curvatures and derive important relationships among these quantities (see also [4], [29], [30], [3l]). 2.1
Chern Connection
The Chern connection on a Finsler manifold is a linear connection on the pull-back tangent bundle. Before we introduce the Chern connection, let us give a brief description of vector bundles and linear connections on a vector bundle. Afc-dimensionalvector bundle over a C°° manifold iV is a C°° manifold V with an onto C°° map TT : V —> N such that for any coordinate domain U C N, T T " 1 ^ ) is diffeomorphic to li x Rfc such that 7r""x(x) is diffeomorphic to {x} x Rfe for any x € U by the restriction of the diffeomorphism. The set Vx '•— K~1(x) is called the fiber at x. We usually denote a vector bundle n : V -> N by V. For a vector bundle V over a manifold N, a section of V is a map X : N —» V such that X(x) £ Vx for any x € N. A local frame of V is a set of local C°° sections {ejjjLj of V defined on some open subset U c M 31
32
Structure Equations
such that for any x € U, the set {ej(a;)}jL1 is a basis for the fiber Vx at x. Given a local frame {ej} of V, any section X of V can be locally expressed by X = X'et. Then X is C°° if and only if all coefficients Xi are C°°. We shall denote by C°°(V) the space of all C°° sections of V.
Figure 2.1 One can view a vector bundle V over a manifold AT as a union of vector spaces Vx indexed on TV, V = \JxeNVx. Let V* denote the dual vector space of Vx. By definition, Vx is the vector space of linear functionals on Vx. Then V* := (JXSAT ^x i s a vector bundle over TV. We call V* the dual vector bundle of V. Let V be a vector bundle over a manifold TV. A linear connection V on V is a family of linear maps V : TXTV x C°°(V) —> Vx, i.e., V : (v, X) G TXN x C°°(V) -> VVX € Vx with the following additional condition: Vv{fX)
= d/(«)X + f(x)VvX,
For any local frame {ej} for V, let X = X'lei.
f € C°°(M). Then
VVX = ^dXi{v) + X'ufiv)}*, where {tOj1} is a set of local 1-forms on TV. {w •*} are called the connection forms of V with respect to {e»}" =1 . By removing v in the above identity, we can express V X : TTTV -> Vx or VX € TX*TV (g) Vx as follows,
VX = \dXl + X-»w/} ® e*,
X - Xlei.
33
Chern Connection
Set
Each SI? is a local 2-form. {fi/} are called the curvature forms of V with respect to {ej}. Let {w1} denote the dual basis of {ej}, then ft := Slj1 uPQei is a well-defined tensor over N, which is a C°° section of T*N®V.
Let M be a connected C°° manifold. Let TMo:=TM\{0} = {|,eTIM
y^O, are M } .
TMO is called the siit tangent bundle over M. The natural projection TT : TMO -> M pulls back TM to a vector bundle n*TM over TMO. The fiber at a point (x, y) € TM0 is defined by
K*TM\iX:y) := {(i.y.t;) ueT.Mj^T.M. In other words, TT*TM|(XJ!/) is just a copy of TXM. ir*TM is called the pui/6acfc tangent bundle. Similarly, we define the pull-back cotangent bundle ir*T*M whose fiber at (x,y) is a copy of T*M. Therefore n*T*M can be viewed as the dual vector bundle of n*TM by setting (x, y, 9)(x, y, v) = 0(v),
(9, v) e T*M x TXM.
Figure 2.2 Take a standard local coordinate system (xl,yl) in TM, where (xl) is a local coordinate system in M and y l 's are coefficients of j/ = j/*^i|xLet {^7,^7} and {dx%,dy%} be the natural local frame and coframe for T{TMO) and T*(TMO), respectively. Then VTM := s p a n { ^ } is a welldefined subbundle of T(TMO), which is called the vertical tangent bundle of
34
Structure Equations
M. n*T*M can be naturally identified with the horizontal tangent cotangent bundle, HT*M := span{dxi} of T*(TMO). Thus HT*M and TT*T*M can be viewed as the dual vector bundle of n*TM. Let
* := ( x '»fl£il0Then {di} is a local frame for ix*TM. The vector bundle n*TM has a canonical section y defined by y(x,y) ==(a;,j/,y)At y = 2/ 8 ^T| X € TXM, y can be expressed as 3> = <M.
(2.i)
Let F be a Finsler metric on M and let 9ij •= g [i?2li/i^(a;' 2/).
c
O-fc : = 4 [ ^ ^ y ^ * ^ . 3/)-
Define g := gij dxi ® dxj,
C := Cijk dxi ® dxj ® dxk.
(2.2)
(2.3) k
dgn = gkjUi + gikUj + 2Cijkw
n+k
,
(2.4)
where un+i:=dyi
+ yiwji,
where y =: ytei, gtj := <7(ej,ej) and Cijk := C(ei,ej,ek).
(2.5)
35
Chern Connection
Proof. We will prove the theorem in a standard local coordinate system (x%,yl) in TMO. Taking e, = di and w* := dx1. The local 1-forms w> can be expressed as Ljji~Tijkdxk
+ U)kdyk.
(2.3) is equivalent to the following equation 0 = d2xi = dxj A (T)kdxk + U)kdyk>). This implies that Wjk - 0 and T)k = Ykj. Thus wj = r*fcdxfc. Let
ATJ:=y m rV
(2.6)
(2.4) becomes dgtj = gimT]idxl + gmjT^dxl
+ 2Cijm (dym + N^dx1}.
(2.7)
Since
(2.7) is further reduced to the following
^ f = gimr]t + 9mjrs + 2CijmNr-
(2.8)
Permutating the indices, one obtains from (2.8) that § • £ = QimTTi + QmiTTj + 2CjmlN™, ^J
(2.9) (2.10)
= gml^Tj + frmTTl + 2CimlN™.
Adding (2.9) and (2.10), then subtracting (2.8) yield . O9l%
®9jl T
~dx
<J9ij
~dx~i~ W
p m 1 ns~<
0
=
9ml
ij
njm , g/"i
2C mlAli
J
Krm
+ 2CimlN
J
nr-i
-
2C
vm"i
Arm
•
From t h e above identity, one obtains r fc
ij
=
i-fcJf ££i£ ,^9H_
2y
I dx*
dxi
d
9H )
dx1 /
-9kl{cjmiNr + CimiNp - CijmNr}-
(2.11)
36
Structure Equations
To determine F*fc completely, one needs to express TVj in terms of F. Contracting (2.11) with y% yields Nk_lki
J
~2
5
+
?9!i_?9ii\i_2kir
\dx*+dxi
dx'P
Cm
9 Cjm G
'
'
r2
12^
[2A2)
where
Contracting (2.12) with \y> yields G
.-V{^ +^-^W« 5
+
l
t
y
~ 4 lac* ari dx r • Plugging (2.13) into (2.12), one obtains a formula for TVj expressed of gij. Plugging them into (2.11) yields a formula for r*.fc expressed of 9ij.
(2 13)
^ - u ; in terms in terms Q.E.D.
With the Chern connection forms {WJ1} with respect to a local frame {e^} for n*TM, one can define a linear connection V on n*TM by VX := ldXi + XjuA
<8) ei,
where X = Xi(x,2/)e! e C°°(-n*TM). Clearly, V is a well-defined linear connection on ir*TM. It is also torsion-free in the following sense:
V^p(F) - Vfp(JT) = p[X, Y], where p : T(TMO) —>
TT*TM
X,Y€ C°°(T(TMO)),
is a vector bundle map defined by
V is called the Chern connection. Using
[^< = t ^ v , [F%vyfc = 2 | ^ y , we can rewrite Gx in (2.13) as follows,
Gi = \gil{[F*\x>ylyk-[F*}xl}.
(2.14)
37
Chern Connection
Using [F2]xl = 2FFxh
[F\kylyk
=
2
l*^gmly™ + 2FFxkylyk,
we obtain Gi=Pyi+Qi,
(2.15)
••=*'£-*>£'
(2 16)
where
Let
G
'
where G* = G'(x,y) are denned in (2.13) with the following homogeneity Gi(x,Xy) = \2Gi(x,y),
A > 0.
(2.17)
G is a well-defined vector field on TMO. We call G the spray induced by F and G% the spray coefficients of F . Any C°° vector field G on TMO in the form (2.16) with homogeneity (2.17) is called a spray on M. Not all spray can be induced by a Finsler metric. Thus sprays are more general geometric structures. See [86] for discussions on sprays. The local functions Nj defined in (2.6) or (2.12) are called the connection coefficients. Differentiating (2.13) with respect to yj yields
*> = w
(2 18)
-
Thus AT] depend on Gi only. Let
J---*L-Nj—
(2 19)
Then HTM := s p a n { ^ } is a well-defined subbundle of T(TMO) and T{TM0) = HTM ® VTM. Note that 2Gi = Njyi. Thus G = yl-£j is a special section of HTM.
38
Structure Equations
The local functions Tl,k in (2.11) are called the Christoffel symbols of the Chern connection. Nj and r*fc are related by (2.6). However, differentiating Nj with respect to yk does not give F*fc. Let
L
v=w~ r ^
(2 20)
-
We obtain a tensor C = Lljkdi(g)dx:> ®dxk. We call C the Landsberg tensor. Note that L*jk are symmetric in j , k. The Landsberg tensor is an important quantity in Finsler geometry. Let r := gikUjk.
(2.21)
Then J :— Jld{ is called the mean Landsberg tensor. It is easy to verify that
Thus if r*fc = r* fc (i) are functions of x only, then Djk — 0. Now let us take a look at the Christoffel symbols of Riemannian metrics. Let F = y/gij(x)yly:i be a Riemannian metric on a manifold M. By (2.13), the spray coefficients G% = Gl(x, y) of F are given by
where (gij(x)) := (^(a;))- 1 . By (2.6) and (2.11), the connection coefficients Nj and the Christoffel symbols Tl-k are given by
Note that the Christoffel symbols r*fe = r*fe(a;) are functions of x only. Thus Ujk = 0. Definition 2.1.2 A Finsler metric F on a manifold M is called a Berwald metric if in any standard local coordinate system {x%,y'1) in TMO, the Christoffel symbols T%-k — F*fc(x) are functions of x e M only, in which case, Gl = ^Tljk(x)yjyk are quadratic in y = yi-^\x. F is called a Lands= berg metric if i^fc 0-
39
Chern Connection
As shown above, Riemannian are Berwald metrics. There are many non-Riemannian Berwald metrics. See Section 4.3 below. G. Landsberg first studied Landsberg metrics [6l], [62]. Since then, Landsberg metrics frequently appear in literatures. By (2.20) or (2.22), we obtain the following Proposition 2.1.3
Every Berwald metric is a Landsberg metric.
It is still an open problem whether or not every Landsberg metric is a Berwald metric. So far no counter-example has been found yet. Let F be a Berwald metric on a manifold M. By definition, in any standard local coordinate system {xl,y%) in TMO, the Christoffel symbols T)k = T)k(x) are local functions of x e M only. Let 6/ := Y)k{x)dxk. Define
DX := {dX* + Xj6^ ® ~ ,
(2.24)
where X = Xj-^ € C°°(TM). Clearly, D is a linear connection on TM. It is also torsion-free in the following sense: DxY-DyX
= [X,Y],
X,Y
£C°°(TM).
D is called the Levi-Civita connection of F. Consider a Riemannian metric g = gij{x)dx% ® dx? on M. Let FJfc = T jk(x) denote the Christoffel symbols. Let D be the Levi-Civita connection defined by (2.24) using Q> := T^k{x)dxk. One can easily verify that X
d61' = ej A 0/ dgij = gik9jk + 9jkeik-
(2.25) (2.26)
(2.25) and (2.26) are equivalent to the following equations in the index-free form: DXY - DyX = [X,Y], y[g(X, Y)} = g(DyX, Y) + g(X, DyY),
(2.27) (2.28)
where y e TXM and X, Y e C°°{TM). One can show that for a Riemannian metric g on a manifold M. D is the unique linear connection on TM satisfying (2.27) and (2.28).
40
2.2
Structure Equations
Structure Equations
In this section we are going to use the Chern connection to introduce several notions of curvatures for Finsler metrics. Let (M,F) be an n-dimensional Finsler manifold. Let {ej} be an arbitrary local frame for ir*TM and {a/} the dual coframe for TT*T*M S* HTM. Express y, Q and C by y = y^i,
Q = gijUJ1 ®w0,
C - Cijku1 ®ujj ®uk.
See (2.1) and (2.2) for definitions. According to Theorem 2.1.1, the Chern connection forms {w/} with respect to {e,} are uniquely determined by dwi = UJJ A a//, k
(2.29) k
dga = gikUj + gkj^i
n+k
+ 2Cijkcj
,
(2.30)
where wn+i:=dyi+y>wji.
(2.31)
Clearly, {usl,u)n+t} is a local coframe for T*(TMO). The curvature forms flj* are defined by Slji:=duji-ujk/\ujki.
(2.32)
Differentiating (2.29), one obtains d2^ = dwj A w / - wj A duj>
= {u,™ A ^ } Ac^ " ^ A {0/+u,/" A < } Since d2u>1 = 0, one obtains the following identity: UJJ
A ft/ = 0.
The above identity is called the first Bianchi identity, ft/ can be expressed in terms of ujl ALJJ, ujlAwn+j and u>n+% Aojn+:>. By the first Bianchi identity, we can express ft/ by ft/ = \R/klcok AJ + P / ^ A wn+\
(2.33)
41
Structure Equations
where (2.34)
R/ki+R/ik=O, i
i
i
Ri kl+Rk iJ+Ri ik
= O,
(2-35) (2-36)
P/u = PkV Let fi* := dun+i - LJn+j A u>.
(2.37)
Differentiating (2.31), we obtain SV = dun+i - wn+i A w / = d2^* + dy* A w / + y»dw/ - un+j
A w/
= [un+i - ymwl} A w/ + 2/ {fi/ + ^
m
A < } - W"+^ A w/
= i/*jy. That is, fi* = y»fi/.
(2.38)
fi* is the essential part of f2 •*. By (2.38), one can express Q,1 in the following form SV = ^fffcjw* A w ' - L'fcIw* A u n + l ,
(2.39)
where «*« + R\k = o, and fl'H := ^fi/fei,
L'w == - V ' P / W
(2-40)
Clearly, y * i r w = 0,
ykL\l
= 0.
(2.41)
Setting
Rik:=Rikllt=ViRjikli/,
(2-42)
42
Structure Equations
we obtain the Riemann tensor. H:=Rikei®wk.
(2.43)
The Riemann tensor 1Z has the following properties. R\yk
= 0,
(2.44)
Rij = Rji,
where Rij := gimR™-. The first identity follows from (2.41) and the second identity will be proved using Bianchi identities in Section 2.4 below. In a standard local coordinate system {xl, y%) in TMO, the natural local coframe {a/,u> n+l } on TMO are given by wn+i = Syl := dy* + N]dxj,
J = dx\
where Nj = j / T j ^ - are defined in (2.6). Plugging u> = Tijkdxk yields j hl
~ JxT~
~5xT
ks ji
into (2.32)
~ ik l"
(
'
where j ~ are defined in (2.19). It follows from (2.40) and (2.45) that _dN}
dN£
Rkl
~-d^-^xT
+N l
dN*k ^ -
dN? N k
(2 4?)
-
^ -
Observe that ^
8T)k dy1
d{T)kyi) 8yl
kl
_0Nl_ri dy1
kl
d*G* ~ dykdyl
z kl
'
We obtain from (2.40) and (2.46) that L k l
~
y
By (2.48), one can see that Lljk (2.20).
~dyT-dy^/~
Tkl
~
(2 48)
-
defined in (2.40) are just those defined in
43
Finsler Metrics of Constant Flag Curvature
Since iVj = |£j-, Rlkl can be expressed directly in terms of G\ tracting (2.47) with yl and using (2.18) we obtain i _odG^ 7 d2Gi k ~ dxk * dxidy*
nrii + 2G
d2Gi dGldGi dyldy^ ~ %*" d^"'
Con-
[2M)
By (2.47) and (2.49), one can verify the following identity directly. Rkl
-3\~d^~~dy^r
( 2 5 0 )
Thus, Rlk and Rlkl can determine each other directly by vertical differentiation. According to (2.49), the Riemann tensor TZ depends only on the spray G = yi -^ - 1Gi^-t of the Finsler metric.
2.3
Finsler Metrics of Constant Flag Curvature
The Riemann tensor 7£ := ii*fcej ® u)k is the most important geometric quantity in Finsler geometry. It measures the curvature of a Finsler manifold. We are particularly interested in evenly curved Finsler manifolds. Let
h\ := Si - F-29kqtftf. We obtain the angular tensor h := h%kei ® uik. Note that h%k has the following properties similar to that of R%k, i.e.,
h\yk = 0. It is natural to compare 7Z with h. A Finsler metric F has constant flag curvature /i if TZ = fj,F2h, i.e.,
R\ = fxF2h\ = PL[F25{ -
(2.51)
By (2.42) and (2.50), we can see that (2.51) is equivalent to the following equation: R'k^^gipy^l-gk^si}. It follows from (2.52) that
•^(Riki)=li{giiSic-gjkSi}.
(2.52)
44
Structure Equations
Assume that F is a Berwald metric. By definition, in any standard local coordinate system (x*,^), Tljk = r*fc(z) are functions of x only. By (2.45),
i* = ~d - ~dr + r*^< - r^rL-
R
(2 53)
-
We see that R*kl = R^kl{x) are local functions of x only. In this case, we have R%
kl ~ - ^ / f c i ^ '
R
jkl
~
~Q^j(R%kl}'
Thus for a Berwald metric F, it is of constant flag curvature \x if and only if Rjki = v{9ji&k-9jkS\)Note that R3mml = (n- l)mi. Thus if \i ^ 0, then gji are independent of y € TXM. We conclude that any Berwald metric of non-zero constant flag curvature must be Riemannian. The case when /i = 0 will be discussed in Theorem 2.3.2 below. Below are some examples of Finsler metrics of constant flag curvature. Example 2.3.1 For a constant /x, let F = aM be the Riemannian metric defined in (1.12), f : =
# T M l ? ) 1 + nlx^
yer,B-(rM)-R",
i
(2.54)
where rM := 1/y/^Ji if /i < 0 and rM := +oo if /i > 0. First we have k
txV
/ 2n(x,y) ~ l + /x|x| 2 ^'
and Fxkylyk - Fxl = [Fxky% - 2FX, = 0. By (2.15), one obtains Ci
G
_
f*(x, y) 2V
'~YTJAXT '
A
,„ __v {
'
Finsler Metrics of Constant Flag Curvature
45
Plugging them into (2.49) yields that R\^^F2h\. Thus F has constant flag curvature /i. One of the fundamental theorems in Riemannian geometry states that every Riemannian metric of constant flag curvature /i can be locally expressed as (2.54) in some local coordinate system. This is a special case of E. Cartan's solution to the local equivalence problem. A Finsler metric F is said to be flat if the induced spray G is flat. More precisely, at any point x in M, there is a standard local coordinate system (xl, yl) in TM in which the spray coefficients G% = 0. F is said to be locally Minkowskian if at any point x € M, there is a standard local coordinate system (xl, yl) in TM in which F = F(y) is a function of (yl) e R™ only. Clearly, any locally Minkowskian metric is flat. The converse is true too. Suppose that a Finsler metric is flat, i.e., Gl — 0 in a local coordinate system (xl,yl). It follows from (2.8) that F 2 = gijy%y> satisfies \F2]
ir
i —
J
7/S;7 — 9 ? / V
A/7™ — 0
\xk — g /,. y y — zy y%m^k
— u-
Thus F is locally Minkowskian. For a Berwald metric, the flatness can be characterized by the curvature equation 1Z = 0. Theorem 2.3.2 (Berwald) Let F be a Berwald metric on a manifold. F is flat if and only if1Z = 0. Here is a sketch of the proof. Assume that F is a Berwald metric. In this case, the Levi-Civita connection D is defined in (2.24). D is a torsion-free linear connection on TM. The curvature condition TZ = 0 implies that D is fiat. Then there is a local coordinate system (x1) in which, F*fc = 0 (see [36]), hence F is flat. Q.E.D. There are many non-Berwaldian Finsler metrics with TZ = 0. See the following example.
46
Structure Equations
Let B"(l) C Rn be the standard unit ball and define
Example 2.3.3
Mtfl2-(l*l2|y|2-(s,i/>2) + (x, y))2 F:=±
2
2
2
2
'
2
(2.56)
(i - |*| )VM -(M M -M )
where y e TXB"(1) 9* R". It is easy to verify that F = F(x, y) satisfies the following equations Fxkylyk
= Fx«
and
Fxkyk = 29F, where ft
V\y\2-(\x\2\y\2-(x,W) l - |xp
+ (x:y)
is the Funk metric on B n (l) which is denned in (1.34). Then by (2.15), we obtain that G% — Qy1. It is easy to verify that 0 = @(x, y) satisfies (1.38),
exk = eeyk. We have Qxivty* = ®x«, tfQxi = ©2,
i ^ ' e ^ ^ = o.
Then by (2.49) and the above identities, we obtain
R\ = 20**1/* -ytfayky*
+ 0 s ^j}
+20^10^,,*^ + eyk5] + eyJ5i}
-{Qyiy' + etyfarf+QSi} = {29 X , - QXJykyj + 2QGyJykyj - 9 9 ^ } ^
+{9 2 -^9^}4 = 0. Thus F has zero flag curvature. The Finsler metric in (2.56) is constructed by L. Berwald [17]. See Example 3.4.7 for a different proof. See also [89] and [75] for related discussion.
47
Bianchi Identities
2.4
Bianchi Identities
In this section we will employ the exterior differentiation method to derive the relationship among curvatures. Let (M,F) be an n-dimensional Finsler manifold. Let {ej} be a local frame for n*TM, {w*,a;n+t} be the corresponding local coframe for T*(TMO) and {c^/} be the set of local Chern connection forms with respect to {ej}. For a scalar function / on TM \ {0}, define / | m and f.m by df = /|fcw* + f.kwn+k.
(2.57)
There is a canonical way to define the covariant derivatives of a tensor on TM0 using the Chern connection. For example, if T = T^J1
:= dTzj - Tkjuk
-
Tikuk.
In a standard local coordinate system (x\ yl), the coefficients, Tij = Tij(x, y), are local functions of (x1, y'), where y = yl-^t\x- Tij\k and Tij.k are given by rp x
i]\k
"•!•%]
rp
— -Q—k
T-IS
rp
J-\S
rp
»rg
J-sjl- ik ~ 1isL jk ~ J *J-s Jv fe i
The covariant differentiation satisfies the product rule. For example, for S = Si^
For the fundamental tensor Q = gijUJ1 ® u J , it follows from (2.30) that 9ij\k — 0;
9ijk = %Cijk-
For the canonical section 3^ = y'e,, it follows from (2.31) that
y\k = o,
y\ = si
Thus [F\m
= 2FF[m =
fly|myV
= 0.
(2.58)
48
Structure Equations Recall the curvature forms in (2.39) Jl* := dujn+i - wn+J A w/ = ^R\^k
A w ' - Vklwk A wn+l.
(2.59)
Differentiating (2.59) yields the following second Bianchi identities, dW = -Qj A LJ/ + u>n+j A
fi/.
(2.60)
It follows from (2.60) that R
\l\j
+ R\j\k
+
Rl
jk\l = -L%jmRmkl
~
%
m
l
-H/JM = ^*fc/-j + ^fei|/ ~ ^ ij\k + L lmL
L%
kmR"lj ~ L\mRmjk, z
• (2.61)
n
(2.62)
- L kmL lj.
k:i
Contracting (2.61) with j / ' yields Ri
k\j -
Ri
i\* + RimmVm = LikmRrr} - L'^Rr
(2.63)
We now use (2.30) to find other relationship between the curvature tensors and the Finsler metric. Differentiating (2.30) yields 0 = 5i fc O/ + 9ki^ik + 2(Cimwl
+ Ciik.iujn+l) A un+k +
2Cijknk.
This gives the following equations: Rjikl + Rijkt + 2CijmRmkl
= 0,
(2.64) m
Pjikl + Pijkl + 2Ciji\k ~ 2CijmL where Rjlki :- 9imRjnki
kl
— 0,
(2.65)
a n d p
j*kl •= gimPj^i-
From (2.34) and (2.35), one obtains 2(Rklji - Rjikl) — (Rklji + Rlkji) - {Rjikl + Rijkl) +(Rkuj + Rikij) + (Rijki + Rjiki)
+(Riljk + Rujk) + (Rjku + Rkju). Applying (2.64) to the above identity, one obtains Rklji - Rjikl = CkimR"^i - CjimRnkl m
+CijmR
kl
+ CkimR™^
+ CUmR
m
jk
+ CjkmRmu.
Let Rij := gikRkj. Contracting (2.66) with yJ and yk yields that ij — ft-ji-
(2.66)
49
Bianchi Identities
This verifies the identity (2.44). We now discuss the Landsberg tensor C = Llklei®u>k Lijk
:= gimLmjk
-ymPmijk.
=
By (2.36) and (2.65), one obtains Lijk = --ymPmijk
-
-ymPjimk
= \ymPimjk + \ymPnmk +
cijk]mym
— ^ymPimjk + -^ymPmijk + Cijk\m = Cijklmym.
(2.67)
Thus Lijk is symmetric in i, j , k. Let Jj := gimJm = gjhLijk. Contracting (2.67) with gjk yields Ji = h\mym-
(2.68)
By virtue of (2.67) and (2.68),
Lijk\mym
Lemma 2.4.1
Jk]mym
= h\p\qypyq.
= --gimRmk.j
- -^gjmRmk.i
= Cijk^yPy",
([72], [74])
Cijk\P\qypyq + CijmRmk
-\gimRmj.k
- \gjmRmi.k
(2.69)
and
h\P\qypyq + imRmk = -\{2Rmk.m
+ Rmm.k}-
(2.70)
Proof. It follows from (2.62) that R
j
kmy
-
K
krajV
+ Lijk\my
- ti kj + H jk +
L
Thus Rjikmy™ = gim{Rmk.j + Rmjk} +
Lm[mym.
jklmV •
50
Structure Equations
By the above formulas for Rjikmym
and (2.64), one obtains
Lijklmym + CijmRmk = -\Qim{Rmk.3 + Rmjk} -\gim{in.i
+ ink}.
(2.7i)
Contracting (2.71) with gij
Jk]mym
+ lmRmk = -{Rmk.m
+ Rmm-k}-
(2-72)
One can rewrite (2.50) as follows Riki = \{Rik.i-Ril.k}-
(2-73)
(Note: the reader can derive other Bianchi identities and verify (2.73) using these identities, then express Rj%ki in terms of R%k, Cijk and Lijk.) Plugging (2.73) to (2.71) and (2.72) yields (2.69) and (2.70). Q.E.D. The identities (2.69) and (2.70) play an important role in the global Finsler geometry. Later on, we will use them to establish some rigidity theorems.
Chapter 3
Geodesies
In this chapter, we are going to introduce the notion of geodesies and discuss Finsler metrics having the same geodesies as point sets. In particular, we will discuss projectively flat Finsler metrics. 3.1
Sprays
Let M be an n-dimensional manifold and TT : TMO := TM \ {0} —» M be the natural projection. A spray G on a manifold M i s a special smooth vector field on TMO in the following form,
where G% = Gl(x,y) are local functions with the following homogeneity: Gi(x,Xy) = X2Gi(x,y):
A > 0.
A curve 7 = j(t) in,TMo is called an integral curve of G if it satisfies 7 = G7. Let 7(i) be an integral curve of G. Then the coordinates of 7(4) satisfy i\t) = »•'(*),
(3.1) (xl(t),yl(t))
2/8W + 2Gi(x(i), 2/(<)) = 0.
Let a(t) :— n("f(t)) be the projection of -j(t) under ir. Then the coordinates 51
52
Geodesies
(x'(t)) of a(t) satisfy a^t) + 2Gi(a(t),&(t)J = 0.
(3.2)
Here we identify a(t) and &(t) = cfl{t)-^r\a(t) with their coordinates (crl(i)) and
(**(<))•
Conversely, given a curve a =
The coordinates of &(t) in TM are (crl(t), &l(t)). It is easy to see that if a(t) satisfies (3.2), then (^(t), ^(i)) = (a*(t), &<(t)) satisfy (3.1). Namely, the canonical lift of a is an integral curve of G. A map a = cr(t) in M is called a geodesic of G if it is a C°° curve, and the canonical lift j(t) := &(t) is an integral curve of G in TMO, i.e., it satisfies (3.1). In a standard local coordinate system, the coordinates (a^t)) of a(t) satisfy (3.2).
G
Every Finsler metric F = F(x,y) on a manifold M induces a spray
= v'-h - 2 G i w
by
( 2 - 14 ) or ( 2 - 15 )' that is-
By the above formula, we can find a formula for the spray coefficients G% of an (a, /3)-metric, which are expressed in terms of the spray coefficients of a and the covariant derivatives of /3. Let F = a<j>(s) s : = - , a
where
a
*' [ 9a ^'' 2 \dxk
dakl
dxi
da k
i | dxl J"
53
Sprays
Let 6i := dxi and 0 / := Tijk{x)dxk.
Define bMj by
6 i ; i 0 j := dbi - bjBJ. We have
Let r « : = 2 (6
s
*i
:=
2 ( ^ ~ **<)'
(3 3
' ^
Since f^- = F ^ , we have - 1 / dt>j dbj \ ~2\dxi dxi)-
Sij
On the other hand, the 1-form ft = 6,y\ which can be expressed as /? = 6jtfxl, has the following the differential, dp = dbj A di* = i f | ^ - - | ^ r W * A daJ. Thus p is closed (d/3 = 0) if and only if s^ = 0. Let jj-.^a^shj,
(3.4)
Sj-^bis'j.
By (2.13) and using a Maple program (see Section A.3 below), we obtain the following relationship between Gl and G%a. L e m m a 3.1.1
r*-r*
The spray coefficients Gl are related to G%a by
x ^ 4>-sV I 4> — sq>
where s = P/a,
s*0 — s
i
i ^
(<£ - sW - s4>4>" 2
^ , s0 := Sij/ 1 , r O o = rijylyj
and b2 : =
a J atjbibj.
A similar formula for the spray coefficients of an (a, /3)-metric is given in [51], [68] and [69]. Note that if 0" = 0, i.e., 4> is linear in s, then ti do not occur in (3.5).
54
Geodesies
Let F — a + (3 be a Randers metric on a manifold M, where a = V^ijJ/V and /? = hy1. F can be expressed in the form F = a(j)(j3/a) where
Pyl + Q\
(3.6)
where P:=^-s0,
Q^as'o,
where eij = r^ + biSj + bjSi, eOo := eijytyj,
s0 •= Siy1 and sl0 :=
(3.7) s%^.
Formula (3.6) is given in [2]. By (3.6), one can show that the following three conditions are equivalent for a Randers metric F = a + (3, (a) F is a Landsberg metric; (b) F is a Berwald metric; (c) (5 is parallel with respect to a, i.e., bi;j = 0. This is a result obtained by several people. See [65], [42], [52], and [95]. As we know that every Randers metric F = a + @ can be expressed in terms of a Riemannian metric h = y/h^yi and a vector field V = Vi g—• by equation (1.39),
h X
{ 'J-V')=1-
By (1.40) and (1.41), we can express F by
where Vo := Vitf and A = 1 - ||V||£. Let
4> := v 1 + s2 - s and
55
Sprays
Then F can be expressed as F = a<j>(s),
s:=£.
Let a = y/a,ijylyi and (3 = biyt. Define r^, sy and s, as in Lemma 3.1.1. We obtain a formula for the spray coefficients G% of F which are expressed in terms of the spray coefficients of a and the covariant derivitives of (3 with respect to a.
Gi
= G«- +*o + ^ r W 2 ^ + roo}{w - £}. (3.9)
We want to express Gl in terms of the spray coefficients of h and the covariant derivatives of V with respect to h. First it is easy to verify that 1 + b2 = \ , A
Let
h2- 2FV0 = XF2.
(3.10)
where "|" denotes the covariant differentiation with respect to h. We shall use h1^ to lift the lower indices and hij to lower the upper indices, such as Vi := hijVj and 5^- := hikSkj, etc. Let U := KjVj,
Kj := V^,
Sj := V^Sy.
j
Note that SjV = 0. We have
\\i = -2{ni + si). The spray coefficients G*a is related to that of h by
Gla = Gi + \(K0 + 6b)i/' - ±-(1V + 5*)/»2, LA
A
where TZ0 := Ti-iy1 and So •= Siy%. Using the above formula for G%a, we get roo = ^ ^ o o
+j2nh2>
**o = 5'o + ^{(Ko + SoW* - (TV+^Vo], so = \s0 + ^ { ( ^ o + 5b)(l - A) - HVoy
56
Geodesies
By (3.9) and the above formulas, we obtain the following Lemma 3.1.3 For a Randers metric F expressed in terms of a Riemannian metric h and a vector field V by (3.8), the spray coefficients Gl of F can be expressed in terms of the spray coefficients Glh of h and the covariant derivatives of V with respect to h as follows:
Gi = Gi-FS'o-^in'+S^ + ^^-V^^Fno-Tloo-F^Y
(3.11)
Formula (3.11) is obtained by C. Robles in a different approach [82], 3.2
Shortest Paths
Every Finsler metric F on a manifold M induces a spray G defined in (2.13). Thus the geodesies of G are called the geodesies of F. We have the following Lemma 3.2.1 If a C°° curve a{t) in a Finsler manifold (M,F) is a geodesic, then it has constant speed. Proof. It follows from (2.8) and (3.2) that
| [F"(*(t)Mt))] = %****'** + 2 § ^ W + 2gij*W = gmjNPcrW +gmN™6-i&i - Ag^GW = 0. Thus F(o(t),&(t)) = constant.
Q.E.D.
A piecewise C°° curve C from a point p to another point q is called a shortest path if dF(p,q) = £F(C). We shall show the following Proposition 3.2.2 For a shortest path in a Finsler manifold, any parametrization with constant speed is a smooth geodesic. Proof. Let C be a shortest path from p to q. Parametrize it by
57
Shortest Paths
piecewise C°° vector field V = V'Wsfrki)
alon
Sa
with v a
( ) = ° = ^(&)-
There is a piecewise C°° map # : [a, b] x (—e,e) —> M such that H(t,O) =
_ ( t , O ) = V(t),
a
and F ( a , s) = a(a),
if(fc, s) = a{b),
\s\ < e.
One may assume that H is C°° on each [£»_i, U] x (—e, e) for some partition
a = to < ti < • • • < tk-i < tk = b. Let £(s) := Cp{Cs) denote the length of Cs parametrized by as(t) := H(t, s).
Figure 3.1 By assumption, C(s) := £F(CS) > CF(C0) =: £(0). Thus C'(0) = 0. To compute the derivative C'(0), we express C(s) as
C(s) = J
F(H(t,s),—(t,s))dt.
Then
k
i
t-
2 + V—fF llv kVk ' [ +
2^2F
*«_,
58
Geodesies
= -j
^gjk{aj + 2&(*,a)}vkdt
where
G* := ^gil(x,y){[F\kyl(x,y)yk
- [F2U(x,y)}.
Note that Gl are just the spray coefficients defined in (2.14). Now, take an arbitrary vector field V(t) = V*(*)g§7|c(t) along a with
V\t) := /(«){**(*) + 2G* (
Figure 3.2 From the above formula for £'(0) and the equation C'(0) = 0, one concludes that a satisfies (3.2) on each (U-i,ti). Then
By choosing a suitable vector field V(t), one immediately sees that &(tf) — &(t~). Thus a(t) is C1 at each U. Since o-(t) satisfies (3.2), one concludes that a is C°° at each tj. Thus any constant speed parametrization of a shortest path is a geodesic. This proves the above claim. Q.E.D. Proposition 3.2.3 Any geodesic a(t) is locally minimizing, namely, at any point to in its domain, there is a small number e > 0 such that the path
59
Shortest Paths
C = {cr(t) | to — e < t < to + e} is the shortest path from p = a(t0 — e) to q = a(to + e). The proof is technical. We omit it here. See [5]. A Finsler metric F on a manifold M is said to be positively (resp. negatively) complete if every geodesic a(t) on (a, b) can be extended to a geodesic defined on (a, oo) (resp. (—00,6)). F is said to be complete if it is both positively complete and negatively complete. There are irreversible Finsler metrics which are only positively complete. For example, the Funk metric on a strongly convex domain in R" is positively complete, but not complete. While the Klein metric is complete. An important fact is that every closed Finsler manifold is complete. We invite the reader to verify the above facts. Let (M,F) be a positively complete Finsler manifold and let a; G M. There is a natural map expx : TXM —> M defined as follows using geodesies. For a vector y 6 TXM, let cry(t) denote the geodesic with ay(0) = x and 6-y(0) = y. Then exp^. is denned by expx(y) := ay{\).
(3.12)
expx is called the exponential map at x. By the homogeneity of the spray, one has
t > 0.
Figure 3.3 The functions /'(i,x, y) := o~y(t) in local coordinates satisfy the following system of second order ordinary equations, d2p -^r(t,x,y)
t +
df \ 2G*(f(t,x,y),-±(t,x,y))=0,
60
Geodesies
with
f(p,x,y)=xi,
^(0,x,y)
= y\
Note that G{ = G'(x,y) are only C 1 in y at y = 0. By the ODE theory, p(t, x, y) is C 1 in y at y = 0 and C°° in y elsewhere. Thus expx is C°° on TXM \ {0} and C 1 at the origin with d(expx)|o = identity [99]. Further, P(t, x, y) is C 2 in j/ at y — 0 if and only if G l = Gl(x, y) are quadratic in TXM. Thus, exps is C 2 at the origin of TXM for all x € M if y = yi^\x€ and only if F is a Berwald metric. This fact is due to H. Akbar-Zadeh [l].
3.3
Projectively Equivalent Finsler Metrics
In this section, we are going to discuss Finsler metrics having the same geodesies as point sets. In particular, we shall discuss those denned on an open subset in R" with straight geodesies. Two Finsler metrics F and F o n a manifold M are said to be projectively equivalent if they have the same geodesies as point sets, more precisely, for any geodesic a(t) of F, after an appropriate oriented reparametrization, t = t(t), the new map a(t) := a(i(t)) is a geodesic of F, and vice versa. We can characterize projectively equivalent Finsler metrics using the induced sprays. Suppose that F is projectively equivalent to F. For any y e TXM \ {0}, let a{t) be the geodesic of F with
2Gi{x,y) = -cr'(O) = -&\0) -1"(0)^(0)
= 2&(x,y) - i"(0)y\
From the above equation, one can see that P := —^P'(O) depends only on (x,y), hence P = P(x,y) is a function of (x,y). Moreover, P has the following homogeneity P(x,\y)
= \P(x,y),
A>0.
Then Gi(x,y)=Gi(x,y) + P(x,y)y\
(3.13)
Conversely, if the sprays of Finsler metrics F and F are related by (3.13), one can easily show that F and F are projectively equivalent.
61
Projectively Equivalent Finsler Metrics
There is another way to characterize projectively equivalent Finsler metrics. First, let us consider F as a scalar function on TM, and denote by F]k and F.k the horizontal and vertical covariant derivatives of F with respect to F, which are defined in (2.57), i.e., dF = Fiku>k +
F.kQn+k,
where {uik,u)n+k} is a local coframe for T*(TMO) determined by F. If Qj denote the Chern connection forms with respect to {cD8}, then u>n+t = dy% + yj&f- In a standard local coordinate system (x%,y%), Qx = dxl. Let wj = T)kdxk. Then u>n+i = dy^Njdx^ where N] = Y)kyk'. The covariant derivatives F;k and F.k are given by F]k = Fxk-N3kFyJ,
F.k = Fyk.
Let Gl = Gl(x, y) and Gl = Gl(x, y) denote the spray coefficients of F and F, respectively, in a common local coordinate system (x*, y1). We have Fxkyk ^F,kyk 2F 2F
Fyi F '
Fxkyiyk - Fxt = F.Myk - F;, +
2&Fy3yi.
Then
^9il{FxWyk -Fxl} = ^9il{F,k,yk - F,} +&-
&^-y\
Using (2.15) and the above identities, we obtain Gl = Gi + Pyi + Q\
(3.14)
where P
= ^
Ql =
\F9U{F;k-iVk-F,}.
Note that when F is the standard Euclidean metric on R n , F-k = Fxk, then (3.14) is reduced to (2.15). Theorem 3.3.1 ([80]) Let F and F be Finsler metrics on a manifold M. F is projectively equivalent to F if and only if F satisfies the following system, F-tk.iyk - F.tl = 0,
(3.15)
62
Geodesies
in which case, their spray coefficients are related by G% = Gi + Pyi where P = £p . Here Flk denote the horizontal covariant derivatives of F with respect to F and F]k.[ = (F]k)yi. Proof. We know that F is projectively equivalent to F if and only if there is a scalar function P = P(x,y) such that Gl = & + Py%, which is equivalent to the following equation: Pyi^Qi = Pyi.
(3.16)
Observe that ylF.kd = F.k. Thus Qij^Q' = \Fyl{F.Myk
- F;l} = 0.
(3.17)
Assume that F is projectively equivalent to F. Then (3.16) holds. Contracting (3.16) with t/i := g^yi yields P = P. By (3.16) again, one concludes that Ql = 0. This implies (3.15). Conversely, if (3.16) holds, then Qi = 0. It follows from (3.15) that Gi = Qi _|_ pyi xhus F is projectively equivalent to F. Q.E.D.
Example 3.3.2 ([42]) Let F = F(x,y) be a Finsler metric and (3 = bi{x)y% be a 1-form on a manifold M. Observe that dfi = dbt A dx* = ^dx> OX3
A dx* = \ A 2 I OX1
- | ^ W
A dx>.
OX3 )
Consider F := F+/3. It follows from (2.58) that F.k = 0. Here the covariant derivatives are taken with respect to F. One has
W-*-/w-* = {£-*},». By Theorem 3.3.1, one concludes that F is projectively equivalent to F if and only if /3 is closed. In particular, when F = a is a Riemannian metric, F = a + (3 is projectively equivalent to a if and only if /? is closed.
Projectively Flat Metrics
3.4
63
Projectively Flat Metrics
Let F = ao(y) be the standard Euclidean norm on R n . The spray coefficients of F vanish, G% = 0. Thus for a Finsler metric F on an open subset i / c R n , the geodesies of F are straight lines if and only if the spray coefficients of F are in the following form
Figure 3.4 Straight lines in U are parametrized by a(t) = f(t)a + b, where a, b € R n are constant vectors and f(t) > 0 is a positive function. We make the following Definition 3.4.1 A Finsler metric F = F(x,y) on an open subset U C R" is said to be projectively flat if all geodesies are straight in U, i.e., a(t) = f(t)a + b for some constant vectors a, 6 € R™. A Finsler metric F on a manifold M is said to be locally projectively flat if at any point, there is a local coordinate system (x4) in which F is projectively flat. By Theorem 3.3.1, a Finsler metric F = F(x,y) on an open subset U C R n is projectively flat if and only if it satisfies the following system of equations, FxWyk-Fxi=0.
(3.18)
This fact is due to G. Hamel [41]. In this case, Gi = Py\ where P = P(x, y) is given by
P=^f-
(3.19)
64
Geodesies
The scalar function P is called the projective factor of F. Note that (3.18) is a linear equation. Thus if F\ and F2 are protectively flat Finsler metrics on U, then F := aF\ + bF<2 is projectively flat on U as long as F is a Finsler metric. If F = F(x, y) is a projectively flat Finsler metric on U, then its inverse F := F(x, —y) is projectively flat on U. Thus the symmetrization F := ^{F + F} is also projectively flat on U. There are many interesting projectively flat Finsler metrics on R™. Below are some examples. Example 3.4.2 Consider the following family of Riemannian metrics denned in (1.12):
" -~
l+n\x\*
V
'
x
^ "
R
'
^-2°)
where rM = 1/y/^jlif \i < 0 and r^ = +cx> if fi > 0. By a direct computation (see Example 2.3.1), one obtains 1 + fj, x
z
Thus aM is projectively flat. The Beltrami's theorem and Cartan's local classification theorem in Riemannian geometry state that every locally projectively flat Riemannian metric is, up to a scaling, locally isometric to aM for some constant /z. However, a projectively flat Riemannian metric may take many different forms. For example, for the family of Riemannian metrics Fa defined in (1.45), a
_ JT^W / l + (a,x)\jm
2(a,y)(x,y) l + {a,x)
(1-\x\*)(a,yy (l + (o,x))2 '
where y € TxRn = R", the spray coefficients are given by l + (a,x)y Thus each Fa is projectively flat. In fact, Fa is isometric to ao- But it is hard to find an isometry between Fa and ao which also maps lines to lines. Example 3.4.3 Let U C Rn be a strongly convex domain defined by a Minkowski norm <\> = (f)(y) on Rn. Let 9 = 6(x, y) denote the Funk metric
Projectively Flat Metrics
65
on U defined by (1.35). By (1.38), one can easily verify that © satisfies (3.18), 9xkyiyk
=Oxi.
Thus 0 is projectively flat on U. By (3.19) and (1.38), the spray coefficients of 0 are given by
The Finsler metric 0 = Q(x,y) defined by (1.34) is called the Funk metric on a strongly convex domain U in a vector space R n . Equation (1.38) is the essential property of 0. Thus we make the following Definition 3.4.4 A Finsler metric © = Q(x, y) on an open subset in R™ is called a Funk metric if it satisfies (1.38).
Example 3.4.5 Let & = O(x, y) be the Funk metric on a strongly convex domain M c R " and let
H:=±[Q(x,y)+Q(x,-y)}. Let 0 := 0(x, — y). H is called the Hilbert metric on U. We are going to show that the Hilbert metric is projectively flat. See also [16], [17] and [77]. By (1.38), 0X* = \\Q\k,
we have ex*=-^[e]yfc.
Then Hxkyiyk = -{©zV
+&xkyijyk
= i{[© 2 ] yV -[e 2 wy = \{Qxi+9x*} = Hxl.
(3.21)
66
Geodesies
That is, H satisfies (3.18). Thus H is projectively flat with Gi = Py\ where P is given by P = ^ . Observe that (8 a »+6,»)y* 2(6 + 6)
_ ([ey-[eyy 4(9 + 9) 9 -62 ~ 2(6 + 6) 2
-i{e-e}. We obtain
P=±{e(x,y)-G(x,-y)}. The Funk metric on the unit ball B n (l) C R" is given by (1.15). e_y/\y\2-Qx\2\y\2_{Xiy)2)
1 - |x|
TxBn^Rn
+ {Xty)
2
'
The corresponding Hilbert metric on B"(l) is given by ti =
j—12
,
y € lxi5
—K.
iJ is the well-known Klein metric on B™(1). Example 3.4.6 ([89]) Let 6 = Q(x,y) denote the Funk metric on a strongly convex domain M c R " (see Example 3.4.3 above). Let a G Rn be a vector. Set
F:=Q(x,y)+ ^f
yeTxU^Rn.
(3.22)
F = F(x,y) is a Finsler metric in a neighborhood of the origin if \a\ is sufficiently small. By (1.38), one can easily verify that F satisfies (3.18). Thus it is projectively flat. It follows from (3.19) and (1.38) that Gi = Pyl where
67
Protectively Flat Metrics
When U = B n (l) is the standard unit ball, F = F(x, y) is a Randers metric given in (1-47), F =
Vlvl2 - (M 2 ly| 2 - (x, y)2) l-|x|2
(x, y) +
l-\x\2
(a, y) l +
{a,x)'
F is projectively flat with G% = Pyl, where 1 f V\y\2 - (M 2 M a - (x, y)2) 2i i-W2
(g, y) i-|x|2
(a, y) l i + (o,i>r
Example 3.4.7 ([43]) Let G = Q(x,y) denote the Funk metric on a strongly convex domain W c R " . Define a function F : TU = U x R" by
F •= e(x,y){l + eym(x,y)xm},
ye
TxU^Rn.
For a point x £U sufficiently close to the origin, Fx is a Minkowski norm on TJU. Thus F is a Finsler metric in a neighborhood of the origin. It follows from (1.38) that Fxk(x,y) = (FQ)yk{x,y). Observe that FxV(x,y)yk
= (FG)ykyl(x,y)yh
= (FQ)y<(x,y) = Fx,{x,y).
Thus F satisfies (3.18) and it is projectively flat. The spray coefficients take the form Gi = Py\ where P = P(x, y) is given by (3.19). Using (1.38), one obtains (Fe)ykyk 2F
=
2FQ ^ 2F
Thus Gi = Q(x,y)yi. When U = B n (l) is the standard unit ball in R™, the Funk metric 9 = &(x,y) on B n (l) is given by (1.15) and the corresponding Finsler metric F = F(x, y) is given by (2.56).
68
Geodesies
It is a natural problem to study locally protectively flat Randers metrics. We have the following Proposition 3.4.8 A Randers metric F = a + f3 is locally projectively flat if and only if a is locally projectively flat and j3 is closed. Proof. Suppose that F = a + P is locally projectively flat. There is a local coordinate system (x%) and a scalar function P such that Gl = Pyl. By Lemma 3.1.2, G* = Gj, + Py* + Q \ where P = eoo/(2F) - s 0 and Ql = as\. We obtain Gi + Pf + Qi = Pyl.
(3.23)
Note that dOm ^-=a-1ymsmo
+ asmm = 0.
Thus ^ r
+ ( n + l ) P = (n + l)P.
By (3.23), one obtains
^o = ^~|pV-GJ,
(3-24)
Note that the right-hand side is quadratic in y e TXM. Thus both sides are identically zero, that is, S
°-°'
G a
~ n + ldy™y-
The first equation implies that /? is closed. The second equation implies that a is projectively flat. The converse is obvious, so the proof is omitted here. See [ll] for related discussion. Q.E.D. Now we consider a larger class of Finsler metrics which contains Randers metrics.
69
Projectively Flat Metrics
Example 3.4.9 Let a = y/aijyiy^ be a Riemannian metric and 8 = biy% be 1-form on a manifold M. Consider an (a, /?)-metric in the following form Q2
F = a + eP + k—, a
where e and k are constants with k ^ 0. Then by (3.5), the spray coefficients Gl of F are given by (sa + 2k8)a2
,
4
-\
X
ea3 - 3kea/32 - 4fe2p3 7
x"^
2F ((1 + 2kb2)a2 - ZkdA I a
+
£ a3
r - 2 ( e a + 2k8)a2
l
- 3kea(32 - 4k283
a
b
5
r-^5
" kP
-i
so + 7"oo r J
I'
(3 25)
'
where G'a denote the spray coefficients of a. Assume that (3 satisfies bi\j = r | ( l + 2kb2)atj - Mbibj},
(3.26)
where r = T(X) is a scalar function on M, then sl0 = 0, so = 0 and
rOo = r { ( l + 2A;62)Q2-3fc/52}. The spray coefficients Gl of F are reduced to
Further, we assume that Gla are in the following form Gi = Oy* - krcx2b\
(3.28)
where 6 = 6iy% is a 1-form, then
G_,=
{
3 2 -4k263} t g + r ea -3kea6 2 2(a + ^ + ^ 2 ) R
f,
In this case, F is projectively flat in the coordinate domain. It can be shown that (3.26) and (3.28) are the necessary condition for F being projectively flat when k ^ 0. The next problem is how to find a and 0 satisfying (3.26) and (3.28). Below is a particular solution.
70
Geodesies
and /3 = biy% be defined by
Let a = sjdijifyi n
^'
=
VM 2 -(M 2 M 2 -foy) 2 ) (1 - |X|2)2
_(x1y)_ (l-|x|2)2'
a is a Riemannian metric and (3 is a 1-form on the unit ball B™(1). We have
bi\j = r | ( l + 2b2)aij - ZbibA and
Gia=T{3pyi-a2bi}, where T := 1 — \x\2. Consider the following family of (a, /3)-metrics,
F = a + 2P+f=^±Pl.
(3.29)
a a By (3.27), the spray coefficients G% of F are given by
-*{» + *£&$•*• Thus F is projectively flat. Note that the metric in (3.29) is the famous Berwald's projectively flat metric with zero flag curvature ([17]). See Example 8.2.8 below for more discussion on this metric. It is clear that there should be many other a = ^Ja,ijy'lyJ and /3 = biy1 satisfying (3.27) and (3.28). Some solutions are given in [75].
Chapter 4
Parallel Translations
Parallel translation is a very natural concept in Finsler geometry. In this chapter, we shall first introduce two kinds of parallel translations, then discuss some basic properties of Berwald metrics and Landsberg metrics. 4.1
Parallel Vector Fields
Let (M, F) be a Finsler manifold. Let c = c(t) be a C°° curve in M and U = Ul{t)-g^i\c(t} be a vector field along c. Define
D6U(t) := {&*(*)
+W{t)Ni(c{t),c{t))cP{t)}^-\c(t),
where iVj := Vjk(x,y)yk are the connection coefficients defined in (2.6). D(.U{t) is well-defined, independent of local coordinate systems. It is easy to verify that D6(U + V)(t) = D6U(t) + D6V(t),
(4.1)
Dc(fU)(t) = f(t)U{t) + f(t)D6U(t).
(4.2)
Since DcU{t) linearly depends on U = U(t), DcU(t) is called the linear covariant derivative of U(t) along c. A vector field U = U(t) along c is called a linearly parallel vector field if it satisfies the equation DcU(t) = 0, i.e., U*{t) + W(t)N} (c(t), c(t)) = 071
(4.3)
72
Parallel Translations
Clearly, for any to in the domain, U linearly depends on the value U(t0).
Figure 4.1 Let a — a(t) be a curve in M. Then the tangent vector field U := &(t) is a special vector field along a. Equation (4.3) becomes a{t) + 2Gi(a{t),a{t))=Q.
(4.4)
Thus a curve a is a geodesic if and only if its tangent vector field U = &(t) is linearly parallel along itself. Let T = Tij(x,y)dxl derivative is denned by
dTij - Tkjuf - Tikuf = Tijlkdxk + Tij.kSy*.
(4.5)
It follows from (4.5) that FfT F¥T r T .3lk , , —Z—lLhk >Clk °T T ~ Ik* ~ ~dy ~~
3
m /V T Nm { ~ m j '
For a non-zero vector y 6 TXM, we obtain a multi-linear form T y : TXM x TXM ^ R denned by T v (u,t;):=ry(i,j/)uV. Let a = a(t) be a geodesic and let U = U(t) and V = V(t) be linearly parallel vector fields along a. Let
T(t) := Ta(U(t), V(t)) = TiMVMtW'WHt)-
73
Parallel Vector Fields
Then using (4.3) and (4.4), we have
T'(t) = fZpidtwv' + ^ifiirvi oxK
—
oyK
+U.-1/V + Tairv'
!i/r fc l7»yj _ 1C.k i IPX" —T-lVi UmVi — T • Ni TPV™ J 1 1 V 3xk dvk UiVmu v t] ^mU
—[
v j-fc _ 2Ck
~ I dxk
j
m J - Tm Nm - Tm 7V j lr7*V '
dyk
° *
I
For the sake of simplicity, we omitted (
T'(t) = TwMtl&iWWWHt).
(4.6)
Lemma 4.1.1 Let a = a(t) be a geodesic in a Finsler manifold (M, F), and let U = U(t) and V = V(t) be linearly parallel vector fields along a. Then for the family of induced inner products ga(t) along a, g
V6U(t) := {?/*(*)+ ^(*)^(c(*),^(*))}^lc(t)-
(4-7)
VcU(t) is well-defined, independent of local coordinate systems. However, this V does not satisfy the linearity (4.1) and (4.2). U is said to be parallel along c if WcU(t) = 0, i.e.,
ir + c?(t)Ni(c(t),U(t))=0-
(4-8)
74
Parallel Translations
Since j/ J AT* = 2Gl, from (4.8), one can see that a curve c = c(t) is a geodesic if and only if its tangent vector field U — c(t) is parallel along itself. Thus we can use either D or V to determine geodesies. Lemma 4.1.2 Let c = c(t) be a piecewise C°° curve in a Finsler manifold (M, F), and let U = U%(t)-J=^\c(t) be a parallel vector field along c. Then F(c(t), U(t)) = constant. Proof. Write F2(c(t), U{t)) = 9ij(c(t),
U(t))u\t)W{t).
Using (2.8), (4.8) and the following fact, Cijk(c(t),U(t)'jUi(t)
= O,
one obtains
Jt[F2(c(t),U(t))} = ^cTUW+
29^(1"
= 2<7ifcAr£<W - 29ikN^cmUl Thus F2 (c(t), U(t)) = constant.
= 0. Q.E.D.
Two Finsler metrics F and F on a manifold M are said to be affinely equivalent if they have the same geodesies as parametrized curves, that is, if a = a(t) is a geodesic of F, then it is also a geodesic of F and vice versa. Let G% = Gl(x, y) and G% = Gl{x, y) denote the spray coefficients of F and F, respectively, in the same standard local coordinate system (z!,?/1) in TM. Clearly, F and F are affinely equivalent if and only if G\x,y)=&(x,y).
(4.9)
Thus if a Finsler metric is affinely equivalent to a Riemannian metric, then it must be a Berwald metric. Assume that Gi = &. By (3.14), we have Py'+Q< = 0.
(4.10)
75
Parallel Vector Fields
As shown in (3.17), JKQ* = o,
where y* := gijU>. Thus
0 = yi(Pyi + Qi) = PF2. We obtain P = 0. Then by (4.10), we get Qi = 0. It follows from P = 0 and Q* = 0 that F,kyk = 0,
Ffe.j/ - F.tl = 0.
Observe that (F]kyk).l =
F;k.iyk+Fli.
Thus F ;i = 0.
(4.11)
Let us remind the reader that F-i denote the horizontal covariant derivatives of F with respect to F. We obtain the following Theorem 4.1.3 Let F and F be Finsler metrics on a manifold M. F is affinely equivalent to F if and only if F satisfies (4-11)-
Example 4.1.4 Let F be a Minkowski metric on a vector space V. In a standard global coordinate system (x\ yl) in TV = V x V, F-k = Fxk = 0. Thus F is affinely equivalent to the Euclidean metric. To study affinely equivalent Finsler metrics, one needs the following Lemma 4.1.5 Let (M, F) be a Finsler manifold. If F is another Finsler metric on M such that for any F-parallel vector field U = U(t) along any curve c = c(t), F(c(t),U(t)\ then F is affinely equivalent to F.
= constant,
(4.12)
76
Parallel Translations
Proof. For any F-parallel vector field U = U(t) along c,
jt[F(c,u)]=FXJ(c,U)cJ
+ Fy,(c,U)Ui
= FXJ(c, U)c? - Fyi(c, C/)ATJ(c, U) (4.13)
= F.J(C,U)C?.
Suppose that F is affinely equivalent to F. Then by Theorem 4.1.3, F-k — 0. Thus for any parallel vector field U = U(t) along c = c(t), F(c(t), U{t)) = constant. Conversely, suppose that (4.12) holds. For any x e M, any nonzero vectors y, u € TXM, let c = c(t) be a curve with c(0) = x, c(0) = y and U = U(t) be a parallel vector field along c with U(0) = u. By assumption, Fy(a:,uy = 0. We conclude that Fy = 0. By Theorem 4.1.3, F is affinely equivalent to F. Q.E.D. 4.2
Parallel Translations
Using parallel vector fields along a curve, one can define parallel translations. Definition 4.2.1 Let c — c(t), a < t < b, be a piecewise C°° curve from c(a) = p to c(6) = g.
Figure 4.2 Define P c : TpM -> T9M by Pc(«) := f/(6),
u 6 TpM,
Parallel Translations
77
where U = U(t) is the parallel vector field along c with U(a) = u. Pc is called the parallel translation along c. By Lemma 4.1.2, the parallel translation Pc is a C°° diffeomorphism from TPM \ {0} onto TqM\ {0}, which is positively homogeneous of degree one, Pc{Xu) = XPc(u),
A > 0, we TPM.
However, Pc is not linear, in general. Parallel translations can be used to define a group on a Finsler manifold (M, F). For a point p € M, denote by C(p) the set of all piecewise C°° closed curves c : [0,1] —> M starting from p = c(0) and ending at p = c(l). Such a curve c is called a loop at p, and the set C(p) is called the loop space at p. For loops c\, C2 € C'(p), define the product C\ * c-z £ C(p) by C1
.,
(c2{2t)
*C2W:=U(2/-1)
if 0 < t < i
if I < t i l .
Figure 4.3 Clearly, PCl*c2=PCloPC2.
(4.14)
For a loop c € C'(p), the inverse c_ = c_(t) of c is also a loop at p defined by c_(t):=c(l-t),
0 < i < 1.
For an arbitrary parallel vector field U = U(t) along c, and let U-(t):=U(l-t),
0 < t < 1.
78
Parallel Translations
One can verify that C/_ = U-(t) is parallel along c_ with U-(0) = C/(l) and t/_(l) =17(0). Thus Pc_ oPc = P c _, c = identity = Pc«c_ =Pco Pc_,
(4.15)
Let Hp := | P C : TpM -> TpM, |c€<7(p)}. By (4.14) and (4.15), one can see that Hp with the multiplication " o " is a group. Hp is called the holonomy group at p. The following proposition shows how to use the holonomy group to construct Finsler metrics which are afflnely equivalent to a given Finsler metric. Proposition 4.2.2 Let (M, F) be a Finsler manifold. Let Hp denote the holonomy group of F at a point p € M. If Fp is a Hp-invariant Minkowski norm on TpM', then Fp can be extended to a Finsler metric F on M by parallel translations such that F is affinely equivalent to F. Proof. For an arbitrary point q € M, let c be a piecewise C°° curve issuing from c(0) = p to c(l) = q. Let Pc denote the parallel translation along c with respect to F. Define Fq : TqM -> [0, oo) by FQ(PC(V))
veTpM.
=FP(V),
If a is another piecewise C°° curve issuing from cr(O) = p to cr(l) = q. Then r := c~l * a is a loop at p. It follows from (4.15) that Pa = P M C - 1 O Pa = P c . c - 1 . , , = Pc O PT.
Since F p is i/p-invariant, one obtains Fq(Pa(v)) = F 9 (P C o PT(v)) = Fp(pr{v))
= Fp(v) = Fq(pc(v)).
Thus Fq is well-defined. From the above construction, one can see that for any F-parallel vector field U = U(t) along any curve c = c(t), F(c(t),U{t)\
= constant.
By Lemma 4.1.5, F is affinely equivalent to F.
Q.E.D.
79
Berwald Metrics
4.3
Berwald Metrics
Recall that a Finsler metric F on a manifold M is a Berwald metric if in any standard local coordinate system (x\ yl) in TM, the spray coefficients Gl = 7j,Tljk(x)yiyk are quadratic in y 6 TXM for all x € M. Riemannian metrics and Minkowski metrics are trivial Berwald metrics. There are many non-Riemannian Berwald metrics. Example 4.3.1 Let / = f{s, t) > 0, s > 0, t > 0, be a C°° function satisfying (1.18). Let (Mi,ai), i = 1,2, be Riemannian manifolds and M = Mi x Mi- Let F(x,y) := ^ / ( [ a a ^ ! , ^ ) ] 2 ,
\a2(x2,y2)]^,
where x = (xi,x2) € M and y = yi ® y2 € TXM S TXlMi © TX2M2. According to Example 1.2.5, F is a Finsler metric if and only if / satisfies (1.20) and (1.21). Let F(x,y) := v /[«i(^i.?/i)] 2 + [a2(x2,y2)]2. F is the standard product of oc\ and a2. By a direct computation, one knows that the spray coefficients of F are split as the direct sum of the spray coefficients of a\ and a2, that is, Ga(x, y) = Ga{xuyx),
Ga(x, y) = Ga(x2, y2).
(4.16)
Let c = (ci(t),c2(t)) be a curve in M and U = Ui(t) © U2(t) be a parallel vector field along c with respect to F. By (4.16), one can see that each Ui must be a parallel vector field along Ci with respect to on, i = 1,2. Thus Oii\Ci(t), Ui(t)) = constant. This implies that F(c(t),U(t))
= constant.
By Lemma 4.1.5, one concludes that F is affinely equivalent to F. The spray coefficients Gl of F are equal to that of F. Since F is a Riemannian metric, G% = ^rtjk(x)y:>yk are quadratic in y, then so are Gl. This shows that F is a Berwald metric.
80
Parallel Translations
Let (M, F) be a Berwald manifold. In any local coordinate system, uiN}(x,y) = ujrjk(x)yk
= y>Y)k{x)uk =
tfNJ&u).
Thus for any curve c = c{t) and any vector field U = V(t) along c, D6U(t) = V6U(t). Therefore, any parallel vector field along a curve linearly depends on its initial value. By Lemma 4.1.2, one immediately obtains the following Proposition 4.3.2 ([47]) Let (M,F) be a Berwald manifold. For any piecewise C°° curve c[t) from p to q in M, the parallel translation Pc is a linear isometry between (TpM, Fp) and (TqM, Fq). We know that if a Finsler metric is affinely equivalent to a Riemannian metric, then it is a Berwald metric. Lemma 4.1.5 gives us an idea to construct Berwald metrics from a Riemannian metric. In fact, every Berwald metric can be constructed from a Riemannian metric in this way. Proposition 4.3.3 ([97]) A Finsler metric F on a manifold M is a Berwald metric if and only if it is affinely equivalent to a Riemannian metric a. In this case, F and a have the same holonomy group Hp at any point p € M. Proof. Assume that F is a Berwald metric. We are going to construct a Riemannian metric a that is affinely equivalent to F. Let D be the LeviCivita connection of F. Let Hp denote the holonomy group of D. Hp acts on TpM leaving the indicatrix Sp of Fp invariant. Let Gp denote the subgroup of all linear transformations 7 : TpM —> TpM that leaves Sp invariant. Gp is a compact Lie group and Hp is a subgroup of Gp [98]. In general, Hp is non-compact. Fix a non-zero vector y0 G TPM and let gVo denote the induced inner product on TPM. Define a Euclidean norm ap on TPM by <*P(V)
- -TTTT /
\/gwo(7w,7v) d/i(7),
v € TPM,
where dji is the bi-invariant Haar measure on Gp. The constant A is chosen so that ap(yo) = Fp(y0). From the definition, one can see that ap is Gpinvariant, hence ifp-invariant. For any point q £ M, let c = c(t), 0 < t < 1,
81
Landsberg Metrics
be a piecewise C°° curve issuing from c(0) = p to c(l) = q. Define a Euclidean norm aq in TqM by a,(v):=a p (K(0)),
ver,tf,
where V = V(£) is the parallel vector field along c with V(l) = v. Since a p is i/p-invariant, aq is well-defined. One obtains a Riemannian metric a(q, v) := aq(v), q 6 M. From the above construction, one can see that for any C°° curve c and any F-parallel vector field U = U(t) along c, a( c(t), U(t) J = constant. By Lemma 4.1.5, one concludes that a is affinely equivalent to F. Q.E.D. According to Proposition 4.3.3, every Berwald metric is affinely equivalent to a Riemannian metric. This observation leads to the classification of Berwald metrics. Theorem 4.3.4 (Local Structure Theorem [97]) Let (M, F) be a Berwald manifold. Then (M, F) can be locally decomposed to a product of locally Minkowski manifolds, Riemannian manifolds and locally irreducible locally symmetric Berwald manifolds of rank r > 2. Since any two-dimensional Berwald manifold does not contain any locally irreducible locally symmetric Berwald manifold of rank r > 2, one obtains the following Corollary 4.3.5 ([97]) Any two-dimensional Berwald manifold is either locally Minkowskian or Riemannian. 4.4
Landsberg Metrics
Recall that a Finsler metric on a manifold is called a Landsberg metric if the Landsberg tensor vanishes, C = 0. See Definition 2.1.2 above. By Proposition 2.1.3, every Berwald metric is a Landsberg metric. But it is still an open problem whether or not there is a Landsberg metric which is not Berwaldian. Landsberg manifolds have some nice properties. For example, all slit tangent spaces with the induced Riemannian metric are isometric in a canonical way. This fact will be proved below.
82
Parallel Translations
Let (M, F) be a Finsler manifold. Take a standard local coordinate system (x\ y%) in TM. At each point x € M, Fx = F\TXM induces a Riemannian metric g x on TXM \ {0}, kx = gijdy1 ® dyi, where gij := \{F2\yiyj (x, y) and {dy1} is the global natural co-frame on the manifold TXM corresponding to the basis {^r| x } for TXM. Proposition 4.4.1 ([48]) Let (M, F) be a Landsberg manifold. Then for any piecewise C°° curve c from, p to q, the parallel translation Pc along c preserves the induced Riemannian metrics on the slit tangent spaces, i.e., Pc : (TPM \ {0}, gp) -» (TqM \ {0}, gg) is an isometry.
Figure 4.4 Proof. Without loss of generality, one may assume that the curve c = c{t),0 < i < 1, is embedded in an open domain U on which there is a non-zero vector field X = Xi^ £ C°°(rM) such that c{t) = Xc(t). Namely, X is an extension of the tangent vector field c = c(t) along c in a neighborhood U of c. The horizontal lift X of X to TMO is defined by
XM:=XHX){^-N^y)^}^. Let Ht and Ht denote theflowsof X and X in U and /n~1U respectively,
*fr{x) = XHt{x), *§?-(x,y) = XBUXty),
H0(x) = x, H0{x,y) = (x,y).
(4.17) (4.18)
83
Landsberg Metrics
By (4.17), Ht(p) - c(t). It is easy to verify that noHt = Ht. For a vector y € TPM, let U(t) := Ht(p, y). Observe that 7r(U(t)) = noHt(p,y)=Ht(p)
= c(t).
Thus U(t) = Ul(t)-^\c(t) is a vector field along c. The local coordinates of Ht(p, y) = U(t) in TM are (c*(t), ^*(*)). By (4.18), -w(P,y)
= Xu{t).
In local coordinates,
^W^jkw + ^W^fli/W =**(*){ J*-^W*),^(*))^f}lc; W . Thus !/*(*)+ c f c (t)^( C (t),^(*))=0. That is, C/ = C/(t) is a parallel vector field along c. This implies that for any r > 0, the map Pr := Hr(p, •) : TPM -
Tc(r)M
is the parallel translation along c r = c(t), 0 < t < r. In particular, we have Pt(c(0)) = Ht(j>,c(0)) = c(t). Let ^ e TPM \ {0} and rj : (-e, e) -> TPM be a C°° curve with 7?(0) = c(0) and ?j(0) = $. Let
We have H(t,0) = ^(p,c(0)) = c(t). The map H is a variation of c = c(i) in TMO. Put X(t,a):=—(t,S),
V ( t , a ) : = — (t,s).
84
Parallel Translations
Since X = X(s, t) is horizontal and V = V(s, i) is vertical, one has w\V) = 0,
un+i(X) = 0,
where u>* = dxl and wn+t = 5y% := dy* + N^dx?. Observe that X[un+i(V)] = du>n+i(X, V) + Vun+i(X)
+ un+i([X, V])
= dun+i(X, V)
= {u;n+iA
|[iV(gc(t))]&O = *[g(V;V0] =
x[gijujn+i(V)uJ^(V)]
= X[ffy]o;n+<(V)a;n+'(V) + 2 fly X[a; n+i (V)]a; n+ J(V) = 2faU™(iK+1(VK+i(y)
= -2flljLi^(iK+!(y)W'1+J(V). By assumption, the Landsberg curvature vanishes, C = 0, one concludes that
,gc(t)) is an isometry for any 0 < * < 1. Q.E.D. According to Proposition 4.4.1, if (M, F) is a Landsberg manifold, then for any loop c at p e M, the parallel translation P c is an isometry of the Riemannian tangent space (TPM \ {0}, gp). Thus Hp is a subgroup of the isometry group G of (TPM \ {0}, gp). The isometry group G is a Lie group [56]. Further, Hp is closed in G. Thus Hp is also a Lie group.
Landsberg Metrics
85
Proposition 4.4.2 ([59], [60]) For any Landsberg manifold (M,F), the holonomy group Hp at any point p € M is a Lie group. A natural question is whether or not the holonomy group of any Finsler manifold is a Lie group. Is there a Finsler manifold whose holonomy group is not the holonomy group of any Riemannian manifold? These problems remain open.
Chapter 5
S-Curvature
On a Finsler manifold (M, F), the indicatrix of Fx := F\TXM on the tangent space TXM can be viewed as the infinitesimal color pattern at x. Thus (M,F) is quite "colorful". If F is a Berwald metric, by Proposition 4.3.2, all tangent spaces (TXM, Fx) are linearly isometric to each other, namely, the infinitesimal color patterns are all the same over the manifold. Intuitively, Berwald manifolds have homogeneous "color". In particular, Riemannian manifolds are entirely "white". It is natural to seek for some quantities which measure the changes in "color" over the Finsler manifold. The Cartan torsion determines the shape of the indicatrix at a point, and the Landsberg tensor is a quantity which measures the changes of indicatrices. The Landsberg tensor is defined on the slit tangent bundle using the Chern connection. In this chapter, we will introduce two new quantities, the distortion and the S-curvature. The distortion is defined on each Minkowski tangent space, which measures the non-Euclidean property of the Minkowski norm, while the S-curvature is the rate of change of the distortion along geodesies. These two quantities are closely related to other quantities. 5.1
Distortion and S-Curvature
The indicatrix of a Minkowski norm on a vector space can be viewed as a color pattern on the vector space. The (mean) Cartan torsion is the (average) rate of tangential change of the color pattern along the indicatrix. Besides the (mean) Cartan torsion, there is another important quantity which also measures certain geometric properties of the indicatrix of a Minkowski 87
88
S-Curvature
norm. Let V be an n-dimensional vector space and let F = F(y) be a Minkowski norm on V. Fix a basis {bj} for V and let Vol(B") ffF
'"~ Vol{(y*)eR"
\F(yibi)
Using ap, we define an n-form on V, up to a sign, by dVF:=aFex
A---A0",
where {61} is a basis for V*, dual to {bj}. Clearly, ap depends on the choice of a particular basis, while dVp does not. dVp is well-defined. li F = yjgijyiyi, where y = ylbi, is a Euclidean norm, then
VolUy') e R" | F^bO < l} = y° 1( ° W \Thus op = yjdet(gij). Then dVF = ^det(gij)9l
A • • • A 8n
is the Euclidean volume form on V. If F is a Minkowski norm, gtj :— \{F2}yiyi{y) y/det(gij(y)), in general. Define
depend on y and ap ^
Wdet (gij(y)j r:=ln-i -.
(5.1)
Op
Both <jp and y/det(gij(y)) are transformed in the same way as the basis {hi} changes. Thus r = r(y) is well-defined, which is called the distortion of F [85], [87]. Observe that
v-^h/l4^]-^-^ where Cijk = \~§iv- Recall that the mean Cartan tensor I = Iidx1 is given by U = gikCijk. Thus Ii=Tyi.
(5.2)
89
Distortion and S-Curvature
By Deicke's theorem (Theorem 1.5.1), one concludes that F is Euclidean if and only if r — constant, in which case, r = 0. Therefore, a Minkowski norm is Euclidean if and only if r = 0. We summarize the above arguments in the following lemma. Lemma 5.1.1 For a Minkowski norm F on a vector space V, the following conditions are equivalent: (a) F is Euclidean, (b) T = constant and (c) T = 0.
Now we consider Finsler metrics. Let F be a Finsler metric on a manifold M. Since the distortion is defined for the Minkowski norm Fx on every tangent space TXM, we obtain a scalar function r — r(x, y) on TM \ {0}. We call it the distortion of F. By Deicke's theorem, F is Riemannian if and only if r = 0. Thus the distortion characterizes Riemannian metrics among Finsler metrics. It is natural to study the rate of change of the distortion along geodesies. For a vector y £ TXM \ {0}, let a = a(t) be the geodesic with
(5.3)
S(x,y) := jt[r{a(t),a(t))]\t=0. S = S(x, y) is positively y-homogeneous of degree one, S(a;,A2/) = AS(x,y),
A > 0.
S is called the S-curvature. In a standard local coordinate system (xl, yl), let CTVF = ap(x)dx1A- • -A n dx denote the volume form and Gl = Gl(x, y) denote the spray coefficients of F. It follows from (2.12) that
Then S=y
dx^'2WG
=
l ml^L 2
dx%
y
i _2IGi _ mj_r y
dxm
\ \
v
')
90
S- Curvature
There is another quantity associated with the S-curvature. Let Elj
:=l-SyiyJ(x,y)= \^-[^-]{x,y).
(5.5)
Then £ := Eijdxiig)dxj is a tensor on TMO. We call it the E-tensor. The Etensor can be viewed as a family of symmetric forms E y : TXM x TXM —> R defined by E y (u,u)
:=Eij(x,y)uivj,
where u = u'^rl*, v = v ^ \ x 6 TXM. Then E := {Ey | y e TM \ {0}} is called the E-curvature or the mean Berwald curvature. From the definition, it is clear that
where u,v G TXM, since S is positively y-homogeneous of degree one. Clearly, if S = 0 , then E = 0. The converse might not be true. But so far, no counter-example has been found yet. Proposition 4.3.2 tells us that every Berwald manifold is modeled on a single Minkowski space. Moreover, the geometry of tangent spaces does not change along geodesies. This observation leads to the following Proposition 5.1.2 ishes, S = 0.
([85]) For any Berwald metric, the S-curvature van-
Proof. Fix an arbitrary point (x, y) € TMO and let a = a(t) be an arbitrary geodesic with cr(O) = x and
By Lemma 4.1.1, gtj(t) — constant. Thus det (gij(t)j = constant. On the other hand, for any (yl) € Rn, the vector field U = ylhi(t) is linearly parallel along a. By Lemma 4.1.2, F(a(t),ylbi(t)J
= constant.
Thus the following convex subset Ut C R n is independent of t,
Ut := {(y>) eRn \ F^o-it)^^))
91
Distortion and S-Curvature
This implies that the coefficient of the volume form dVp is a constant,
(rF{a(t)) = ^
Y = constant.
Therefore, the distortion must be a constant along a, i.e.,
r U *(*)) = In V^(yW) =
constant
Thus S = 0 by (5.3).
Q.E.D.
A Finsler metric F on an n-dimensional manifold M is said to have almost isotropic S-curvature if there is a scalar function c = c(x) on M such that S = (n + l ) | c F + »j|,
(5.6)
where 77 = r)i(x)yl is a closed 1-form. F is said to have isotropic S-curvature if 77 = 0. F is said to have constant S-curvature if 77 = 0 and c = constant. Similarly, F is said to have isotropic E-curvature if there is a scalar function c = c(x) on M such that E=^-(n + l)cF-1h.
(5.7)
Here h is a family of bilinear forms h^ = hij(x, y)dx% ^dx^ on TXM, which are defined by hij := FFyiyj. It is clear that if F satisfies (5.6), then it satisfies (5.7). But the converse might be false although no counter-example has been found.
Example 5.1.3 ([89]) Let 6 = O(x,y) denote the Funk metric on a strongly convex domain W c R " (see Example 3.4.3). Let { , ) denote the inner product on R n and a 6 R" be a constant vector. Let
F:=Q(x,y)+
{
°"V) 1 + (a,x)
yeTxU^Rn.
Since F(0, y) = 0(0, y) + (a, y) and 6(0, y) is a Minkowski norm, by continuity, one can see that F is a Finsler metric in a neighborhood of the origin
92
S-Curvature
in R™ for sufficiently small vector o. According to Example 3.4.6, the spray coefficients of F are given by G* = Py%', where
Thus F is projectively flat. A direct computation using the homogeneity of P gives dGm — = (n + l)P. Observe that S = (n + l ) P - j , ™ ^ ; ( l n ^ )
= (n + l){±F + #}, where
f:=-1n[(l
+ (a,x))
(5.8)
Thus the S-curvature is almost constant. It follows from the above formula for S that
E= ^(n+lJF- 1 ^ Thus the E-curvature is constant. 5.2
Flanders Metrics of Isotropic S-Curvature
First of all, let us take a look at the following example. Example 5.2.1 Let F:=
_____
+ u(^)'
^
where a € R™ is an arbitrary constant vector with \a\ < 1. F is a Randers metric denned on the whole unit ball B"(l) C R". By Example 5.1.3,
93
Randers Metrics of Isotropic S-Curvature
G% = P(x, y)y% where P = P(x, y) is given by 1 f V\y\2 - (|a:|2M2 - {x, y)2) + (x, y)
21
l-l^l
2
{a, y)
}
l + (o,x)J"
According to Example 1.3.1, aF{x) =
(l + (a!x])n^
Plugging it into (5.8) yields that h(x) = —In-^/l — \a\2. Hence
Thus F has constant S-curvature.
It is a natural problem to study and characterize Randers metrics with isotropic (or constant) S-curvature. Let F = a + (3 be a Randers metric on an n-dimensional manifold M, where a = ^aij{x)yiyi and /3 = bi(x)yl. Let P:=\n^\-\\f3x\\l.
By (1.28), the volume forms (WF and dVa are related by dVF = e ( n + 1 ) "( x ^F Q . We continue to use the same notations as in Lemma 3.1.2. The spray coefficients G* — Gz(x, y) of F and the spray coefficients Gla — Gla(x, y) of a are related by (3.6) and (3.7),
& = Gi + Py* + Q\ where p
-=%-
s
o,
Ql = <*s\,
where eOo := eijylyj, SQ •= styl and s\ := sl^yj are denned in (3.3) and (3.4). Since Sij + Sji = 0, we have that SQO := s^y^y1 = 0 and
94
S-Curvature
sli = al^Sij = 0. By the homogeneity of P and the anti-symmetry of Sij, we have d(Pvm) dP -j±- = a'hoo + as™m = 0. Since a is Riemannian, the following holds,
^=^»i
= ^(V/d^)=l/"'^(ln4
where fljkdenote the Christoffel symbols of a. By the above identities, we obtain
dGZ -dym+
d(Py^) dym
dQ™ + dym
dp
d /
N
& + l)V Qxm V Qxm {lll(T«j
= (n + l){P- Po } = (n + l){j^-(so+Po)},
(5.10)
where p0 := pxi(x)y\ Lemma 5.2.2 ([25]) For a Randers metric F = a+/3 on an n-dimensional manifold M, the following are equivalent (a) S = (n + l)cF, (b) E = \{n + \)cF-^, (c) eOo = 2c(a2-(3*), where c = c{x) is a scalar function on M. Proof. From the definitions of S and E, it is obvious that (a) => (b). (b) => (c): The condition (b) implies that
S = (n + l){c(x)F + 7?}, where rj = r]i(x)yt is a 1-form on M. By (5.10), (b) is equivalent to the following e00 = 2cF2 + 26F,
95
Randers Metrics of Isotropic S-Curvature
where 6 := so + Po + V- This implies that e00 = 2c(a2 + 01) + 20(3,
0 = 4c0 + 20.
Solving for 6 from the above equation on the right, then substituting it into the equation on the left, we obtain (c). (c) =*- (a): Substituting eOo = 2c(a2 - j32) into (5.10) yields
S = (n + l){c(a-/?)-(s o + po)}.
(5.11)
On the other hand, contracting etj = 2c(a,ij — bibj) with b> gives Si + pi+ 2cbi = 0. Thus so + po = -2c/?. Substituting it into (5.11) yields (a).
Q.E.D.
Example 5.2.3 Consider the Randers metric F = a + ft on Rn, where a and (3 are defined by yj{l-e>)(x,y)*+e\y\*{\+e\x\>) l+e\x\2 ft '
P
=
Vi - g2 (x, y) l + e\x\2 '
where e is an arbitrary constant with |e| < 1. Note that (3 is closed. Thus Sij = 0 and s, = 0. By computing frj.j, one obtains eij
~
(l +
e\x\2)(e+\x\*)ij-
On the other hand, aij bibj:=
-
l+£e\x\*Sij-
Thus e^j = 2c(a,ij — bibj) with ° := 2{e + |x| 2 )" By Lemma 5.2.2, F has isotropic S-curvature and E-curvature, i.e., S = (n + l)cF,
-E=\cF~lh.
96
S- Curvature
Since a is not projectively flat and /? is closed when e ^ 0, by Proposition 3.4.8, one can see that F is not projectively flat. It is known that a Finsler metric F = F(x, y) on a manifold M is a Randers metric if and only if it is a solution of the following equation:
where h = y/hij (x)yxyi is a Riemannian metric and V = ^ l ( ^ ) ^ r is a vector field with h(x, -Vx) = v //i i j (x)y i (x)^'(x) < 1. F is given by (3.8), F=VWT%_v»
A
(512)
A
where Vo := Vtf and A = 1 - \\V\\\. Let Tlij :=. ±(ViU + V^), Ilj := V^j, U := ViTlijV:i be defined as in Lemma 3.1.3. By a direct computation, we obtain from (3.11) that
Let dVp = crpdx1 • • -dxn and dVh = ah denote the volume form of F and h respectively. By Lemma 1.4.2, dVp = dVh, i.e., uF = oh. Since h is a Riemannian metric, we have
Then it follows from (5.4), (5.13) and (5.14) that
S = ^ { 2 ^ o - Koo " F2K}.
(5.15)
Let
Since \\V\\h < 1, the vector f := C^\x by (3.10), it is easy to verify that hijCe =h2-
€ TXM can be arbitrary. Moreover,
2FVQ + (1 - A)F2 = F2.
By (5.15) , we obtain the following
(5.16)
Randers Metrics of Isotropic S-Curvature
97
Lemma 5.2.4 For an n-dimensional Randers metric F expressed by (5.12), the S-curvature is given by S__n F~
+ 2
l^e|i hijW
(5J7)
By (5.17), we immediately obtain the following Proposition 5.2.5 Let F = a + ft be an n-dimensional Randers metric expressed in terms of a Riemannian metric h — \fhij{x)y%yi and a vector field V = V^xjgfr by (5.12). Then F has isotropic S-curvature, S = (n + l)c{x)F if and only ifV satisfies Kij = -2c(x)hij.
(5.18)
By Lemma 5.2.2, the condition S = (n + l)c(x)F is equivalent to the following equation eoo = 2 c ( i ) ( a 2 - / ? 2 ) .
(5.19)
It is implicitly shown in [8l], [9] and [7] that (5.19) is equivalent to (5.18). But the arguments are embedded in the proof of the theorem on Randers metrics of constant flag curvature. Under the constant flag curvature condition, (5.18) and (5.19) hold for a constant c. A direct proof of Proposition 5.2.5 was first given by then an undergraduate student at Beijing University, H. Xing, shortly after he received the paper [7]. Xing's argument does not use (5.17). In what follows, we are going to give explicit formulas for V = V*^r and c = c(x) satisfying (5.18) when the underlying Riemannian metric h is of constant sectional curvature [94]. First we prove the following Lemma 5.2.6 Let h = ^/h^y^ be a Riemannian metric on an ndimensional manifold M. Let V = V%-^ be a vector field on M satisfying (5.18) for some scalar function c = c(x) on M. Then Vfcj.jj- = 2{c)fc/ly - C^hjk - cyftfci} - VmR/li,
(5.20)
where C|» = cxi are the usual partial derivatives of c in local coordinate systems and Rjmki denote the coefficients of the Riemann curvature tensor
ofh.
98
S- Curvature
Proof. First, differentiating (5.18) and exchanging the indices, we obtain Vi\j\k + Vj\i\k = -4c| f c /iij,
(5.21)
Vjiku + VkW = -4e\ihjk,
(5.22)
Vk\i\j + Vmi
(5.23)
= -icyhki.
Adding (5.22) and (5.23) together, then the sum being subtracted by (5.21), we obtain (ViWj - Vim) + (Vmi - VMk) + (VkW - Vmj) + 2Vmj = 4c\khij - Ac\ihjk - AcyhkiUsing the Ricci identity, Vk\%\j\ — Vk\j\% = VmR-iPij' w e obtain VmRzmkj + VnR^i
+ VrnRfii + 2VkWj = 4c, f c ^ - 4c\ihjk -
By applying the Bianchi identity, R^kj+R^ji+Rflk
= °>
we
AC]jhki.
obtain (5.20). Q.E.D.
By Lemma 5.2.6, we obtain another identity on c and V. Lemma 5.2.7 Let (M, h) be an n-dimensional Riemannian manifold. Let V = V ^ r be a vector field on M and c — c{x) be a scalar function on M satisfying (5.18) for some scalar function c = c(x). Then c and V satisfy 2Jc|i|j/ijfc +c\j\khuj - 2|c|j||/ifei + c\i\khji|
+Vm\jRimki - Vm]iR3mkl + V^RRj
- V^kR^j.
(5.24)
Proof. By (5.20), V
i\i\k = ^{c\ihjk - c{jhki - clkhij} - VmRftj.
(5.25)
Differentiating (5.25) yields Vi\j\k\i = 2{c|i|jftifc - cmhki
- cwhij}
- VmliRkmi:j - VmRkmm.
(5.26)
Exchanging the indices k and I yields Vi\j\i\k = 2{c|i|fc/ijj - c{j\khu - C|j|fc/iij} - V^kR^j
- VmR^.
(5.27)
Flanders Metrics of Isotropic S-Curvature
99
Note that c = c(x) is a scalar function, thus c\ky = c^k. It follows from (5.26) and (5.27) that Vi\j\k\i - Vi\j\i\k = 1\c\i\ihjk + cyikhiij - 2^c\j\ihki + c\i\khjij + Vm\kRt™j - Vm\lRk\j + Vm{Ri™j\k - Rk"ij\ljApplying the Ricci identity V^|jlfc|( - Vi]mk = V^jR^ identity (5.18) to the above identity, we obtain
+ V^mR^
and the
^
= 4cRijkl + Vm{RkmiJit -
R^j]k}
+VmiJRi™[ - VmiiR/^i + Vm\iRk"\j - VmftRi™}. This gives (5.24).
Q.E.D.
Lemma 5.2.8 Let h = y/hijylyi be a Riemannian metric of constant curvature K^ = (j, and V = V1 -g~ be a vector field on an n-dimensional manifold M satisfying (5.18) for some scalar function c = c(x). Then c satisfies the following equations c\i\j + nchij = 0, Ac + 2/ic = 0,
{n > 2) (n = 2)
(5.28) (5.29)
where c\i\j denote the covariant derivatives of c with respect to h and Ac := h%ic\i\j denotes the Laplacian of c with respect to h. Proof. By assumption, h has constant curvature,
By (5.18), we obtain from (5.24) that \c\i\ihjk + c\j\khuj - \c\j\ihki + c\nkhjij = 2/j.c^hjihik - hjkhzij. (5.30) For the sake of simplicity, we may choose an orthonormal basis at a point so that hij = Sij. In (5.24), letting fe = j and I = i (i ^ j) yields C]ill+cmj
+ 2fic = 0
(i^j).
(5.31)
100
S-Curvature
When n > 3, it follows from (5.31) that C|<|i + fic = O.
(5.32)
For any i, I, there is m ^ i, 1. In (5.24), letting j = k = m, one obtains c\i\i + c]m\m8u + 2iicSa = 0.
(5.33)
By (5.32), C|m|m = -fie. Substituting it into (5.33) yields (5.28). In dimension two, (5.29) follows from (5.31) directly. Q.E.D. Next we are going to find an explicit formula for the vector field satisfying (5.18) when the Riemannian metric h is of constant curvature K/, = /x. It is known that h is locally isometric to the following metric h^ = \/hij-yiy^ on the ball B"(rM) C R n , where rM := +oo if /x > 0 and rM := ir/y/^Jl,
y€rxB-(rM)-R-.
h^VW±^WZEm,
(5.34)
The metric coefficients hy are given by ,
._ ij
yx%xi
Sjj
'" 1 + fi\x
2
(l + n\x\2)2'
The inverse matrix of (hij) is given by
/^ = (l+/i|i| 2 ){^ + ^zV}. Then the Christoffel symbols of 7J-fe of /tM are given xHik+xk8ij
__
i ljk
~ "M
I + MM2
•
Thus for a tensor T = Tidxi, the covariant derivative DT = Ti\idxi ® dxj is given by
T j=
^
dxi+fi-mw
(5>35)
Lemma 5.2.9 / / a scalar function c — c(x) satisfies (5.28) on the Euclidean ball (B ra (r /i ),/i M ), then
c=±±h±, /i
i
I
19
(5.36) ^
'
101
Randers Metrics of Isotropic S-Curvature
where X is a constant and a € Rn is a constant vector. Proof. By (5.35), we obtain _ X%Cxj + XjCxi C\i\j - Cxixi + H j + / i | x | 2 ,
where cxi = ^ and cxixj = dxi£xj denote the partial derivatives of c. Let / := \Jl + n\x\2 c. We have fxixi = \ / l + /i|a:| 2 |c|i|j + nchijj = 0. Thus f = \ + {a,x), where A is a constant and a € R™ is a constant vector. We obtain the explicit formula (5.36) for c. Q.E.D. For the above scalar function c in (5.36), we can solve (5.18) for V. Proposition 5.2.10 ([94]) Let h = h^ be defined in (7.38) and V be a vector field on the Euclidean ball (B"(r^), h^). If n > 3, V = V1-^ satisfies (5.18) for some scalar function c = c(x), then c is given by (5.36) and V is given by
V = - 2 {(AyTM^+ (a,x))x -
lf!!l
} (5.37)
+xQ + b + fi(b, x)x,
where Q = (g •*) is an antisymmetric matrix and b = (bl) G R" is a constant vector. Conversely, for any dimension n > 2, the vector field V = (V1) in (7.40) satisfies (5.18) for the scalar function c = c(x) in (5.36). Proof. It suffices to solve (5.18) for the scalar function in (5.36). We divide the argument into two cases. Case 1: n = 0. Let Pi:=Vi-\x\2ai
+ 2(\ + (a,x))xi.
Then (5.18) is equivalent to dxi
dxi
102
S-Curvature
By an elementary argument (see [7]), we get
pt = aSq/ + b\ where Q = (q^1) is an antisymmetric matrix and b = (bl) e R n is a constant vector. We obtain
Vi = Vi = -2{(A + (a,xfix* - ^-o}) + xjq/ + b\ Case 2: fi^O. Let P. — V- — —r •
M Then Pi satisfy Pib- + P, H = 0.
(5.38)
Using (5.35), we can rewrite (5.38) as follows: i
Pj+xjPi_0 * l+{i\x\2
9Pi , 9P, dxi dx*
2*
Let Hi := (1 + »\x\2)Pi. We obtain dHi
dHi
.
.
l2 .
2
i-aPi
5P 7
^ 7 + ^ f = (i + MM ){^- + ^
^ Pi + xiPi)
+^ ^ ^ f )
= o-
By an argument similar to that for P» in the case when \i = 0, we obtain
where Q = ( ? / ) is an antisymmetric matrix and v = (v%) G R n is a constant vector. Thus
Pi = (l+/,|z|2)-1{:rV + ^}A direct computation yields Cl
-
ai
_M(A + ( a , x ) y
103
Randers Metrics of Isotropic S-Curvature
We obtain Vi = Pi + -c\i /-,
i
2s i f •• i
2 a i
it
2(X +
(a,x))xi
Finally, we completely determine V = h*-7'^-. 0 = 2^1+n\x\i{ii~1ai
- A*'} + a^'g/ + v* + ^(v, i>x*.
We express
M"Vl +A*|a;|2 =
M"1{\/1
+M^I2 - l} + M'1
X\2
_x
Let
We obtain
^ = -2(AVTT7^F+ (a,x))^ + / J ! f ^
1
Q.E.D. Assume that a vector field V satisfies (5.18) on the Euclidean ball (B"(rM),/iM) for some constant c. By Lemma 5.2.8, we see that c = 0 if yu T^ 0. Then (5.28) always holds regardless of dimension. By Lemma 5.2.9, c can be expressed by _ A + (a, x) c
~
V I + MM 2 '
where A = 0 and a = 0 if (j, ^ 0, and A = c and a = 0 if pi = 0. By the argument of Proposition 5.2.10, we obtain the following Corollary 5.2.11 Let h — h^ be defined in (5.34) and V be a vector field on the Euclidean ball (B n (r^),/i^). IfV - Vi^T satisfies (5.18) for some
104
S- Curvature
constant c, then c = 0 i / / i ^ f l , and V is given by J-2cx + xQ + b \xQ + b + fi{a,x}x
if fi = 0 if»^o>
^•^>
where Q is an anti-symmetric matrix and a,b G Rn are constant vectors. Conversely, ifV is a vector field satisfying (5.40), then V satisfies (5.18) with c(x) =c(=0ifn^0). 5.3
An Equation on the S-Curvature
The main purpose of this section is to establish an equation on the Scurvature. This equation together with (2.70) gives a relation between the S-curvature and the Riemann tensor. Lemma 5.3.1
([24], [73]) S.k\mym m
S.klmy
- S|fc = Ik\P\qypyq
+ ImRmk,
(5.41)
m
(5.42)
m
- S, t = -\{2R k.m
+ R m.k}-
Proof. Equation (5.42) follows from (2.70) and (5.41). Thus it suffices to prove (5.41). Take a natural local frame {e, = di} for ir*TM. Let {cj\ujn+l} = {dxl,5y1} be the corresponding local coframe for T*(TMO). Write dr = T\HJ + T.iujn+i.
(5.43)
r.i = h.
(5.44)
It follows from (5.2) that
Differentiating (5.43) and using (5.44), we obtain 0 = d2r = {r{kliujl + Tlk.njn+l} Auk
+ [ik]lJ
+ ik.tLjn+l}Aujk
+
imnn+m.
This yields the following Ricci identities r\k\i = TWk + ImRmki, m
r\k-i = h\k - ImL kl.
(5.45) (5.46)
An Equation on the S-Curvature
105
From the definition of the S-curvature, we have S = T{mym.
(5.47)
Contracting (5.46) with yk and using (5.47), we obtain S.fe = (T| m y m ). fe = Tlm.kym + r\k = h\mym-IiLlmkym m
= h\my
+ T\k
+ r\k = Jk+ r\k.
(5.48)
It follows from (5.48) that S.fc|j = T)fc|i + Jfc|,.
(5.49)
By (5.45) and (5.49), we obtain
S-k\mym - S|fc - {S.fc,m -
S.mlk}ym
= y]k\m - T\m\k}ym + | Jk\m ~ Jm\k}ym = ImRmk + Jk\mymSince Jk\mym
= h\P\qypyq,
we obtain (5.41).
Q.E.D.
Equations (5.41) and (5.42) reveal some important relationships among the S-curvature and the Riemann tensor. We shall use them to establish some global rigidity theorems for Finsler metrics.
Chapter 6
Riemann Curvature
In the previous chapter, we discussed some non-Riemannian quantities. These quantities all vanish for Berwald metrics. In particular, they vanish for Riemannian metrics. In Riemannian geometry, the Riemann tensor is the most important quantity that measures the "curvature" of the space at a point. The Riemann tensor is first extended by L. Berwald to Finsler metrics [14], [15]. Our goal is to understand the geometric meaning of this important quantity. 6.1
Riemann Curvature
Let (M, F) be an n-dimensional Finsler manifold. Let {t>,} be a local frame for TM and {0*} be the local coframe for T*M. Then {e* := (x,y,bi)} is a local frame for ir*TM and {ui1 := TT*81} is the dual local coframe for ir*T*M. The Riemann tensor R = R1^ <8>u;fe is denned in (2.43) using the Chern connection. It can be viewed as a family of linear maps
x e M},
K={-Ry\y€TxM\{0},
where Rj, = fl^b* ® 6k : TXM -> TXM is denned by TLy{v) := R'^x, y)vk bu
v = w'bj e TXM.
R is called the Riemann curvature. We have Rv(y) = 0,
RXy = \2Ry, 107
A>0.
(6.1)
108
Riemann Curvature
By (2.44), R y is self-adjoint with respect to gj,, gy(Ry(u),
v ) = gy(u, Ry(vfj,
u,ve TXM.
(6.2)
(6.1) and (6.2) are the basic equations of the Riemann curvature. For a tangent plane P CTXM containing y, let
K(P,y) := —,
r-7
rrl
U2>
(6-3)
where u € P such that P = span{y, u } .
Figure 6.1 By (6.1) and (6.2), one can easily verify t h a t K ( P , y) is independent of the choice of a particular vector u € P such t h a t P = span{y, u}. K = K ( P , y) is called t h e flag curvature. In dimension two, P = TXM is t h e tangent plane. Thus the flag curvature K = K ( x , y) is a scalar function on T M \ { 0 } , which is called the Gauss curvature. Let n
Ric^^y'g^R^bO.b,-), where { b j is a basis for TXM, gij := g 2/ (b i , bj) and (gtJ) = (gij)"1- Ric is a well-defined scalar function on TM\ {0}. We call Ric the i?icd curvature (or .Ricci scalar). In a local coordinate system, Ric = ^ % = i ? ^ .
109
Riemann Curvature
Proposition 6.1.1 Let F be a Riemannian metric on a manifold M. For any tangent plane P C TXM, the flag curvature K(P,y) = K(P) is independent ofyeP\ {0}. Proof. Since F is Riemannian (Cijk — 0), gij = gij(x) and Rfkl = Rj*kl(x) are functions of x € M. Thus Rjiki := gimRfki a r e functions of x € M. It follows from (2.34) and (2.64) that Rjikl = -Rijkl = Rijlk-
This implies that Rik(x,y)uiuk
= Riiki{x)yjyluiuk
- Rijlk(x)uiukyjyl
= Rji(x, u)y>yl.
Thus for any tangent plane P = span{y, u} C TXM, K
( p «) = fiifc^.yy"* {9ji(x)gik(x) - gi:j(x)gki(x)}y^yluluk =
RjiMvjyl {9jl{ )9ik{x) - gij(x)gki(x)}y3yluluk x
= K ( p u)
That is, the flag curvature K(F, y) = K(JP) is independent of y € P \ {0}. Q.E.D. For Riemannian metrics, the flag curvature K = K(P) is called the sectional curvature of the section P C TXM. In dimension two, the Gauss curvature K = K(x) is a scalar function on M. Definition 6.1.2 Let F be a Finsler metric on an n-dimensional manifold M. JP is said to be of scalar (flag) curvature if K.(P,y) = ~K(x,y) is a scalar function on TM \ {0}. F is said to have isotropic flag curvature if K(P, y) = K(i) is a scalar function on M. F is said to have constant flag curvature if K(P, y) = constant. F is called an Einstein metric if there is a scalar function K = K(x) on M such that Ric = (n - 1)KF 2 .
It follows from the above definition that a Finsler metric F = F(x, y) on a manifold M is of scalar flag curvature with flag curvature K = K(x, y)
110
Riemann Curvature
if a n d only if for any y,u € TXM \ {0},
Ry(u) = K {gy(y, y)u- gy(y, u) t/}. This is equivalent to the following equation:
R\=KF2h\, where h\ := Svk - F-2gkqyqy\ Compare (2.51). There are plenty of Finsler metrics of scalar flag curvature. First, let us prove the following Proposition 6.1.3 ([17]) Any locally projectively flat Finsler metric is of scalar flag curvature. Proof. Let F be a locally projectively flat Finsler metric on a manifold M. By definition, at any point x e M, there is a standard local coordinate system (x\yl) in TM such that the spray coefficients are in the following form G% = Py\ Substituting them into (2.49) yields
™ ^oW) k dxk
r
j&JPy1) , Q , P ^ W ) dxWyk KW)dyidyk
= [P2 - yiP^yi
djPy^djPyi) dyi dyk
+ {2Pxu - tfPxiyk - PPyk}y\
We obtain R\=S5i+rky\
(6.4)
where
E:=P2-Pxkyk,
Tk =
3(Pxk-PPyk)+Zyk.
We have rkyk = -E
(6.5)
and
Rjk •= 9ijR\ = Zgjk + ngav1By (2.44), Rjk = Rkj. This implies rk9ijVl = Tjgikyl-
(6.6)
111
Riemann Curvature
Contracting (6.6) with y> and using (6.5) yields TkF2 =
-Egikyi.
Namely, T^ = —EF~1Fyk. Then we obtain
R\ = SSI + ny* = -{% - F-'Fyty*}.
(6.7)
Thus F is of scalar flag curvature and its flag curvature is given by 5 P 2 - Px,yk F2~~ F2 '
^
;
Q.E.D. Example 6.1.4 Q =
Consider the following family of Riemannian metrics: y € T l B (r ) = R
"-
fT^F
'
^
-
According to Example 3.4.2, the spray coefficients Gl = Pyl, where i + /i|i| 2 '
By a direct computation, we get p
fc__
xl=y
~
|y| 2 (l+Ai|»l 2 )-2/i(x,y) 2 M ( I + MI^I 2 ) 2
By (6.8), we obtain that K = /i. Thus aM has constant sectional curvature. In Riemannian geometry, E. Cartan's local classification theorem asserts that any Riemannian metric of constant sectional curvature /J, is locally isometric to aM. Example 6.1.5 B n (l) C R n , p
F
=
Consider the following Randers metric on the unit ball
V\y\2~(\x\2\y\i-{x,y)i) + {x,y)
r^p
+
(a,y)
r?M'
(6 9)
-
112
Riemann Curvature
where a e Rn with \a\ < 1. By Example 5.2.1, the spray coefficients of F are in the form G% = Py%, where p
_ l f V M 2 - ( N 2 | y | 2 - ( g , y ) 2 ) + (g,i/) 21 1-kl2
(a,y) ] l + (a,x>r
Thus, F is projectively flat. By a direct computation, p
k X V
"
l(W\y\2-(\x\2\y\2-(x,y)*j+(x,y))2 ~2l (1-kl2)2
+
(a,y)* -, (l + ( a , z ) ) 2 / '
By (6.8) we obtain that K = -1/4. See [88] for more details. The Finsler metric in (6.9) is a special Randers metric. Let us take a look at more general Randers metrics. Let F = a + f3 be an n-dimensional Randers metric on a manifold M, where a = \faij(x)yiyi is a Riemannian metric and /3 = bi(x)yl is a closed 1-form. We shall continue to use the same abbreviations as in Lemma 3.1.2. The spray coefficients Gl of F and the spray coefficients Gla of a are related by G% = Gla + Py% + Q\ where r>
e
P
00
-=^p~
s
o,
r\i
i
Ql = as\.
In general, the expression for the Riemann curvature is very complicated. Thus we assume that the 1-form f3 is closed. Then Sij — 0 and Si = 0. In this case, Gl = Gla + Py\ where F
-2F~SO~2F-
Substituting Gi = G^ + Pyi into (2.49) yields
^ = ^ + ( 3 ( ^ ) 2 - i ) { « i - ^ } + ^yS
(6-io)
where $ := &iy-j/Y,
* := 6ia-;fci/Vi/*,
Tk = j (bi.j.k - bi.M.J)yiy>. (6.11)
We can use the above formula to calculate the Riemann curvature when (3 is closed. Example 6.1.6 ([24]) Let (S n ,a) be the standard unit sphere in R n + 1 and f(x) := sxl, where e is an arbitrary nonnegative number and xl is
113
Riemann Curvature
one of the position functions
U,j = -tijf,
where the covariant derivatives of / are taken with respect to an orthonormal frame of a. It follows from (6.12) that S :— f(x)2 + \dfx\2 is a constant. We choose a small e such that S < 1. Define
F := a(x,y) + P(x,y), where /3 = bi{x)y% is given by b
/;»(*)
Note that /3 is closed. According to Example 3.3.2, F is projectively equivalent to Q. Since a is locally projectively flat, F must be locally projectively flat. Moreover, the geodesies of F are great circles. Since (3 = dip is the differential of the scalar function, ip — - arcsin(/(x)), the great circles have F-length of 2n. To see this, take an arbitrary great circle C parametrized by c = c(t), 0 < t < 2n, where t is the arclength parameter with respect to a. Then
F(c(t),c(t)) =
l+v'(t),
where tp{t) :=
Jo
F{c(t),c(t))dt=
Jo
(l+
This proves our claim. Now let us compute the S-curvature. Let
By (6.12), we obtain Po •= Px*Vl = — /
2 /?-
114
Riemann Curvature
On the other hand,
, USij + f2(fj6ik + frSjk) (1 + 2/ 2 )/ ; i / ; j / ; f c °i;i;k ^ _ f2y/2 ^ _ /2^5/2 •
(0.13)
This gives
By (5.10), we obtain
Thus the S-curvature is isotropic. Since F is projectively flat, by Proposition 6.1.3, it is of scalar flag curvature. By (6.10), the flag curvature is given by
•<=iM<M50'-£}-
<6»»
Since (3 is closed, by (6.13), the following holds
and
.-££(•>-*> Substituting them into (6.14) yields 3 4(1-/(x)2)
F(g,-y) F(I,J/)
+
1 4'
We see that the flag curvature always satisfies the lower bound K > 4.
Second Variation of a Geodesic
115
Finally, let us take a look at the Hilbert metric. Example 6.1.7 Let H = H(x,y) be the Hilbert metric on a strongly convex domain U C R n ,
where 0 = Q(x, y) is the Funk metric on U and © := Q(x, -y). © satisfies (1.38) and 0 satisfies (3.21). The Hilbert metric is a reversible Finsler metric. Usually H can't be expressed in terms of elementary functions unless U is denned by a Randers norm. According to Example 3.4.5, H is projectively flat and its projective factor P is given by
,-i{e-S}. Substituting the above expression for P into (6.8) and using (1.38), one obtains that (0 - 9) 2 - 2(0^ -Sxk)yk (0 + 0)2
^ (0-0)2-([0V+[0V)yfc (0 + 0)2 _ (0 - e) 2 - 2(02 + 6>2) Thus H has constant flag curvature. The Hilbert metric on the unit ball B"(l) C R™ is just the Klein metric discussed above. It is also an interesting problem to study Finsler metrics with K(P, y) = K(P) independent of y € P for every tangent plane P C TXM. Obviously, Finsler metrics with constant flag curvature have this curvature property. There are other Finsler metrics having this curvature property. Further investigation is needed. 6.2
Second Variation of a Geodesic
The geometric meaning of the Riemann curvature lies in the second variations of a geodesic. There are two types of variations for a geodesic a: (i)
116
Riemann Curvature
the variation of the length function of curves in a neighborhood of a (fixing the endpoints) and (ii) the variation of geodesies in a neighborhood of a. First we consider the variation of the length function. Let (M, F) be a Finsler manifold and let a = cr(t), a < t < b, be a geodesic in M. Let H : [a, 6] x (—e, e) —> M be a C°° variation of a, H(t,0)=a(t),
a
Figure 6.2 Let C(s) denote the length function of the curves crs(t) :— H(t, s), a
Ja
F(
Assuming that H fixes the endpoints, H(a, 0) = a{a),
H(b, s) = a(b),
\s\ < e.
Let
and F x (t) denote the orthogonal complement of V(t) with respect to ga(t), V±(t) :=V(t) -
g&{t)(a(t),V(t)).
Since a is a geodesic, £'(0) = 0 (see Section 3.2). By a direct computation, we obtain C
"W=f
{&*n(D>VJ-(t),D&VJ-(t)) -g*(0 ( ^ ( 0 ( ^ ( 0 ) . ^X(*))}dt,
(6.15)
where D&V1-(t) denotes the linear covariant derivative of V1-(t) along a. By (6.15), if the flag curvature is negative and VL{t) ^ 0, then £"(0) > 0 and
117
Second Variation of a Geodesic
hence any geodesic a has minimal length among curves in its neighborhood with the same endpoints. Now we discuss another type of variation of a geodesic: the geodesic variation. Let a = cr(t), a < t < b, be a geodesic in a Finsler manifold (M, F). Let H = H(s, t) be a variation of a such that each curve as(t) := H(t,s), a < t < b, is a geodesic. Such a variation is said to be geodesic. Let J(t) :=—(t,0). We are going to show that J(t) satisfies D&D&J(t) + R&{t)(J(t))
= 0.
(6.16)
By assumption, each as is a geodesic. Thus ^
+ 2 G
.(if,f)=,
For simplicity, let &r» '
dx% '
dt '
ds '
Figure 6.3 Equation (6.17) becomes —
+ 2G*(H,T) = 0.
Observe that ds
dsdt
dt
(6.18)
118
Riemann Curvature
and
|[G'( 5 ,T)]=^g(F,T) + fiV-(if,r), 9 dN \NUH T)] Tk dN * (H T) + ***—^r{H,T) * (H T) -[Nj(H,T)\=T —(H,T)
dN* dNi k = T ^(H,T)-2G (H,T)^(H,T). k
(6.19)
Differentiating (6.18) with respect to t yields
Using the above identities, one obtains
One can express the Riemann tensor R%k denned in (2.49) as follows k
"~ dxk
* dxi +
dyk
>
V
Thus (6.20) can be expressed as D T D T t/ + R r ( ^ ) = 0.
(6.21)
Equation (6.21) restricted to a = a0 is just (6.16). The above proof is due to [57], [58]. Any vector field J(t) satisfying (6.16) is called a Jacobi field along a. Jacobi fields play an important role in Finsler geometry [87]. For a geodesic a = a(t),t > 0, issuing from a point
t > 0, \s\ < e.
Second Variation of a Geodesic
119
Figure 6.4 The variation field J(t) :=-^(t,0)
= d(expx)\ty(tv)
is a Jacobi field along a. Under certain condition on the flag curvature or the Ricci curvature, one can estimate the zeros of J(t), hence the singularity of dexpx on TXM. The sphere theorem is a classical global result in Riemann-Finsler geometry. It states that a simply-connected and closed manifold with a Riemannian metric satisfying 1/4 < K < 1 is homeomorphic to the n-sphere [54]. P. Dazord extended this theorem to reversible Finsler manifolds [32] [33] (see also [87]). For a general Finsler metric F on a manifold M, let A :=
sup
F{x, -y) -^r r-.
Obviously A > 1 and A = 1 if and only if F is reversible. Thus the number A is called the reversibility of F by H.B. Rademacher. Recently, he proves the following sphere theorem for (not necessary reversible) Finsler manifolds. Theorem 6.2.1 ([78]) Let (M, F) be a simply-connected, closed manifold of dimension n > 3 with reversibility A. Suppose the flag curvature satisfies
Then M is homotopy equivalent to the n-sphere. The sphere theorem is proved earlier for reversible Finsler manifolds in [87]. Rademacher has overcome the irreversibility obstruction. He first proves that under the condition in Theorem 6.2.1, the length of any closed geodesic is at least 7r(l + j). Then the theorem follows from a Rauch
120
Riemann Curvature
comparison argument and the Morse theory of the energy functional on the free loop space. We will not go into the technical part in this monograph. 6.3
Nonpositive Flag Curvature
The sign of the Riemann curvature has great implication on the geometry and topology of the manifold. In this section we are going to prove the Cartan-Hadamard theorem for nonpositively curved Finsler manifolds and a metric rigidity theorem for nonpositively curved Finsler manifolds with constant S-curvature. Let (M, F) be a Finsler manifold. F is said to have nonpositive flag curvature if K < 0. It is said to have negativeflagcurvature if K < 0. The Riemann curvature is a family of self-adjoint linear maps R y : TXM —> TXM with respect to gj,. By (6.3), one can see that F has nonpositive flag curvature K < 0 if and only if for any non-zero vectors y, v € TXM, gy(Ry(v), v) <0. Since R y (t/) = 0 for any y e TXM \ {0}, F has negative flag curvature K < 0 if and only if for any non-zero vectors y,v £TXM with gy(y, v) = 0,
gy(Ry(v), v) <0. Assume that (M, F) is positively complete. Then at any point x € M, the exponential map exp^ : TXM —> M is defined on the whole tangent space. For non-zero vectors y, v € TXM \ {0}, let H(t, s) := expx[t(y + sv)],
0 < t < oo, \s\ < e.
H = H(t, s) is a geodesic variation of the geodesic a{t) := ex.px(ty). Then BH J{t) •= -^-(*.°) = d(expx)\ty{tv),
0 < t < oo,
(6.22)
is a Jacobi field along a, that is, it satisfies (6.16). J = J(t) is C°° at t = 0 with J(0) - 0,
D& J(0) = v.
(6.23)
Conversely, for any y,v e TXM \ {0}, the Jacobi field J = J(t) along the geodesic a = expx(ty) with (6.23) is given by (6.22).
121
Nonpositive Flag Curvature
From (6.22), one can see that expx is singular at ry € TXM, where r > 0 and y ^ 0, if and only if there is a non-zero Jacobi field J = J(t) along the geodesic a := expx(ty), 0 < t < r, with J(0) = 0 = J(r). Thus, to prove the regularity of expx at ry, it suffices to prove that any Jacobi field J(t), t > 0, with J(0) = 0 and D& J(0) / 0, does not vanish at t = r. Theorem 6.3.1
(Cartan-Hadamard [3]) Let (M, F) be a positively com-
plete Finsler manifold. Suppose that the flag curvature K < 0. Then for any point x E M, the exponential map exp x : TXM —> M is non-singular.
Proof. Let y, v € TXM \ {0} and let J(t) be the Jacobi field along the geodesic a = expx(ty) satisfying (6.23). Let f(t):=g*(t)(j(t),J(t)),
t>0.
By (6.16), one obtains
lnt)
=
s&(D&j(t),j(t))
= f {Sa (D&J, D&j} + Sa (D&D&J, J) )dr
= J {g
122
Riemann Curvature
are going to prove a rigidity theorem for complete Finsler manifolds with nonpositive flag curvature and constant S-curvature. Theorem 6.3.2 ([92]) Let (M,F) be a complete Finsler manifold with nonpositive flag curvature. Suppose that F has constant S-curvature and bounded mean Cartan torsion. Then F is weakly Landsbergian (J = 0) with RJ,(IJ,) = 0. Moreover, F is Riemannian at points where the flag curvature is negative. Proof. Let {bj} be a local frame for TM and {e^ := (x, y, hi)} be the corresponding local frame for n*TM. Let {9Z} be dual to {b^} and {u/ :— TT*91} be dual to {ej}. The mean Cartan tensor 1 = IiLJ1 can be expressed as a family of vectors, Iy = P(x, y)bj, where P := gl-*Ij, and the Landsberg tensor J = Jitj1 can also be expressed as a family of vectors, J^ = Jl(x, y)bi, where J* := gijJj. Thus 11,(1,) =
R^I™^.
By assumption, S = (n + l)cF for some constant c. Thus
S.k\mym - S|fc = (n + l)c{F. fc | TO y m - F, fc } = 0. It follows from (5.41) that J\mym+RimIm
(6.24)
= 0.
Let
3[t) :=
Jk{a{t),a{t))bk\a(t).
Thus
D*I(t) = &*>(t)Ik]p(o-(t), &(t))bk\a{t)
= 3{t)
(6.25)
and D*J(t) = &p(t)j)p(a(t),
&{t))bk\a{t) = D*D*I(t).
(6.26)
Equation (6.24) restricted to a(t) becomes D^D^I(^) + R* (t) (I(t)) = 0. Thus, the mean Cartan torsion is a Jacobi field along any geodesic.
(6.27)
123
Nonpositive Flag Curvature
Let p(t):=g* W (l(i),I(t))It follows from (6.25), (6.26) and (6.27) that
+2g* w (D*I(0,D*I(t))
= -2g&(t)(RHt)(I(t)),I(tj)
+ 2g<w(j(t),J(t)).
(6.28)
By assumption, K < 0. It follows from (6.28) that f"(t) > 0. Thus ip(t) is convex and nonpositive. Suppose that
*<*«•
If
p(0 >¥>(*»)+v'(O(*-*»)> *> toOne can see that lim t _ +00 y(i) = oo or limt-^-oo y(t) = oo. This implies that the mean Cartan torsion is unbounded, that contradicts the assumption. Therefore, tp'(t) = 0 and hence ip"(t) = 0. It follows from (6.28) that R* ( t ) (I(t))=0,
J(t)=O.
Since a is arbitrary, one can conclude that Ry{ly) = 0 ,
Jy = 0.
Assume that F has negative flag curvature at a point x € M. Since the vector Ij, is orthogonal to y with respect to gy, and R y (I y ) = 0, one concludes that Iy = 0 for all y e TXM\ {0}. By Deicke's theorem (Theorem 1.5.1), F is Riemannian. Q.E.D. Any Finsler metric on a closed manifold is complete with bounded Cartan torsion. One immediately obtains the following Corollary 6.3.3 Let (M, F) be a closed Finsler manifold of negative flag curvature. If F has constant S-curvature, then it must be Riemannian.
124
Riemann Curvature
It is not difficult to show that every complete Finsler surface with nonpositive Gauss curvature, constant S-curvature and bounded mean Cartan torsion is either Riemannian or locally Minkowskian. First one sees that such a Finsler surface must be weakly Landsbergian (J = 0). Then the conclusion follows from a global rigidity theorem in [6]. The details are left to the reader for an exercise. Example 6.3.4 Let / : [0, oo) x [0, co) -> [0, oo) be an arbitrary C°° function satisfying (1.18), (1.20) and (1.21). Let (Mi,ai), i = 1,2, be arbitrary Riemannian manifolds and M = M\ x M2. Let
Mz2,2/2)]2),
F--=yJf([ai(xuyi)]^
where x = (ii,x 2 ) € M and y = yi © y2 € T(Xl)X2)(Mi x M2) = TXxMx © TX2M2. By Example 1.2.5, F is a Finsler metric. By Example 4.3.1, the spray coefficients of F are given by Ga(x, y) = Ga(Xl,yi),
Ga(x, y) = Ga{xu yi),
(6.29)
where Ga and Ga are the spray coefficients of ot\ and a2 respectively. From (6.29), we can see that F is a Berwald metric. Thus by Proposition 5.1.2, S = 0. By a direct computation, one obtains the following formula for the Riemann tensor of F:
(*<)-(*,)•(*'
a
*,)•
a
where R b and R p are the coefficients of the Riemann tensor of a.\ and a2 respectively. Let Rtj := gikRkj, Rab •= Sac&b and Ra0 := gaiFC0- Using (1.19), one obtains (p \ - (fsRab
{"**)-{ For any vector v = v'-^\x
0
0 \
ftRaJ-
e TXM,
g v (R y (v), v) = fsRabVavb + ftRa0vav0.
(6.30)
Assume that a± and a2 both have nonpositive sectional curvature. Then it follows from (6.30) that F has nonpositive flag curvature.
125
Nonpositive Flag Curvature
Using (1.22), we can compute the mean Cartan torsion. First, observe that
^hv^M=^[in^([ai]2'H2)]-
u= We obtain
T — hi-
T
— hi-
where ya := gabVb and y a := gapyP • Clearly, the mean Cartan torsion is bounded. Since yaR% = 0 and yaRap = 0, we have
gy(Ry(Iy),Iy) = h&jV = \yaR\Ih
+ jyaRa0I0 = 0.
Thus the Riemann curvature vanishes on the mean Cartan torsion. Since F is a Berwald metric, by Proposition 2.1.3, it is a Landsberg metric, i.e., J = 0. Thus the conclusion in Theorem 6.3.2 holds. The completeness in Theorem 6.3.2 can't be dropped. See Example 6.1.5 and the following Fish tank metric. Example 6.3.5 ([90]) Let n > 2 and U:= j p = ( s , t , p ) € R 2 x R " - 2
s2 + « 2 < l } .
Define F = F(x,y) J( -tu + sv) + \y\2(l -s2 -t2) - (-tu + sv) F(*,y) == J 1^2—T2 ' where y = (u,v,y) € TXU = R" and x = (s,t,p) € fi. One can verify that F is a Finsler metric on Q with vanishing flag curvature K = 0 and vanishing S-curvature S = 0. One can verify that J ^ 0. Thus Theorem 6.3.2 does not hold if the assumption on completeness is dropped. From the above discussion, we may ask the following question: is there any non-Berwaldian Finsler metric of dimension n > 3 satisfying the following conditions: K = 0,
S = 0,
J = 0?
126
Riemann Curvature
This problem remains open. Note that in dimension two, the conditions, S = 0 and J = 0, imply that the metric is a Berwald metric. Berwald metrics with K = 0 are locally Minkowskian.
Chapter 7
Finsler Metrics of Scalar Flag Curvature
In this chapter, we are going to discuss Finsler metrics of scalar (flag) curvature. We have seen that every locally projectively flat Finsler metrics are of scalar flag curvature (Proposition 6.1.3). There are Finsler metrics of scalar flag curvature which are not locally projectively flat. This shows the complexity and richness of general Finsler metrics. It is our goal to reveal the relationship between the flag curvature and other non-Riemannian quantities for Finsler metrics of scalar flag curvature. 7.1
Some Basic Properties
Let (M, F) be a Finsler manifold. Assume that F is of scalar flag curvature, that is, the flag curvature K = K(x,y) is a scalar function on TM \ {0}. This curvature condition is equivalent to the following identity in any standard local coordinate system,
R\ = KF2 h\ = K{FH{ - sfc,yV},
.
(7-1)
where h\ := 5{ - F~2gkqyqyi- It follows from (2.73) and (7.1) that
&kl = \K.IF2 h\ - l-K.kF2 h\ + K{glp5i - gkp5\)yv.
(7.2)
To study the relationship between the flag curvature and other nonRiemannian quantities, we will need some identities. Differentiating (7.1) yields
R\d = K , F 2 h\ + K{2glpyPS? - gkr>yn\ 127
Sfc,y*}.
(7.3)
128
Finsler Metrics of Scalar Flag Curvature
Let hij := gikhkj = FFyiyi By (2.69), (2.70) and (7.3), one obtains Cijk\P\qypyq = ~F2{K.ihjk
+ K.jhik + K.khij + 3KCijk}
(7.4)
and h\P\qypyq = -\F2{(™
+ i)K.fc + 3K/ fc }.
(7.5)
Recall the Matsumoto torsion defined by Mijk •= Cijk
—llihjh
+ Ijhik + hhij \.
It follows from (7.4) and (7.5) that Mijk\P\qypyq + KF2Mijk = 0.
(7.6)
This is an important equation for Finsler metrics of scalar flag curvature. By (7.6), one can show that for a Landsberg metric of scalar flag curvature on a manifold of dimension > 3, if the flag curvature K ^ 0, then it is Riemannnian ([76]). Since F is of scalar flag curvature, (2.63) can be simplified to
Substituting (7.1) and (7.2) into (7.7) yields
Ku/^ - Kifcfc*, - \{K.ilmh\
-
K.kimhil}ym
-F-2Kimym{glp5i-gkP6t}y»
= 0.
Taking the summation over i = k in the above identity yields (n - 2){K|, - F-2KlmymglpyP
- ±K.l{mym}
= 0.
(7.8)
On the other hand, the equation d2K = 0 yields the following Ricci identity: K.(|m = K|m./ + K.fcL
ml.
Contracting the above identity with ym yields K.llmym
= Klm.iym = (K| m y m )., - K,,.
129
Some Basic Properties
Assume that n > 3. Substituting the above equation into (7.8) yields 4F 3 K,, - (F 3 K | m 2 / m )., = 0.
(7.9)
Then we obtain the following Lemma 7.1.1 (Schur Lemma [18]) Let (M,F) be a Finsler manifold of dimension n > 3. Suppose that the flag curvature is isotropic, i.e., the flag curvature K = K(x) is a function of x £ M only. Then K = constant. Proof. Since K\k = K x fc(x), it suffices t o prove t h a t K| fc = 0. Since K is a scalar function on M, K | m = K x m is a function o f i e M only. Hence K | m . ( = 0. It follows from (7.9) t h a t
F2K{j = (Klmym) gjvf.
(7.10)
Differentiating (7.10) with respect to yk yields K | m y m gjk = 2Kb- gkpyp - K|fc gjpyp.
(7.11)
Let u — ul-£^\x e TXM be gy-orthogonal to y, namely, g,y(y,u) = 0. Contracting (7.11) with u^ and uk yields K| m y m gy(u,u) = 2 ^ ^ ' gj,(y,w) - K\kuk gy(y,u) = 0. Thus K\mym
= 0. By (7.10), one concludes that K b = 0.
Q.E.D.
Many known Finsler metrics of scalar flag curvature have almost isotropic S-curvature. Thus it is natural to study Finsler metrics of scalar flag curvature with isotropic S-curvature. Proposition 7.1.2 ([24]) Let (M, F) be an n-dimensional Finsler manifold of scalar flag curvature with flag curvature K = K(ar, y). Suppose that the S-curvature is almost isotropic, i.e., S = (n + l){cF + jj},
(7.12)
where c = c{x) is a scalar function and n — rji(x)yt is a closed I-form on M. Then there is a scalar function a = cr(x) on M such that the flag curvature is in the following form,
K = 3 ^ - + a. r
130
Finsler Metrics of Scalar Flag Curvature
Proof. Substituting (7.3) into (5.42), one obtains s.klmym-Slk
=
-~^K.kF2.
By (7.12), one obtains
S.k\mym ~ S|fc = (n + l)[c\mymF.k - c\kF + [r,k]m - r)m\k]ym} = (n +
l){c]mymF.k-c]kF}.
Thus c\mymF.k - clkF = - ^ K . ^ 2 .
(7.13)
Here c = c(x) is viewed as a scalar function on TMO and its covariant derivatives are defined in the usual way, i.e., c^ = cxi are the usual partial derivatives with respect to xl. Rewriting (7.13) as follows
one concludes that the following quantity 3cxmym is a scalar function on M. This proves the proposition.
Q.E.D.
Corollary 7.1.3 ([73]) Let F be an n-dimensional Finsler metric of scalar flag curvature with flag curvature K = K(ar, y). If F has constant S-curvature, i.e., S = (n + l)cF, where c = constant, then K = K(a;) is a scalar function on M. According to Lemma 7.1.1, when n = dimM > 3, K = K(x) if and only if K = constant. We can use Proposition 7.1.2 to compute the flag curvature if the Finsler metric is of scalar flag curvature and isotropic S-curvature. First using the scalar function c = c(x), one can determine the first part of the flag curvature, then choose a special vector y to determine the scalar function a = a(x) since it is independent of y.
131
Global Rigidity Theorems Example 7.1.4 be given by
Q :
For an arbitrary number e with 0 < £ < l , l e t F = a + / 3
_ y/(l - e2){su + tv)2 + e(u2 + v2)(l + e(s2 + t2)) ~ l + s(s2+t2) '
P : =
\/l — s2(su + tv) l+e(s* + t2) '
where x = (s,t) € R2 and y = (u,v) £ TXR2 ~ R2. F is a Randers metric on R2. This is just Example 5.2.3 in dimension two. Thus S = 3cF, where ° ~ 2(e + s2 +12)' Note that /3 is closed. Thus F is protectively equivalent to a. However, the Gauss curvature K of a is not a constant, T>
^
2£ e
+
S2
+
1-s t2
^
(£
+
S2
+
2 t2)2 •
According to the Beltrami theorem in Riemannian geometry, a (hence F) is not locally projectively flat. Thus F is not projectively flat. Using a Maple program, we obtain the following expression for the Gauss curvature: 3-y/l ~£2{su + tv) (e + s2 + t2)2F
7.2
+
2e e + s2 + t2
+
7(1 - e2) 4{s + s2 + t2)2'
Global Rigidity Theorems
In this section we are going to discuss some global rigidity properties of Finsler metrics of scalar flag curvature. We first find a special equation on the flag curvature and the Cartan torsion or the Matsumoto torsion, such that the restriction of it to an arbitrary geodesic is a second order ordinary differential equation. Then using basic comparison techniques, we show that under certain growth conditions on the Cartan torsion (resp. the Matsumoto torsion), the Cartan torsion (resp. the Matsumoto torsion) must vanish. These growth conditions are always satisfied if the manifold is closed.
132
Finsler Metrics of Scalar Flag Curvature
Let (M, F) be a complete Finsler manifold. Let M be the Matsumoto torsion defined in (1.53). The norm of M at a point x e M is denned by ||M||,:=
sup F(x,y)\My(u,v,^=^ «,«,w,«;erxA/\{o} V8v( u ' u )8s( u i l ')6s(«'! u ')
(7.14)
The Matsumoto torsion grows sub-exponentially at rate offc> 0 if for any point x € M
M{x,r):=
sup
min(d(z,z),d(x,z))
||M||2 = o(ekr),
(r - +oo).
The Matsumoto torsion grows sub-linearly if for any x e M, lim r~ 1 M(x,r) = 0,
(r -> +oo).
Similarly, we define the norm of the (mean) Cartan torsion at a point x € M as follows,
11,. = -
sup
j/,«eT,M\{0}
liril • -
\\\*s\\x • —
sun
SUp
M M
Vgy(W,W] /
F(x,y)\Cy(u,v,w)\ ,
,
.
.
,
.•
y.u.^eT^MUO} y/gy{u, U)gy(V, V)gy(w, W)
Let I{x,r)~
sup min(d(z,x),d(a:,z))
||I|| X ,
sup
C(x,r):=
min(d(z,i),rf(a:,z))
||C|| X .
Then we can define the growth rate for I and C as above. From the definition of the Matsumoto torsion in (1.52), we have
||M|U<||C|U + -^-||I|U<3||C|U. Thus if the Cartan torsion grows sub-exponentially at rate of k, then the Matsumoto torsion grows at the same rate. When M is closed, F is always complete and all mentioned geometric quantities are bounded. Theorem 7.2.1 ([74]) Let (M, F) be an n-dimensional complete Finsler manifold of scalar flag curvature with flag curvature K = K.(x,y) < — 1 (n > 3). Suppose that the Matsumoto torsion grows sub-exponentially at rate of k = 1. Then F is a Randers metric.
Global Rigidity Theorems
133
Proof. According to Proposition 1.5.3, in dimension n > 3, a Finsler metric is a Randers metric if and only if the Matsumoto torsion vanishes. Thus it suffices to prove that the Matsumoto torsion vanishes under the assumption. Suppose that this is not true. Then M.y(u, u, u) ^ 0 for some y,u €.TXM\ {0} with F(x, y) — 1. Let a = a(t) be the unit speed geodesic with a(0) = x and
= Mijk(v(t),&(t))ui{t)U*(t)Uk(t).
Figure 7.1 It follows from (7.6) that M"(t) + K(t)M(t)=0,
(7.15)
where K(t) := Kf v(t),&(t) J < —1. We are going to estimate M.{i) by comparing it with the following function: Mo(t) := M(0) cosh(t) + M'(0) sinh(t). Note that M.o{t) satisfies M'&t) - Mo{t) = 0.
(7.16)
Let (a, b) be the maximal interval on which A4(t) / 0. Let
It follows from (7.15) that /'(t) + /(*) 2 = - K ( t ) > l .
(7.17)
134
Finsler Metrics of Scalar Flag Curvature
Let (a,/?) be the maximal internal on which Mo(t) / 0. Let
/.(«):-£§. «<«„ It follows from (7.16) that fo(t)+fo(t)2 = l.
(7.18)
We claim that tp(t) := \M(t)/Mo(t)\ attains its minimum tp(O) = 1 at t — 0. To show this, consider the following function,
h(t) := {/(*) - fo(t)} exp { j[f(t) + fo(t)]dt}. By (7.17) and (7.18), we have
h'it) = {[/'(t) + f{tf] ~ lf'o(t) + fo(t)2}} exp { J[f(t) + fo(t)]dt} > 0. Note that h(0) = 0. Thus h(t) < 0 for t < 0 and h(t) > 0 for i > 0. Since h(t) has the same sign as f(t) — fo(t), we conclude that
This implies that
(7.19)
Clearly, (a,/3) C {a,b). Since rf(a(-^),x) < t and d(x,a(t)) < t for any t > 0, one gets M{x, r) > max (M(t)
\t\ < r\.
135
Global Rigidity Theorems
Figure 7.2 Suppose that /A'(0) = 0 or it has the same sign as A^(0). Since Mo(t) ^ 0 for all t > 0. Thus f3 = oo and M{x,r)>
M{r) > M(0) cosh(r) + M'(0) sinh(r),
r > 0.
Suppose that M'(0) has the opposite sign as M(0). Since M0{t) ^ 0 for all t < 0. Thus a = —oo and M(x,r)>
M(-r)
> M(0) cosh(r) + A^'(0) sinh(r),
r > 0.
In either case, l i m i n f M ( x ' r ) > U\M(0)\ + |M'(0)|| > 0. r—»oo
e
Z I.
)
But M(x,r) grows exponentially at rate of k = 1. This is a contradiction. Thus the Matsumoto torsion vanishes. By Proposition 1.5.3, F must be a Randers metric. Q.E.D. Consider a locally protectively fiat complete Finsler metric F on a manifold of dimension n > 3. First, by Proposition 6.1.3, it is of scalar flag curvature. Assume that the flag curvature satisfies K < - 1 and the Matsumoto torsion grows sub-exponentially at rate of k = 1, then F = a + f3 is a Randers metric. Moreover, by Proposition 3.4.8, the Riemannian metric a is locally protectively flat and the 1-form /3 is closed. Since any Finsler metric on a closed manifold is complete with bounded Cartan torsion (and hence bounded Matsumoto torsion), one obtains the following
136
Finsler Metrics of Scalar Flag Curvature
Corollary 7.2.2 Let F be a Finsler metric on a closed manifold M of dimension n > 3. Suppose that F is of scalar flag curvature with negative flag curvature, then it is a Randers metric. In particular, if F is locally protectively flat with negative flag curvature, then it is a locally projectively flat Randers metric. Let a be a Riemannian metric of negative constant sectional curvature on a closed manifold M. According to Examples 3.4.2 and 6.1.4, a is locally projectively flat. Let /3 be an arbitrary closed 1-form on M. Then for sufficiently small e, Fe := a+e/3 is a locally projectively flat Randers metric, since it is projectively equivalent to a by Proposition 3.4.8. Therefore Fe is of scalar flag curvature by Proposition 6.1.3. By continuity, the flag curvature of F£ is negative for small e. If we impose the reversibility condition on the Finsler mertric, we obtain the following Corollary 7.2.3 Let F be a reversible Finsler metric on a closed manifold M of dimension n > 3. Suppose that F is of scalar flag curvature with negative flag curvature, then it is a Riemannian metric of constant negative sectional curvature. The reversibility condition in Corollary 7.2.3 can be dropped if we assume that the flag curvature is isotropic, i.e., K = K(x) is independent of directions. See Corollary 7.2.5 below. First, we prove the following Theorem 7.2.4 Let (M, F) be a complete Finsler manifold with isotropic flag curvature K = K(x). (a) / / K < —1 and I grows sub-exponentially at rate of k = 1, then F is Riemannian. (b) / / K < 0 and C (resp. I) grows sub-linearly, then F is Landsbergian (resp. weakly Landsbergian). Further F is Riemannian on any open subset where K < 0. Proof. By assumption, K.jt = 0. It follows from (2.68) and (7.5) that Ii\p\qypy"+KF2Ii=0.
(7.20)
Let y, u G TXM \ {0} be arbitrary vectors with F(x, y) = 1. Let a = a(t) be the unit speed geodesic with <J(0) = x and a(0) — y, and let U = U(t)
137
Global Rigidity Theorems
be the parallel vector field along a with f/(0) = u. Set l(t) := I* (t) (t/(i)) = Ji(<7(t),(r(t))tf*(t). It follows from (7.20) that I"(t) + K(t)J(t) = 0,
(7.21)
where K(t) := K(tr(t)). Suppose that K < — 1 and the mean Cartan torsion I grows subexponentially at rate of k = 1, i.e., I(x, r) = o(er) for any x £ M. By the same argument as in Theorem 7.2.1, one can show that 1(0) = Ii(x, y)u% = 0. Since y and u are arbitrary, one concludes that the mean Cartan torsion 1 = 0. Therefore F is Riemannian by Theorem 1.5.1. Suppose that K < 0 and the mean Cartan torsion grows sub-linearly. We first show that J = 0. We prove this by contradiction. Assume that Jy( u ) 7^ 0 f° r some y,u e TXM \ {0}. Let I(t) be defined as above for y,u so that 1(0) = Iy(u) and I'(0) = Jj,(u). Since J'(0) / 0, I(e) ^ 0 and X'{e) ^ 0 for small e > 0. Thus we may assume that 1(0) ^ 0 and I'(0) 7^ 0. We are going to show the following inequality, l{t) > I(0)+I'(0)t,
a
(7.22)
where (a, /3) is the maximal interval containing 0 on which the function on the right of (7.22) is not equal to zero. Let
lo(t):=
1(0)+I'{0)t.
Let
m
m
-
- m.
i(t)'
t) f (h[t)
w.
-jo{ty
We have f{t) + f{t)2 > 0,
fo(t) + fo(t)2
= 0.
Consider
hit) := {fit) - foit)} exp { Jlfit) + foit)}dt}.
(7.23)
138
Finsler Metrics of Scalar Flag Curvature
By (7.23), we obtain
h'(t) = {[/'(t) + f(t)2} - [f'o(t) + fo(t)2}} exp { J[f(t) + fo(t)]dt} > 0. This h(t) is a non-decreasing function. Note that h(0) = 0. Thus h(t) < 0 for t < 0 and h(t) > 0 for t > 0. We obtain
l l n io(t)IJ ~ We see that In
lit)
m
fo<
ti-\
>o if < > o •
attains its minimum at f = 0, that is,
Equivalently |j(t)| > \Ut)V By a simple argument, one can see that the above inequality holds for t in the maximal interval (a,/3) on which To{t) ^ 0. This proves (7.22). If X'(0) has the same sign as X(0), then I(x,t)
> l{t)
> 1 ( 0 ) + I ' ( 0 ) t,
t>0.
Thus /3 = +oo. Letting t —> oo yields that I(t) grows at least linearly. This is impossible. If I'(0) has the opposite sign as 1(0), then I(x,-t)
> J ( 0 | > J(0) - X'(0) t,
t<0.
Thus a = —oo. Letting t —» -oo yields that I(t) grows at least linearly. This is impossible. Therefore, X'(0) = 0, that is J y (u) = 0. This is a contradiction. We have shown that J = 0. Equation (7.5) is reduced to K(x)Ik = 0.
(7.24)
From (7.24), one can see that 4 = 0 at i where K(x) < 0. Hence Fx is Euclidean by Deicke's theorem.
139
Randers Metrics of Scalar Flag Curvature
Now assume that K < 0 and the Cartan torsion grows sub-linearly. By (7-4), Cimp\qypyq + KF2cijk
= o.
By a similar argument, one can show that Lijk = Cijk\iVl Landsbergian.
(7.25) =
0- Thus F is Q.E.D.
Let (M, F) be a closed Finsler manifold. Then F is always complete and the Cartan torsion is bounded. Further, assume that F has isotropic flag curvature K = K(x). By Theorem 7.2.4, F is Riemannian when K < 0 and F is Landsbergian when K = 0. In the case when K = 0, one can show that F is actually Berwaldian. Hence it is locally Minkowskian. We state this corollary without further details. See [86]. Corollary 7.2.5 ([l]) Let (M,F) be a closed Finsler manifold with flag curvature K = K.(x). Then (a) if K < 0, then F is Riemannian. (b) if K = 0, then it is locally Minkowskian. We know that every Berwald metric is a Landsberg metric (L = 0). Thus we may ask the following question again: is there any non-Berwaldian Finsler metric satisfying K = 0,
L = 0 (orJ = 0)?
If such a metric exists, then by the above theorem, either it is incomplete or it has unbounded (mean) Cartan torsion. 7.3
Randers Metrics of Scalar Flag Curvature
It is one of the important problems in Finsler geometry to classify Finsler metrics of constant/scalar flag curvature. However, this problem still remains open even in the constant flag curvature case We shall first study Randers metrics of scalar flag curvature with isotropic S-curvature. Let F be a Randers metric expressed in terms of a Riemannian metric h = y/hijytyi and a vector field V = V^ by (5.12), i.e., f
^ 4 ,
K-.-Vj,
(7.26)
140
Finsler Metrics of Scalar Flag Curvature
where A := 1-/1(1, V)2. Let fty := | ( % + % ) , Uj := V*fty, ft := 7 ^ , <Stj := 5(Vi|j — Vj\i) and<Sj := VlSij. By Lemma 3.1.3, the spray coefficients Gl of F can be expressed in terms of the spray coefficients G\ of h and the covariant derivatives of V with respect to h as follow:
Gi = Gl-FS\-l-F\ni+Si)+l-[ti-Vi^\2Fn0-Tl00-F2ny
(7.27)
By (7.27), one can express the Riemann curvature Rlk of F in terms of the Riemann curvature Rlk of h and the covariant derivatives of V. Recall the formula (2.49): k
d& 8xk
#& dxmdykV
&& dymdyk
d& 8CT" dym dyk '
[
'
Let
G^Gi + Q*, where Qi := _Fs*0
-
IF2(IV
+ si) + \{j-
V^^FTZQ
- n00 - F2n}.
Then
R\ = R\ + 2Q\k - [Q)m}ykym + 2Qm[Q%myk - [ Q V [ Q " V
(7-29)
Here "|" denotes the horizontal covariant differentiation with respect to h. From now on, we assume that F is of isotropic S-curvature, i.e., S = (n + V)cF for some scalar function c = c(x). By Proposition 5.2.5, Tloo = -2ch2.
(7.30)
Then the spray coefficients Gl are reduced to the following expression: G'^Gi
+ QK
where
Q* := -FS\ - l-F2Sl + cFy\ By (5.20), we have K|j|fc = 2(cxihjk - cxjhik - cxkhij) - RkPijVp.
(7.31)
141
Randers Metrics of Scalar Flag Curvature
Then S\\0 = 2{himcxmyk - c^j) 5'0|fc = 2(/l™<wfc -
M
RZmqVmy\
-
" ^ ) +
R;kqtfV\
S\k = 2 ^ - S*mS™ + 2(
5' |0 = 1cS\ - S^S™ + lic^V^ S*o]o = 2(/iimc,m/i2 - qoj/') -
- h cxmV0) -
^ ^ W , R;mqV"Vym,
R;mqyWVm,
where Rp kq denotes the Riemann curvature tensor of h such that Rlk := R\ ypyg. Further, we need the following formulas 2cF(yk-FVk) + F(FSk + Sk0)
P =
2
^
'
F |o = 2cF2 + ^-So, (Fy~)\o = (^S0 +2c^){yk -FVk} - ^S0Vk - ^Sk0. where A := y'A/i2 + V^2. By (??) and the above identities, we first obtain the following very simple formula:
R\ = RP\qvpyq - FR;kqv*tf 2
+F Rp\qVW +( ^ 1
FR;kyv>
- Fy,R;myy"Vm
_C2 -2cxmVm){F25i~
+ FFykR;mqyWVm FFy^y
(7.32)
It is surprised that all the terms with Sl or Slk do not occur in (7.32). We can rewrite (7.32) as follows Rik =
RPikq(vp-FV')W-FV'>) -Fy,R;mg(yp - FV?)(yq - FV«)Vm
+ (^pl
- c2 - 2cxmVm) {FH{ - FFV^}.
(7.33)
Let e^yi-F^V*, Let h := h(x,£) = ^/hpq^i
tk:=hik?.
= V^fc? a n d ^o := V^. By (5.16), we have
142
Finsler Metrics of Scalar Flag Curvature
h = F. Thus yi=?+hV\ Observe that \h = \F = y/\h2 + Vo - Vo = y/\h? + V0- ViiC + W1) = x/A/i2 + VQ -Vo- h{\ - A). This gives y/\h* + V{i = h + V[i. By the above identities, we obtain pvk -
^k h + V0'
F25l - FFyrf = hH\ - &? - rLrr^(h2Sl, -
^)V.
h + V0
Let R\ '•= Rp' kg£P£q-
By (7.33), we obtain the following Lemma 7.3.1 ([27]) Let F = a + (3 be a Randers metric expressed by (7.26). Suppose that it has isotropic S-curvature, S = (n + l)cF. Then for any scalar function fj, = fi(x) on M,
R\-(^^+»-c>-2cxmV™){FHi-FFykyi} = Ri- ^5{-ikc)
- j^{R\~
^(hHi-ip^}v^7.M)
By Lemma 7.3.1, we immedately obtain the following Theorem 7.3.2 ([27]) Let F be a Randers metric on n-dimensional manifold M defined by (7.26). Suppose that S = (n + l)cF where c = c(x) is a scalar function. Then F is of scalar flag curvature if and only if h is of
143
Randers Metrics of Scalar Flag Curvature
sectional curvature K = \i, where (i = n(x) is a scalar function (=constant if n> 3). In this case, theflagcurvature of F is given by K =^ where a \= fj, — c 2 —
-
(7.35)
+ a,
2cxmVm.
Proof. Assume that F is of scalar flag curvature, then Proposition 7.1.2, the flag curvature is given by K-
6cx™y f
+ a,
where a — o~{x) is a scalar function on M. Let li:=a + c2 + 2cxmVm. It suffices to show that h has sectional curvature K = \i. It follows from (7.34) that
k\ - Jh26i -&£•') - --^ZJR^-Jtfsj,-zP?)}y v
'
ft
+ Vo
*•
^
'
J
= o.
Clearly, we have
R\=^(h2Si-ae)-
(7.36)
Thus /i has sectional curvature K — /j.. By the Schur lemma (Lemma 7.1.1), /i = constant in dimension n > 3. Conversely, iffthas sectional curvature K = /x, then (7.36) holds. By (7.34) again, we get
R\=(^f-+
(7.37)
where
Q.E.D.
By Proposition 5.2.10 and Theorem 7.3.2, we obtain the following Theorem 7.3.3 Let F = a + (3 be a Randers metric on a manifold M of dimension n > 3, which is expressed in terms of a Riemannian metric ft and a vectorfieldV by (7.26). F is of isotropic S-curvature S = (n+ l)cF
144
Finsler Metrics of Scalar Flag Curvature
and of scalar flag curvature, K = K(x, y), if and only if at any point, there is a local coordinate system in which h, c and V are given by
h
V|y|2 + M(|s|2|y|a-(s,y)2)
=
C=
iTM2 4+M = ,
(
'
}
( 7 .3 9 )
\/l+M|z| 2
V = -2{(8y/T+^
+ (a,x))x - -7=ptr—
}
+a;g + 6 + /u(fe,2:)a;, (7.40) where S, fi are constants, Q — (q^) is an anti-symmetric matrix and a, & 6 R" are constant vectors. In this case, the flag curvature is given by K = ^
-
+ a,
(7.41)
where a = /j, — c2 — 2cxm Vm.
Proof. By assumption, the dimension of M is not less than 3. First we assume that F = a + j3 is of isotropic S-curvature and of scalar flag curvature. By Theorem 7.3.2, the flag curvature of F is given by (7.35) and h has constant sectional curvature K = /J,. At any point, there is a local coordinate system in which h is given by (7.38). By Proposition 5.2.10, if S = (n + l)c.F, then c and V are given by (7.39) and (7.40) respectively in the same local coordinate system. Conversely, assume that there is a local coordinate system in which h, c and V are given by (7.38), (7.39) and (7.40) respectively, then by Proposition 5.2.10, S = (n + l)cF. Since h has constant sectional curvature K = /i, by Theorem 7.3.2, F is of scalar curvature with flag curvature given by (7.35). Q.E.D. Let us take a look at the following special example. Example 7.3.4 b = 0, we get
([93]) In (7.38)-(7.40), letting fi = 0,6 = Q,Q = 0 and
h = \y\,
c=(a,x),
V = ~2{a,x)x+ \x\2a.
(7.42)
145
Randers Metrics of Scalar Flag Curvature
The Randers metric F = a + (3 defined by (7.26) is given by F_V(1
- \a\2\x\4)\y\2 + (H 2 (a, y) - 2{a, x)(x, y)f \-\a\2\x\* \x\2(a,y) -2(a,x)(x,y)
l-|a|2M4
The above defined Randers metric F is of isotropic S-curvature and scalar flag curvature, i.e., S = (n + l)(a, x)F,
K= ^
^ + 3(a, xf -
2\a\2\x\\
This example satisfies the assumption and conclusion of Proposition 7.1.2. Now we study the Ricci curvature of a Randers metric with isotropic S-curvature. Let Ric and Ric denote the Ricci curvature of F and h respectively. They are defined by Ric := Rmm,
R £ := Rmm.
Let ST^. .
D"1
D " i CPC1
m e .= n m = np mqt, t, , where f := yl - FW\ Clearly, Ric = (n - l)fih2 if and only if Ric = (n -
l)nh2.
First we have the following Lemma 7.3.5 ([27]) Let F = a + (3 be a Randers metric expressed by (7.26). Suppose that it has isotropic S-curvature, S = (n+ l)cF. Then for any scalar {unction fi = /J,(X) on M, R i c - ( n - l ) ( 3 C ^ m + / i - c 2 - 2 c : E m y m ) F 2 = Ric-(n-l)M 2 . (7.43) Proof. Observe that £mR™ = t.mRip'jCfl3 = C^RimpjC^ = 0 and Cm (h25™ - ZpT) = h2^ - iPh2 = 0.
Then (7.43) follows from (7.34).
Q.E.D.
146
Finsler Metrics of Scalar Flag Curvature
From Lemma 7.3.5 we immediately obtain the following Theorem 7.3.6 ([27]) Let F be a Randers metric on n-dimensional manifold M defined by (7.26) and let c = c(x) and fi — y{x) be scalar functions on M. Suppose S = (n + l)cF. Then Ric = (n — l)/jh2 if and only if
Ric = (n - 1 ) { ^ ^ - + t*. - c2 - 2cxmVm}F2.
(7.44)
In general, a Randers metric of scalar flag curvature does not necessarily have isotropic S-curvature. However, we have the following Lemma 7.3.7 ([9]) Let F be a Randers metric on n-dimensional manifold M defined by (7.26) using a Riemannian metric h and a vector field V. If the Ricci curvature is constant, Ric = (n — 1)KF2 where K is a constant, then there is a constant c such that n00 = -2ch2. The proof of Lemma 7.3.7 is technical, so is omitted. Now we can give a complete list of explicit formulas for Randers metrics of constant flag curvature. Let F be a, Randers metric defined by (7.26). Suppose that F has constant flag curvature. Then by Lemma 7.3.7, V satisfies 7^oo = —2ch,
c = constant.
That is, F has constant S-curvature. By Theorem 7.3.2, h has sectional curvature K = y, where /i = /x(x) is a scalar function. In this case, the flag curvature of F is given by K = //-c2. Since K = constant by assumption, we conclude that // = constant. Namely, h has constant sectional curvature K — \x. By chooing a local coordinate system, we may assume that h — h^ is defined in (7.38),
_ V|y| 2 + /*(|s| 2 |y| 2 -(s,y) 2 )
.....
147
Randers Metrics of Scalar Flag Curvature
By Corollary 5.2.11, V is given by (-2cx + xQ + b \xQ + b + li(b,x)x
if> = 0 if (j. ^ 0 ,
l
''4
J
where <3 = (qf) is a skew-symmetric matrix and b = (bl) is a constant vector with |6| < 1. This completes the proof of the following Theorem 7.3.8 ([7]) Let F be a Randers metric on a manifold M, whichis expressed by (7.26) in terms of a Riemannian metric h and a vector field V. Then F has constantflagcurvature if and only if h is a Riemannian metric and V is a vector field on M with the following property: at any point in M, there is a local coordinate system {xl) with xl{p) — 0 such that h is locally expressed by (7-45) and V = (V1) is given by (7.46). In this case, S = (n + \)cF and K = \x - c2. According to Theorem 7.3.8, any Randers metric with K = 0 is given by V\y\2 - (\y\2\xQ + fr|2) - (xQ + b, y)*) l-\xQ + b\2
(XQ + b, y) l - | x Q + fe|2'
One can show that F is positively complete if and only if Q = 0, in which case, F is Minkowskian. This fact can also be proved in a different way without using the above classification result (see Theorem 1.2 in [90]). By Theorem 7.3.8, one can show that there are non-Riemannian Randers metrics of constant flag curvature K = 1 on any standard unit sphere Sn C R n+1 , regardless of the dimension. See [7] and [13] for discussions on special Randers metrics of K = 1 on S". The geodesic properties of some examples in Theorem 7.3.8 have been discussed in [50] and [103]. See [82] for more recent work on geodesies. The classification problem of Randers metrics with constant flag curvature was first attempted by Yasuda-Shimada [101] and Matsumoto [67], Motivated by Yasuda-Shimada's result, Bao-Shen constructed a family of Randers metrics on S3 with K = 1 [10]. Bao-Shen's examples satisfy the conditions listed in [101] and [67] for Randers metrics of positive constant flag curvature. It was believed that Yasuda-Shimada's result was completely true, until some new counter-examples were constructed [90] and
148
Finsler Metrics of Scalar Flag Curvature
[91]. Shortly after these examples were found, a correct version of YasudaShimada's result was obtained by Bao-Robles [8], meanwhile, independently by Matsumoto-Shimada [7l]. From their results, one learns that YasudaShimada's conclusions are still true under an additional condition. This additional condition is satisfied by the example in [10] (see also [12] [13]). We have seen many irreversible Finsler metrics of constant flag curvature. We know that the Hilbert metric H = H(x, y) on a strongly convex domain U is reversible complete with K — — 1. Is there any reversible Finsler metrics with positive constant flag curvature on S"? In [84], the following theorem is proved: if (M, F) is an n-dimensional closed simply connected reversible Finsler manifold with K = 1, then M is diffeomorphic to Sn and for any point p £ M there is a unique point p* with d(p,p*) = TT such that any unit speed geodesic from p must pass p* and form a closed geodesic of length 2n. In [53], Kim-Yim prove that any reversible Finsler metric on S n with K = 1 and S = 0 must be Riemannian. The proof requires a volume comparison theorem on the metric balls in a Finsler manifold [85] and a volume comparison theorem on the unit sphere bundle of a Finsler manifold [35]. Recently, Foulon claims that any reversible Finsler metric on S2 with K = 1 is Riemannian, provided that its geodesies are great circles [38]. Most recently, R. Bryant announces that any reversible Finsler metric on S2 with K = 1 must be Riemannian. Bryant's proof is a combination of his earlier results [20] with a fundamental (and deeper) result of LeBrun and Mason [63].
Chapter 8
Projectively Flat Finsler Metrics
It is Hilbert's Fourth Problem to characterize the (not-necessarily-reversible) distance functions on an open subset in Rn such that straight lines are geodesies [44]. Regular distance functions with straight geodesies are projectively flat Finsler metrics. They are characterized by a system of partial differential equations (3.18). However, it is still difficult to understand the local metric structure of such metrics. In this chapter, we will discuss projectively flat metrics with special curvature properties. 8.1
Projectively Flat Randers Metrics
According to Proposition 3.4.8, a Randers metric F = a + (3 is locally projectively flat if and only if a is locally projectively flat and (3 is closed. By the Beltrami theorem in Riemannian geometry, a is locally projectively flat if and only if it is of constant sectional curvature K a = p,. In this case, a is locally isometric to the following metric defined on a ball B"(rAI) c R n .
a :=
"
TTW
'
( 1}
Moreover, if F is of constant flag curvature or isotropic S-curvature, /3 can be completely determined. First, let us consider projectively flat Randers metrics with constant flag curvature. Proposition 8.1.1 ([88]) Let F = a + (3 (/3 ^ OJ be a locally projectively flat Randers metric on an n-dimensional manifold M. Suppose that it has 149
150
Projectively Flat Finsler Metrics
constant Ricci curvature Ric = (n — 1)XF2. Then A < 0. / / A = 0, F is locally Minkowskian. If X — —1/4, F is given by P F =
V\u\2 - (M 2 M 2 - (x, y)2) ± (x, y) _,_
±
rq^
where a e R" is a constant vector. In this case,
K = -i,
(a, y)
rr^)'
(8 2)
-
S = ±i(n+1)F.
Proof. One may assume that a has constant sectional curvature K a = \i and /3 is closed. Let $ = bi^yl\f and \P = bi.ij-^ylyiyk be the homogeneous functions defined in (6.11). Since /3 is closed, $ = eooIt follows from (6.10) that
That is, lic?{a + (3)2 + ^ $ 2 - l * ( a + /3) = A(a + /3)4. This gives rise to two equations ^$2
=
1 ^
+
(A
_
M)a4 +
(gA
_ ^^2^2
+ A/g4j
i * = (2/x-4A)a 2 / 9-4A^ 3 .
(g 3)
(8.4)
Substituting (8.4) into (8.3) yields ^ $ 2 = (A - /i)a 4 + (2A + /u)a2^2 - 3A/34.
(8.5)
Differentiating (8.5), one obtains 1$ bi-j-kyY = 2(2A + ^ a 2 ^ hiifcy* - 12A/33 b^y1.
(8.6)
Contracting (8.6) with yk yields ^ $ $ = 2{(2A + /x)a 2 -6A/3 2 }$/3. Substituting (8.4) into (8.7) yields 4(n - 4A)$a2/?2 = 0.
(8.7)
151
Projectively Flat Randers Metrics
We assert that ^ = 4A. If not, from the above equation, $/3 = 0. Then on the open subset U := {x € M \ (5X ^ 0}, $ = 0. Thus (3 is parallel with respect to a and \I> = 0. It follows from (8.4) that fi = 4A = 0. It is a contradiction. Substituting /tz = 4A into (8.5) yields $2 = -4A(Q2-/32)2.
It follows that A < 0. Let c := ±\/—A. The above identity becomes e00 = 2c(a2 - (32).
(8.8)
Now we are going to find an explicit formula for /? using (8.8). Assume that A = 0. By the above argument, /x = 4A = 0 and c = ±\/—A = 0. Thus a is flat and eoo = 0. Since /? is closed, we get bij = 0, namely, /3 is parallel with respect to a. In this case, Gl(x,y) = Gla(x,y) quadratic in y. F is a Berwald metric with zero flag curvature. Thus it is locally Minkowskian by Theorem 2.3.2. Assume that A = —c2 < 0. In this case, a has negative constant curvature // = —4c2. By scaling, we may assume that a has constant curvature /j, — — 1 (c = ± | ) , hence A = —1/4. Thus a can be expressed as a_i, a_i =
,
y € 1XB (1) = R .
It follows from (8.8) that bi-j = e(a,ij - bibj),
e = ±1.
(8.9)
1
We can express /3 = biy in the following gradient form, R_
(x, y) 1 - \x\2
dfx(y) f(x)
where f(x) > 0 is a scalar function on B"(l) and e = ±1 is the same as in (8.9). It follows from (8.9) that fxixi = 0. Thus / is a linear function f = S(l + {a,x)),
5>0,
and
" = erF+erTM>
yeT^d)^. Q.E.D.
152
Projectively Flat Finsler Metrics
Next we are going to study projectively flat Randers metrics with almost isotropic S-curvature. Recall that for a locally projectively flat Randers metric F = a + (3, the Riemannian metric a = aM must be of constant sectional curvature K a = [i and the 1-form (3 must be closed. We have the following Proposition 8.1.2 ([24]) Let F = a + (3 be a locally projectively flat Randers metric on an n-dimensional manifold M. Suppose that F has almost isotropic S-curvature S = (n + l ) | c F + 7j},
(8.10)
where c = c(x) is a scalar function on M and n = r)i{x)yl is a closed 1-form on M. Then the flag curvature is given by
<811»
K3
' -TSW+^2+"
where cxk denotes the partial derivative of c with respect to xk. Moreover, (A) if fi + Ac(x)2 = 0, then c(x) = c is a constant and the flag curvature K = - c 2 . In this case, F — a + j3 is either locally Minkowskian (c = 0) or, up to a scaling (c = ±1/2), locally isometric to the metric in (8.2); (B) if ii + 4c(x)2 ^ 0, then F = a + (3 is locally given by
where aM is given by (8.1) and c(x) :— cfl(x) is given by c
f ( H < a , x ) ) ^ ( 1 + ( , N ^ ( A + ^ z/M^O
c,>{x)-<
±1
^
2y/X+2(a,x)
+ \x\2
J
^
(8.14)
'
where a € R" is a constant vector and X is a constant number. Proof. Let a = i / a ^ V and (3 = hy*. Let $ = b^yi and "J = bi;j;kyiy:iyk be the homogeneous functions denned in (6.11). Since (3 is closed, $ = eOo- By Lemma 5.2.2, $ = e00 = 2C(Q 2 - /32) = 2c(a - 0)F.
(8.15)
Protectively Flat Randers Metrics
153
Using (8.15), one obtains from a-k = 0, $ = /3;kUk and * = $:kyk that * = 2c,kyk{a2 - (32) - 4C/5/?;fc/ = 2cxkyk(a2
-/32)-4c$f3
2cxkyk(a2-p2)-8c2(3(a2-f32)
=
= 2(cx./-4c2/3)(a-/?)F. By (6.10) and the above formulas, one obtains
= ^a 2 + 3c2(a - /3)2 - (c^y* - 4c2p)(a - (3).
(8.16)
On the other hand, by Theorem 7.1.2, the flag curvature is in the following form K
=
^
+ *.
(8.17)
where <j = a(x) is a scalar function on M. Combining (8.16) and (8.17) yields
2{2c x ,/ + (a + c2)p}a + \2cxkyk + (a + c2)p}p
+ {a - 3c2 - /x}a2 = 0. This gives 2cxkyk + (a + c2)/3 = 0,
(8.18)
a - 3c2 - /i = 0.
(8.19)
From (8.19), one obtains that a = 3c2 + /x. Substituting it into (8.18) yields (fi + 4c2)/3= -2cxkyk.
(8.20)
2
Substituting a = 3c + /j, into (8.17) yields (8.11). Using (8.20), one can rewrite (8.11) as (8.12). Case 1: Suppose that /j + 4c(x)2 = 0. Then c(x) = constant. It follows from (8.11) that K = 3c2+M = -c2.
154
Projectively Flat Finsler Metrics
Then Theorem 8.1.2 (A) follows from Proposition 8.1.1. Case 2: Suppose that fi + 4c2 / 0 on an open subset U C M. Then by (8.20), 0 is given by 2cxkyk TT~2fi + Ac2
P =
(8.21)
Note that /3 is exact. Let cti := cxi and c-ti.j denote the covariant derivatives of c with respect to a. It follows from (8.15) and (8.21) that l2cC;iC-j 2 C;i;j = -C(H + 4C ) a y + ^ + ^ •
The above equation can be converted to the following equation in a local coordinate system: y,{xlcxj +xJcxi) _ l + /z|z|2 -
°xixl+ ,
9, f
u,xxxi
Sa
i
-^+4C){TT^-(TT^T}
12ccTiCTj
+
7^^-
(8 22)
-
To solve (8.22) for c = c(x), let
{
2 C y^l+ M |ip
,
V±(M+4e^) U ^ ^ U (8.23) ^ if/« = 0 where the sign ± depends on the value of c such that ±(/i + 4c2) > 0. Then (8.22) is reduced to the following equation: _ f0 if n ^ 0 <*>*<** -
\SSij
if/i = 0 '
One immediately obtains _
( A + (a, x)
if /j, y£ 0
^ ~ \4(A + 2(a,a;) + |a;|2) if0 = 0 ' where a G R™ is a constant vector and A is a constant number. Solving (8.23) for c gives the formula in (8.14). Q.E.D. By Proposition 8.1.2, one immediately obtains the following
155
Projectively Flat Metrics with Constant Flag Curvature
Corollary 8.1.3 Let F = a + /3 be a locally projectively flat Randers metric on an n-dimensional manifold M. Suppose that F has constant Scurvature S = (n + l)cF. Then F is locally Minkowskian, or Riemannian with constant curvature, or up to a scaling, locally isometric to the metric in (8.2). Proof: Let /i be the constant sectional curvature of a. First assume that fi + 4c2 = 0. Then by Proposition 8.1.2 (A), F = a + (3 is either locally Minkowskian or, up to a scaling, locally isometric to the metric in (8.2). Suppose that y, + Ac2 ^ 0. Then by Proposition 8.1.2 (B), F = a + (5 is given by (8.13). Since cxk = 0, /? = 0 and F = a is a Riemannian metric. Q.E.D.
8.2
Projectively Flat Metrics with Constant Flag Curvature
The first set of Finsler metrics of constant flag curvature were discovered by Hilbert, Berwald [17] and Funk [39], [40]. All these metrics are locally projectively flat. In the past ten years, R. Bryant made a significant progress in the study of Finsler metrics of constant flag curvature. In particularly, he has classified projectively flat Finsler metrics on S" with constant curvature K = 1 [19], [20], [2l]. In this section, we shall study and characterize projectively flat Finsler metrics of constant flag curvature. Such metrics can be described using algebraic equations or using Taylor expansions. We shall also discuss various examples. Lemma 8.2.1 Let F = F(x, y) be a Finsler metric on an open subset U C R n . Then F is projectively flat with constant flag curvature K = A if and only if there are positively y-homogeneous functions P = P(x, y) on TU^U x R" such that Fxk = (PF)yk,
(8.24)
PXK = PPyk - XFFyk,
(8.25)
in which case, P = jpFxmym
\F\
is the projective factor of F and P2—Pxkyk =
156
Projectively Flat Finsler Metrics
Proof. Assume that F = F(x, y) is projectively flat on U. Then it satisfies (3.18) and the projective factor is given by P := jpFx™ym. Observe that (PF)y- = \(Fx™ymV
= \(F*™y«ym + F*>) = \(Fx- + Fxk) = Fxk.
Thus F satisfies (8.24). Comparing (7.29) and (6.7) yields p, fXk
PP — FFyk
-
—
( K F 3 )y= —
—
X P P
-APtyk.
Thus P = P(x,y) satisfies (8.25). Conversely, suppose that (8.24) and (8.25) hold for some positively yhomogeneous functions P = P(x,y) and a constant A. First by (8.24), one obtains Fxkyiyk
= (PF)ykylyk
= {PF)yi = Fxh
±Fxky* = ±(PF)y^
= P.
By Theorem 3.3.1, F is projectively flat and P is the projective factor. Contracting (8.25) with yk yields that P2 - Pxkyk
= XF2.
By (6.8), the flag curvature is a constant, i.e., K — A.
Q.E.D.
Before we go on, let us take a look at the Funk metric. Example 8.2.2 Let <j> = <j>(y) be a Minkowski norm on R™ and U be the domain enclosed by the indicatrix of (j>. The Funk metric G = G(x, y) on U is defined by e(x,y) =
(8.26)
y&TxU.
It satisfies (1.38), i.e., Oxk = QQyk. By this PDE, one can easily verify that F := 0{x,y) and P := \®{x,y) satisfy (8.24) and (8.25) with A = —1/4, respectively. Thus G is projectively flat with constant flag curvature K = - 1 / 4 . When U = B n (l) is the standard unit ball in R", G is given by
e =
V(i-|*l 2 )|y| 2 + (*,y)2 , (x,v)
T^P
+
TH^P
(
, 897 x }
Protectively Flat Metrics with Constant Flag Curvature
157
We can construct projectively flat Finsler metrics of negative constant flag curvature using algebraic equations. Theorem 8.2.3 ([89]) Let ip = ip(y) be an arbitrary Minkowski norm on R™ and
(8.28)
F:=i{* + (i,y)-
(8.29)
V±(x,y)=
is a projectively flat Finsler metric with constant flag curvature K = — 1 and F(0, y) = ip(y), and its protective factor P — P(x, y) is given by
P=^{y+(x,y)+t>_(x,y)}
(8.30)
withP(0,y) = ip(y). We first prove that the functions ^ ± defined in (8.28) exist and satisfy the following equations: (*±) l f c = * ± ( * ± ) v * . Then, using these equations, we can show that the functions F and P defined in (8.29) and (8.30) have the desired properties.
Lemma 8.2.4 Let
f(x,y)=4>(y + f(x,y)xy
(8.31)
Moreover, f satisfies /*<•=//„*•
(8-32)
158
Projectively Flat Finsler Metrics
Proof. Fix y e TxRn S Rn with y ^ 0. Let h(t) := t - <j)(y + tx). By the homogeneity of , there is a small S > 0 such that if |x| < <$, then at any t with y + tx ^ 0, h'{t) = l-<j>yk{y + tx)xk>
-.
Note that /i(0) < 0. Thus there is a unique to > 0 such that ft(io) = 0. Setting f(x, y) := i 0 , one obtains the unique solution. Differentiating (8.31) with respect to xk and yk gives (l-tfymZ"1)/^^*/
Since
Q.E.D.
Proof of Theorem 8.2.3: Let \I>± = *±(x, y) be the functions defined in (8.28). By Lemma 8.2.4, * ± satisfy (8.32), i.e., (*±)«* = * ± ( * ± V
(8.33)
with *±(0,j/) = ^>±(y) = y>(r/) ± V(j/). It follows from (8.33) that (*±Wy f e = (*±)x«This implies that F := | { * + - * _ } satisfies (3.18). Hence F is projectively flat by Theorem 3.3.1. Observe that
Fxkyk = i{* + (* + ) y t - *_(*_)„*}j/fc = \{&+ - &_}. Thus the projective factor P = ^F~1Fxkyk is given by
By a similar argument, one obtains
Thus the flag curvature K = - 1 by (6.8).
Q.E.D.
By the formulas in Theorem 8.2.3, one can construct some projectively flat Finsler metrics with K = — 1.
Projectively Flat Metrics with Constant Flag Curvature
159
Example 8.2.5 Let
V-
2
. _ 4>{y) + s
where e = ± 1 and <5 is a constant chosen so that V> is a Minkowski norm on R n . Note that
$H£y)-
Let * + = *+(x,y) and ^ _ = * _ ( x , y ) be the solutions of (8.28) with 4>+(y) = <j>(y) and 4>-{y) := 5<j>{ey), respectively. Let B = O(x, t/) denote the Funk metric of <j> defined in (8.26). Then *+(i,2/) = 0(x,y),
* - ( x , y ) = (5e(<Jex,£j/).
By Theorem 8.2.3, one concludes that the following function
F:=^{e(x,y)-5Q(6ex,£y)} is a projectively flat Finsler metric on its domain with K = — 1 and its projective factor is given by
P=^{G(x,y)+6O(6ex,£y)}. When 6 = —1 and e = —1, * + ( i , y) = 0 ( i , y),
* _ (x, y) = -B{x, -y).
By the above argument, we know that
F:=±{e(x,y)+G(x,-y)} is a projectively flat Finsler metric on U with K = — 1 and its projective factor is given by
P=±{G(x,y)-e(x,-y)}. F is the Hilbert metric on U (see Example 3.4.5). We now consider the special case when <j> — \y\ + (a, y) where a 6 R™ with \a\ < 1. The Funk metric denned by
160
Protectively Flat Finsler Metrics
a(x, y) + /3(x, y), where a = a(x, y) and (3 = (3(x, y) are given by
^[(1 -.(a,x)f - |x|*] [|j,|2 - (a,y)*] + [(1 - (a,x))(a,i/) + (x,y)]2 Q : =
P
-
(l-(a,x»2-|x|2
'
(l-(a,x))(o,y) + <x,y) (l-(a,x))2-|x|2 •
Let e = ±1 and <J be a constant with <5 < 1 and |1 — e5\\a\ < 1 — 5. Then F := ^\a(x, y) - Ja(Jea;, j / ) | + -|/3(a;, ?/) - 6s/3(6£x, y)} is a projectively flat Finsler metric near the origin with K = - 1 and its projective factor is given by
P = \{a(x,y) +5a{5ex,y)} + ^{p{x,y)+ 8ep(8ex,y)}. Note that F is no longer of Randers type. Example 8.2.6 Let <j> = <j>(y) be an arbitrary Minkowski norm on R" and O = Q(x, y) denote the Funk metric of <j>. For a constant vector a G R n , let V> := ^ (
V := 2 ( ^ ( ^ ~ ( a ' V))'
such that v(y) + ip(y) = 4>{y),
Let * + = *+(x,y) and $ _ = *_(x,y) be defined by (8.28) with
*
(a
-
'^
By Theorem 8.2.3, one knows that the following function
is projectively flat with K = - 1 and its projective factor is given by
P=1-{&(x,y)V Ul 21
{a y)
]
}.
l + {a,x))
161
Projectively Flat Metrics with Constant Flag Curvature
When <> / = |y|, the Funk metric @ on the unit ball B"(l) is given by (8.27). Thus for any constant vector a e Rn with |o| < 1, the following function -,
x i f v % l 2 - ( l g | 2 M 2 - f o y ) * ) + (g,y) ,
(a,y) t
+
l + {a,x)J
t
l-|xP
^y>-2\
is projectively flat on B"(l) with K = —1. See Examples 5.1.3 and 6.1.5 above. Now we are going to construct a projectively flat Finsler metric of zero flag curvature for any given pair {, V}Theorem 8.2.7 ([89]) Let xp = ip(y) be a Minkowski norm on R" and ip =
(8.35)
+ P{x,y)xy
Let F:=^(y + P(x,y)x){l+Pym(x,y)xm}.
(8.36)
Then F = F(x, y) is a projectively flat Finsler metric of zero flag curvature with F(0, y) = ip{y) and its projective factor is P = P(x, y) with P(0, y) =
Pxk = PPyk with P(0, y) =
( 8 - 38 )
P lV =^(PV=^V
Differentiating (8.36) with respect to xk and using (8.37), one obtains Fxk = %m(y + Px)[Pd% + PxkXm}{\ +ip(y + Px){pyk
+
+ Pyixl]
PymxkXmy
By (8.37), one obtains PF = i>{y + Px){P + PxmXmy
(8.39)
162
Projectively Flat Finsler Metrics
Differentiating (8.39) with respect to yk yields {PF)yk = ipym(y + Px){d? + Pyuxm}{p +iP(y + Px){Pyk +
+ Px.x1}
PxmykxmY
By (8.37) again, one obtains
{p5Z + Pxkxm){l + Pyix1} = {S? + Pykxm}{p+Pxlx1}. Together with (8.38), one can see that F satisfies (8.24). Note that (8.37) is equivalent to (8.25) with A = 0. Thus F is a projectively flat Finsler metric on its domain with K = 0 and its projective factor is P. Q.E.D. By Theorem 8.2.7, one can construct several projectively flat Finsler metrics with K = 0. Example 8.2.8 Let (j> = (y) be a Minkowski norm on R" and 0 = 8(x, y) denote the Funk metric of (f> defined in (8.26). Let ip := (j)(y) + (a, y) and ip :- <j>{y). Let P = P(x,y) and F = F{x,y) be denned in (8.35) and (8.36) respectively. Clearly, P = 9. Observe that
Thus ip{y + Px) = Q + (a,y) + (a,x)G. The function F defined in (8.36) is given by
F = {(l + (a,x)^Q(x,y) + (a,y)}{l + 9yk{x,y)xk}.
(8.40)
By Theorem 8.2.7, one knows that F is projectively flat with K = 0 and its projective factor P = Q(x, y). Note that when a = 0, the Finsler metric in (8.40) is reduced to
F = Q(x,y)[l + eyk{x,y)xk]
= G(x,y) + Qxk(x,y)xk.
When ip = \y\ + (a,y) and ip = \y\, the Finsler metric in (8.40) can be expressed by
I
VM 2 -(|z| 2 |#-(z,2/} 2 ) +
Projectively Flat Metrics with Constant Flag Curvature
(V\y\2-(W2\y\2-(x,y)2) + (*, y))J * x± 2 2 2 (l-|x|2)Vl2/| -(W |y| -(x,y)2)
163
(8.41)
Clearly, F is not locally Minkowskian. When a = 0, the Finsler metric in (8.41) is reduced to (V\y\2-(\x\2\y\2-(X'y)2) F=±
2
2
2
+ (x,y))2 ' 2
(i-W )VI#-(N li/l -<*.y> )
(8.42)
This is just the projectively flat Finsler metric constructed by L. Berwald [17]. The Finsler metric F in (8.42) is positively complete, i.e., every unit speed geodesic on (S, T) can be extended to a geodesic on (5, oo). The Finsler metric defined in (8.42) is positively complete, but not complete. To construct a complete one, one should construct a reversible metric F = F(x,y) by (8.35) and (8.36). It suffices to assume that ip =
Hxk = HHyk
with H(0, y) = ip(y)+iip(y). Express H = P + iF with P(0, y) = tp(y) and F(0,y) = i>{y). By continuity, F = F(x,y) is a Finsler metric for x close to the origin. It follows from (8.43) that Pxk - PPyk + FFyk + i ^
- (PF)yk}
= 0.
164
Projectively Flat Finsler Metrics
That is, Fxk = (PF)yk,
Pxk=
PPyk - FFyk.
By Lemma 8.2.1, F = F(x,y) is projectively flat with constant flag curvature K = 1 and its projective factor is P = P(x,y). Thus, the remaining problem is how to find complex-valued functions H = H(x, y) satisfying (8.43) with H(0,y) = f(y) + iip(y). We might construct such functions using a Taylor expansion. Let H = H(x,y) be a positively homogeneous function of degree one satisfying (8.43) with H{0,y) =
Hxil...x<m =-±—\Hm+1]
. . .
m+ll
Thus
Jy'l-y'm
Hxil...xim(O,y) = —!—[(^ + iYOm+1] . m + ll
. (y).
Jy'l—y'm
If H = H(x, y) is analytic in x at x = 0, then
= E 7^TTv^[( v ' ( 2 / +te)+ iV'(2/ +
H
m=0 ^
'
te) m+1 1 0
) ]^-
(8 44)
'
However, for a given pair {
z =
Then the solution H := z satisfies (8.43) with H(0, y) =
=ie~ia\y\.
165
Protectively Flat Metrics with Constant Flag Curvature
Note that \y + zx\ = y/(y + zx,y + zx) can be defined for 2 € C Equation (8.45) becomes z = ie-iay/\y\2
+ 2(x,y)z + \x\2z2.
One obtains a formula for H :— z, H =
-(x,y) +W(e*i<* + \x\*)\y\* e2ia +
ja-jZ
(x,W •
\ •
)
Prom (8.46), one can see that F is a complex-valued function on xR n . We can express it in the form H := P + iF. By the above argument, F = F(x, y) is a projectively fiat Finsler metric of constant flag curvature K = 1 and P = P(x, y) is the projective factor of F. To express F and P in terms of elementary functions, let A : = (cos(2a)M 2 + (|x|2|y|2 - (x,y)2))2 + (sin(2a)|2/| 2 ) 2 , B: = cos(2a)\y\2 + U :-
(\x\2\y\2-(x,y)2),
sin(2a)(x,y),
V:= (cos(2a) + |a;|2)(a;,?/), E: = | i | 4 + 2cos(2a)|x|2 + l. For an angle a with 0 < a < TT/2,
^(ea- + |,|2)|y|2 _ {x>y)2 =
]J^A±I.+i]j^Y£.
From (8.46), one obtains
In dimension two, one can verify that the Finsler metric in (8.47) is a Bryant metric [19], [20].
166
Projectively Flat Finsler Metrics
According to (8.44), one can also express if as a power series,
Thus F and P can be expressed as power series, ^
^ - 2^ m=0
sin[(m+l)(7r/2-a)] dm r
(^Tl)i
v
m
d* F
p _ V cos[(m + l)(7r/2-a)]d-r
^-A,
( m + i)!
dt-L
+ tX|
|y+fx|
+1i
J |«=o ,
W
Projectively flat Finsler metrics of constant flag curvature K = 1 constructed above are all local in a sense that they are defined on an open convex domain in R™ and hence they are incomplete. The Finsler metrics in Example 8.2.9 can be pulled back to Sn by two maps ip± to form complete irreversible projectively flat Finsler metrics of constant flag curvature K = 1. The maps ip± are defined in Example 1.2.4. Recently, R. Bryant has completely determined the global structures of Finsler metrics with K = 1 on S n , whose geodesies are great circles. His approach is very nice and different from the above one. See Bryant's work [19] and [20], [21]. 8.3
Projectively Flat Metrics with Almost Isotropic S-Curvature
In this section we are going to study and characterize locally projectively flat Finsler metrics with almost isotropic S-curvature. Recall that a Finsler metric F = F(x, y) is said to have almost isotropic S-curvature, if S = (n + l ) | c F + ^ | ,
(8.49)
where c = c(x) is a scalar function and r) = r)i(x)yl is a closed 1-form on M. We know that a Randers metric has isotropic S-curvature if and only if it has almost isotropic S-curvature. In Proposition 8.1.2, we have completely classified projectively flat Randers metrics with isotropic S-curvature.
167
Projectively Flat Metrics with Almost Isotropic S-Curvature
According to Example 5.1.3, the following Finsler metric F is projectively flat with constant flag curvature and almost isotropic S-curvature, F := Q(x,y) +
y^TxU = Rn,
{ }
y 1 + (a, x)
where © = 0 ( x , y) is the Funk metric on a strongly convex domain U C R n . When U is the standard unit ball B"(l), F is of isotropic S-curvature. A natural problem arises: are there other types of projectively flat Finsler metrics of almost isotropic S-curvature. The answer is that if the metric is not a Randers metric, then, after a scaling, it must be in the above form or its reversed form. More precisely, we have the following Proposition 8.3.1 ([26]) Let F = F(x,y) be a projectively flat Finsler metric on an open subset U C R n . Suppose that F has almost isotropic S-curvature, i.e.,
S = (n + l){cF + 7?},
(8.50)
where c = c(x) is a scalar function and n = •qidxt is a closed 1-form on U. Then F is determined as follows. (a) If K ^ -c{x)2 + Ca>((3f^"* at every point x e U, then F = a + 0 is a Randers metric on U. and it is determined in Proposition 8.1.2 (B); (b) 7 / K = — c(x)2 + Cxp(*y) ' then c{x) = c is a constant, and either F is locally Minkowskian (c — 0) or up to a scaling, F can be expressed as
\e(x -v){
y)
i+
(c--x-y (<-—
2/
{
'
where a e R" is a constant vector and © = &(x,y) is a Funk metric defined by (1.38). Proof. By assumption, F is projectively flat. Thus the spray coefficients are given by Gl = Py%, where
P:=^ff-
(8-52)
168
Protectively Flat Finsler Metrics
By (6.8), the flag curvature of F is given by P2 K
— P kiik
=
(8-53)
•
F2
The S-curvature is given by
S = (n + l){cF + r?} _ dGm
d(\naF)
Qym
=<.
I*
+
Qy-m
.>r-^.
Since 77 = ry(x, y) is closed on U, it can be expressed in the form 77 = dhx(y) where h = h(x) is a scalar function on U. From the above identities, we obtain (8.54)
P = cF + dip, where
III[(TF{X)}.
It follows from (8.52) and (8.54) that
Fxmym =2FP = 2F{cF + yxmyrny
(8.55)
Substituting (8.54) into (8.53) and using (8.55), one obtains
K
=
\cF + Vxmym}
- |c xm {x)y m F + cFxmym + —
fxixiyiyj}
-c2F2 - cxmymF + {
(8.56)
On the other hand, since F is of scalar flag curvature, by Proposition 7.1.2, the flag curvature of F is given by K = 3 £*^r + f f >
(857)
where a = a(x) is a scalar function on U. Comparing (8.56) with (8.57) yields
{a + c 2 }F 2 + 4cxmymF + { y w -
(8.58)
169
Projectively Flat Metrics with Almost Isotropic S-Curvature
Assume that K ^ -c(x)2 + ^p/^'Ji at every point x € U. Then, by (8.57), for any x e U, there is a non-zero vector y e TJA such that , s
, >Q
2cx™(x)ym
,
We claim that a(x) + c(x)2 ^ 0 for any x G if. If not, there is a point x0 £ U such that cr(xo) + c(xo)2 = 0. The above inequality implies that dc ^ 0 at i o . Then (8.58) at xo is reduced to Acxm(xo)ymF{xo,y) + {
= 0.
fj, + 4c(x)2 ^ 0. By Proposition 8.1.2 (B), F is given by (8.13). We now assume that K = -c(x)2 + y a(x)+c(x)2+2Cxm^ym
^ • It follows from (8.57) that s 0
.
This implies that c{x) = c is a constant, hence a(x) = —(? is a constant too. In this case, the flag curvature is given by K = —c2. The equation (8.58) is reduced to ¥ W -
170
Protectively Flat Finsler Metrics
where a 6 R n is a constant vector and C is a constant. If c = 0, then K = - c 2 = 0. It follows from (8.54) that the projective factor P = d
F=l{*(a:,y)-^(tf)} = l{*(x,y) +
I
^ } .
It follows from Lemma 8.2.1 that Pxk = PPyk + c2FFyk.
Fxk = (PF)yk, We have
*xi=Pxi+cFxi = PPyi + c2FFyi + c(PF)yi = {P + cF}{Pyi+cFyi} = Wyi. Since c might be negative, F and \I> have opposite signs when a is sufficiently small. Thus we introduce another function as follows, e
=
r^(x,y)
ifc>0
\ -
if c < 0
Then 9 = Q(x,y) satisfies (1.38), i.e., Qxi
=QQyi.
By definition, 0 is a Funk metric. Choosing c = ± | , we obtain (8.51). Q.E.D.
Appendix A
Maple Programs
In Finsler geometry, the computations of geometric quantities are usually very complicated. However, using a Maple program, we can quickly check whether or not an expression is correct even though we can derive it manually; we can simplify the expressions in the computation without making mistakes. A.I
Spray Coefficients of Two-dimensional Finsler Metrics
In this section, we shall compute the spray coefficients for two-dimensional Finsler metrics. For simplicity, we denote a point (a:1, a;2) in R2 by (x,y) and a tangent vector J^gfr + y1^ at (x 1 ,^ 2 ) by (x,y;u,v). Let F = F(x, y, u, v) be a Finsler metric on an open subset U C R2. Let L{x,y;u,v) := -F2(x,y,u,v). The spray coefficients, G := Gl(x, y; u, v) and H := G2(x, y; u, v), are given by (~,
yJ-'x'-'vv ~ LiyLjuv)
— \LXV — LyU]Lv
/ , .>
2[LUULVV — (Luv) \ rr \~LXLUV H :=
+ LyLuu) + (Lxv — LyU)Lu -; ^ .
2\LUULVV
, . „> (A.2)
— (Luv) J
Below is a Maple program for computing G and H. Our testing example 171
172
Maple Programs
is the Klein metric on the unit disk in R2. It is denned by s/{u2 + v2) - ((a:2 + y2)(u2 + v2) - (xu + yvj2) ~ l-(x2+y2) • It takes a while if one computes the spray coefficients G% by hands. Now we can use a Maple program to find its spray coefficients within a couple of seconds once the metric function is entered. >
restart;
> > >
GH:=proc(F) local L,Lx,Ly,Lu,Lv,Lxv,Ljru,Luu,Luv,Lvv,Numl)Num2, Den.G.H;
>
L:=F~2/2;
>
Lx:=diff(L,x);
>
Ly:=diff(L,y);
>
Lu:=diff(L,u);
>
Lv:=diff(L,v);
>
Lxv:=diff(L,x,v);
>
Lyu:=diff(L,y,u);
>
Luu:=diff(L,u,u);
>
Luv:=diff(L,u,v) ;
>
Lvv:=diff(L,v,v);
>
Den:=2*(Luu*Lvv-Luv*Luv);
>
Numl:=(Lx*Lvv-Ly*Luv)-(Lxv-Lyu)*Lv;
>
Num2:=(-Lx*Luv+Ly*Luu)+(Lxv-Lyu)*Lu;
>
G:=Numl/Den;
>
H:=Num2/Den;
>
RETURN([G,H]);
>
end:
>
F:=sqrt(yy-(xx*yy-xy"2))/(l-xx); _ y/yy -xxyy + xy2 1 — xx
>
xx:=x~2+y~2:
>
yy:=u"2+v"2:
Spray Coefficients of Two-dimensional Finsler Metrics
173
> xy:=x*u+y*v: > M:=GH(F): > G:=M[1]: > H:=M[2]: > G:=simplify(G);
>
_ '~
u(xu + yv) - 1 + x2 + y2
_ ''~
v(xu + yv) -l + x2 + y2
H:=simplify(H);
In the above program, we define a function GH(F). The input is a Finsler metric F and the output is the matrix [G, H] formed by the spray coefficients. For the Klein metric, one gets G and H which can be expressed in the form G = Pu and H = Pv. Thus the Klein metric is projectively flat on the unit disk. The above Maple program can be used to compute the spray coefficients for general two-dimensional Finsler metrics. The Bryant metrics on the upper (or lower) hemisphere can be pulled to projective metrics defined on R2 using the same standard map ip± in Example 1.2.4. Thus they can be expressed as metrics on R2.
where A: = B2 + (l-e2)(u2 + v2)2, B : = e(u2 + v2) + \(x2 + y2)(u2 + v2) - (xu + yv)% U : = V l — e2 (xu + yv), V : = [e + (x2 + y2)] (xu + yv), E: = l + 2e(x2 + y2) + (x2 + y2)2. Below is a portion of a Maple program for computing G and H of a family of Bryant metrics on S2. We omit the head part of the above Maple program denning GH(F).
174
Maple Programs
>
F:=sqrt((sqrt(A)+B)/(2*E)+(U/E)~2)+U/E;
i / VA + B >
P:=sqrt((sqrt(A)-B)/(2*E)-(U/E)-2)-V/E;
p.^\f^A-B
" 2V
>
Tip u
E
UP2 V E
E
A:=(e*yy+(xx*yy-xy~2))"2+(l-e"2)*yy~2:
> B:=e*yy+(xx*yy-xy"2): >
U:=sqrt(l-e~2)*xy:
>
V:=(e+xx)*xy:
>
E:=l+2*e*xx+xx~2:
> xx:=x"2+y"2: > yy:=u~2+v"2: >
xy:=x*u+y*v:
>
M:=GH(F):
>
G:=M[1]:
>
H:=M[2]:
>
x:=-l/2;y:=-3/5;u:=-l;v:=4/5;e:=l/4; -1 -3 u := -1 4
6:
_ 1 ~4
> simplify(G/u-H/v); 0 > simplify(G/u-P); 0 > simplify(H/v-P); 0
Spray Coefficients of Two-dimensional Finsler Metrics
175
In the above Maple prgram, we verified two facts: 1) F is projectively flat, and 2) G = Pu and H — Pv where P is given by ly/A-B
(U\*
V
One can also use equation (3.18) to check whether a Finsler metric is projectively flat. Equation (3.18) in dimension two is given as follows, Fxuu + Fyuv = Fx,
Fxvu + Fyvv = Fy.
(A.3)
If a Finsler metric F = F(x, y; u, v) satisfies (A.3), then by Theorem 3.3.1, it is projectively flat. In this case, the spray coefficients are in the form G = Pu and H — Pv, where P = ^ > .
(A-4)
Below is a Maple program by which we verify that a Bryant metric F in (8.47) satisfies (A.3), hence it is projectively flat. We also compare the function P in (8.48) with the function P in (A.4) using fractional numbers. They are always sufficiently close. Thus we conclude that they are equal to each other. A tip is to select small values for the variables, otherwise the computation will go beyond the capacity of Maple. >
restart;
>
F:=sqrt((sqrt(A)+B)/(2*E)+(U/E)~2)+U7E;
>
P:=sqrt((sqrt(A)-B)/(2*E)-(U/E)~2)-V/E;
>
1 ly/A-B 4U2 V 2 V E ~ E2 ~E A:=(e*yy+(xx*yy-xy~2))"2+(l-e~2)*yy~2:
>
B:=e*yy+(xx*yy-xy~2):
>
U:=sqrt(l-e~2)*xy:
>
V:=(e+xx)*xy:
>
E:=l+2*e*xx+xx"2:
>
xx:=x"2+y"2:
>
yy:=u~2+v"2:
176
Maple Programs
> xy:=x*u+y*v: > simplify(diff(F,x,u)*u+diff(F,y,u)*v-diff(F,x)); 0 > simplify(diff(F,x,v)*u+diff(F,y,v)*v-diff(F,y)); 0 > PP: = (diff(F,x)*u+diff(F,y)*v)/(2*F): > e:=l/4;x:=l;y:=-l/2;u:=l/3;v:=l/5; 1 p - ^
4 x:=l -1 I
1
V "-— —
>
simplify(PP-P); 0
A.2
Gauss Curvature
The flag curvature in dimension two is called the Gauss curvature. For a Finsler metric F = F(x,y;u,v), the Gauss curvature K = ~K(x,y;u,v) is given by K : = ^ { 2 G X + 2Hy -G\-Qxu-Qyv
H2V - 2HUGV
+ 2GQU + 2HQV\,
(A.5)
where G = G(x, y; u, v) and H = H(x, y; u, v) denote its spray coefficients and Q = Gu + Hv. The formula (A.5) is in fact a formula for the Ricci scalar Ric divided by F2. In dimension two, the quotient Ric/F 2 is the Gauss curvature. Below is a Maple program for the Gauss curvature, by which we compute the Gauss curvature of the Funk metric on the unit disk. As we know, the Gauss curvature of the Funk metric is equal to —0.25. However, our PC does not run fast enough to complete the symbolic computation. Thus we
Gauss Curvature
177
randomly select some point (x, y) in the unit disk and a direction (u, v). If we always obtain a value sufficiently close to —0.25, then we can conclude that the Gauss curvature is equal to —0.25 and look for a rigorous proof. >
restart;
> > >
GC:=proc(F) local L,Lx,Ly,Lu,Lv,Lxv,Lyu,Luu,Luv,Lvv,Numl,Num2, Den.G.H.Gx.Hy.Gu.Gv.Hu.Hv.Q.Qx.Qy.Cju.qv.M.N.K;
>
L:=F*2/2;
>
Lx:=diff(L,x);
>
Ly:=diff(L,y);
>
Lu:=diff(L,u);
>
Lv:=diff(L,v);
>
Lxv:=diff(L,x,v);
>
Lyu:=diff(L,y,u);
>
Luu:=diff(L,u,u);
>
Luv:=diff(L,u,v);
>
Lvv:=diff(L,v,v);
>
Den:=2*(Luu*Lvv-Luv*Luv);
>
Numl:=(Lx*Lvv-Ly*Luv)-(Lxv-Lyu)*Lv;
>
Num2:=(-Lx*Luv+Ly*Luu)+(Lxv-Lyu)*Lu;
>
G:=Numl/Den;
>
H:=Num2/Den;
>
Gx:=diff(G,x);
>
Hy:=diff(H,y);
>
Gu:=diff(G,u):
>
Gv:=diff(G,v):
> Hu:=diff(H,u): > >
Hv:=diff(H,v): Q:=Gu+Hv:
>
Qx:=diff(Q,x):
>
qy:=diff(Q,y):
>
Qu:=diff(Q,u):
178
Maple Programs
>
Qv:=diff(Q,v):
> M:=Qx*u+Qy*v: > N:=G*Qu+H*Qv: >
K:=(2*Gx+2*Hy-Gu~2-Hv-2-2*Hu*Gv-M+2*N)/F~2:
> RETURN(K); >
end:
>
F:=sqrt(Y-(X*Y-Z~2))/(l-X)+Z/(l-X); y/Y-XY 1-X
>
+ Z2
Z Y^~X
+
X:=x"2+y"2:Y:=u~2+v~2:Z:=x*u+y*v:
> K:=GC(F): >
x:=0.3:y:=-0.6:u:=2.1:v:=0.5:
>
K:=simplify(K); K := -.2499999933
In the above program, we define a function K:=GC(F). The input is a Finsler metric F and the output is the Gauss curvature K. Note that the first half of the program is for computing the spray coefficients G and H. For a projective flat metric F = F(x,y;u,v), one can first find the projective factor using (A.4), then use the projective factor to compute the flag curvature by the formula in (6.8), i.e., K
A.3
P2 - PXU - PyV =
F~2
•
Spray Coefficients of (a,/3)-Metrics
In this section, we shall find a formula for the spray coefficients Gl of an (a,/3)-metric in any dimension. This formula is given in (3.5) without detailed computation. We shall start with computing gij, then find a formula for g%d, by which we derive a formula for Gl. In each step, we introduce some new variables which are expressed in terms of previous variables. We leave them without simplification until we obtain a formula for Gl. Then we simplify all the coefficients involved in the formula of G% using Maple.
Spray Coefficients of (a, 0)-Metrics
179
Let s = -. a
F = a
9ij = pan + pobibj + pi (biCtj + bjati) + p2<*i<Xj,
(A.6)
where ccj := a^y3 /a and
p: = <^|^-s^'j, po : = H " + 4>'
+
pa:=s{s(M>" Rewrite gtj as follows:
9ii = p{Aii+nYjrj} where Yi : = cti + ebi, Aij : = OJJ 4- S bibj, . . _ Pi £ . — s
,
P2 ^ po - e 2 /9 2 p
The inverse (Alj) := (A^)" 1 is given by Aij =aij
-TVV,
where S
(A.7)
180
Maple Programs
Using the formula for Al:>, we can find a formula for the inverse (gij) : = OK;)" 1 gij = p-1^
- jjY^yj} = p '
1
^ - r bV - r, F ^ } ,
where
yi.. = Aijy
VL + Xtf^ a ',,
£ - (5s
The geodesic coefficients Gl are given by
G'=GL + ^
+ f/{F ;fc ,/-F ;i },
where F ; j denote the covariant derivatives of F with respect to a and F;j.^ = [F ; i] y j. We use the above identity to find a formula for G \ Observe that F;k = t'bifiV' F.Myk
=
(bl-Sal)
Thus F;kyk = 4>'h;kyiyk = 0VOO. F-k-iyk - F.jt = (6, - saW— Then we obtain the following formula for G%.
a
+ 2
Gi = Gia + Pyl + Q\ where P — Es0 + a~1QrOo
Qi = Qas\ + ($as0 + $r 0 oV
Spray Coefficients of (a, f})-Metrics
where
s U " (b2 - s2)r)\
(/>' ~ 2(f>
2p
2p
P
* = ^ { l - ( & 2 - s 2 ) ( r + r,A2)}. We use Maple to simplify Q and 0 as follows, ^
e
=
-S(/>'
_
<W - s(4>4>" + W) 2
s2)<j>")
We also use Maple to find the following identity,
Let x :=
e•
X is given by x
A
-
^
00' - s(00" + 4>'(j>')'
Finally, we obtain the following formula for G%.
Gi = Gia + Qs\ + e { - 2Qs0 + roo}{^ + x&*}. Below is a Maple program for the above computation. >
restart;
>
f:=phi(s):
>
fs:=diff(f,s):
181
182
Maple Programs >
fss:=diff(f,s,s):
>
rho:=f*(f-s*fs):
>
rhoO:=f*fss+fs"2:
>
rhol:=-(s*(f*fss+fs~2)-f*fs):
>
rho2:=s*(s*(f*fss+fs~2)-f*fs):
>
epsilon:=rtiol/rh.o2:
>
delta:=(rhoO-epsilon~2*rho2)/rho:
>
mu:=rho2/rho:
>
tau:=delta/(l+delta*b2):
>
lambda:=-tau*s+epsilon-epsilon*tau*b2:
>
Y2:=1+(lambda+epsilon)*s+lambda*epsilon*b2:
>
eta:=mu/(l+Y2*mu):
>
Xi:=-(eta*lambda/rho)*f*fs:
>
Thetal:=fs/(2*f)-s*f*fss/(2*rho):
>
Theta2:=-(b2-s"2)*eta*lambda*f*fss/(2*rho):
>
Theta:=Thetal+Theta2:
>
Q:=f*fs/rho:
>
Phi:=-(f/rho)*(tau+eta*lambda"2)*fs:
>
Psil:=(f/(2*rho))*(l-(b2-s~2)*(tau+eta*lambda~2))*fss:
>
Q:=simplify(Q);
g.=
Theta:=simplify(Theta);
> Q
>
im
-4>(S)+S(&
_ 1 S fa) ( g r
>
simplify(Xi/Theta+2*Q);
Spray Coefficients of (a, /?)-Metrics
0 > chi:=simplify(Psil/Theta); X
=
•
(&*(*))#*) 8
183
Bibliography
[1] H. Akbar-Zadeh, Sur les espaces de Finsler a courbures sectionnelles constantes, Bull. Acad. Roy. Bel. Cl, Sci, 5e Serie - Tome LXXXIV (1988) 281-322. [2] P. Antonelli, R. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic Publishers, 1993. [3] L. Auslander, On curvature in Finsler geometry, Trans. Amer. Math. Soc. 79(1955), 378-388. [4] D. Bao and S. S. Chern, On a notable connection in Finsler geometry, Houston J. Math. 19(1)(1993), 135-180. [5] D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer, 2000. [6] D. Bao, S. S. Chern and Z. Shen, Rigidity issues on Finsler surfaces, Rev. Roumaine Math. Pures Appl. 42(1997), 707-735. [7] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom. 66(2004), 391-449. [8] D. Bao and C. Robles, On Randers metrics of constant curvature, Rep. Math. Phys. 51(2003), 9-42. [9] D. Bao and C. Robles, On Ricci curvature and flag curvature in Finsler geometry, In "A Sampler of Finsler Geometry" MSRI series, Cambridge University Press, 2004. [10] D. Bao and Z. Shen, Finsler metrics of constant curvature on the Lie group S3, J. London Math. Soc. 66(2002), 453-467. [11] S. Bacso and M. Matsumoto, On Finsler spaces of Douglas type. A generalization of the notion of Berwald space, Publ. Math. Debrecen, 51(1997), 385-406.
[12] A. Bejancu and H. R. Farran, Finsler metrics of positive constant flag curvature on Sasakian space forms, Hokkaido Math. J. 31(2002), 459-468. [13] A. Bejancu and H. R. Farran, Randers manifolds of positive constant flag curvature, Int. J. Math. Sci. 18(2003), 1155-1165. 185
186
Bibliography
[14] L. Berwald, Untersuchung der Krummung allgemeiner metrischer Raume auf Grund des in ihnen herrschenden Parallelismus, Math. Z. 25(1926), 40-73. [15] L. Berwald, Paralleliibertragung in allgemeinen Raumen, Atti Congr. Intern. Mat. Bologna 4(1928), 263-270. [16] L. Berwald, Uber eine charakterische Eigenschaft der allgemeinen Raume konstanter Krummung mit geradlinigen Extremalen, Monatsh. Math. Phys. 36(1929), 315-330. [17] L. Berwald, Uber die n-dimensionalen Geometrien konstanter Krummung, in denen die Geraden die kurzesten sind, Math. Z. 30(1929), 449-469. [18] L. Berwald, Ueber Finslersche und Cartansche Geometrie IV. Projektivkriimmung allgemeiner affiner Raume und Finslersche Raume skalarer Krummung, Ann. Math. 48(1947), 755-781. [19] R. Bryant, Finsler structures on the 2-sphere satisfying K = 1, Finsler Geometry, Contemporary Mathematics 196, Amer. Math. Soc, Providence, RI, 1996, 27-42. [20] R. Bryant, Projectively flat Finsler 2-spheres of constant curvature, Selecta Math., N.S. 3(1997), 161-204. [21] R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math. 28(2) (2002), 221-262. [22] H. Busemann, Intrinsic area, Ann. Math. 48(1947), 234-267. [23] E. Cartan, Les espaces de Finsler, Actualites 79, Paris, 1934. [24] X. Chen, X. Mo and Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. London Math. Soc. 68(2) (2003), 762-780. [25] X. Chen and Z. Shen, Randers metrics with special curvature properties, Osaka J. Math. 40(2003), 87-101. [26] X. Chen and Z. Shen, Projectively flat Finsler metrics with almost isotropic S-curvature, Acta Math. Sci., to appear. [27] X. Cheng and Z. Shen, Randers metrics of scalar flag curvature, preprint, 2005. [28] S. S. Chern, On the Euclidean connections in a Finsler space, Proc. National Acad. Soc, 29(1943), 33-37; or Selected Papers, vol. H, 107-111, Springer, 1989. [29] S. S. Chern, Local equivalence and Euclidean connections in Finsler spaces, Science Reports Nat. Tsing Hua Univ. 5(1948), 95-121. [30] S. S. Chern, On Finsler geometry, C. R. Acad. Sc. Paris 314(1992), 757761. [31] S. S. Chern, On the connection in Finsler geometry, Chin. Ann. Math. 23B:2(2002), 181-186. [32] P. Dazord, Varietes finsleriennes de dimension S-pincees, C. R. Acad. Sc. Paris 266(1968), 496-498. [33] P. Dazord, Varietes finsleriennes en forme des spheres, C. R. Acad. Sc. Paris 267(1968), 353-355. [34] A. Deicke, Uber die Finsler-Raume mit A{ = 0, Arch. Math. 4(1953), 45-51.
Bibliography
187
[35] C. E. Duran, A volume comparison theorem for Finsler manifolds, Proc. of A.M.S. 126(1998), 3079-3082. [36] L. Eisenhart, Non-Riemannian geometry, American Mathematical Society, 1922. [37] P. Finsler, Uber Kurven und Fldchen in allgemeinen Ra'umen, (Dissertation, Gottingen, 1918), Birkhauser Verlag, Basel, 1951. [38] P. Foulon, Curvature and global rigidity in Finsler geometry, Houston J. Math. 28(2002), 263-292. [39] P. Funk, Uber Geometrien bei denen die Geraden die Kurzesten sind, Math. Ann. 101(1929), 226-237. [40] P. Funk, Uber zweidimensionale Finslersche Ra'ume, insbesondere uber solche mit geradlinigen Extremalen und positiver konstanter Krummung, Math. Z. 40(1936), 86-93. [41] G. Hamel, Uber die Geometrien in denen die Geraden die Kurzesten sind, Math. Ann. 57(1903), 231-264. [42] M. Hashiguchi and Y. Ichijy5, On some special {a, (i)-metrics, Rep. Fac. Sci. Kagoshima Univ. 8(1975), 39-46. [43] M. Hashiguchi and Y. Ichijyo, Randers spaces with rectilinear geodesies, Rep. Fac. Sci. Kagoshima Univ. (Math. Phys. & Chen.) 13(1980), 33-40. [44] D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 37(2001), 407436. Reprinted from Bull. Amer. Math. Soc. 8 (July 1902), 437-479. [45] D. Hrimiuc and H. Shimada, On the L-duality between Finsler and Hamilton manifolds, Nonlinear World 3(1996), 613-641. [46] D. Hrimiuc and H. Shimada, On some special problems concerning the Lduality between Finsler and Cartan spaces, Tensor, N. S. 58(1997), 48-61. [47] Y. Ichijyo, Finsler spaces modeled on a Minkowski space, 3. Math. Kyoto Univ. 16(1976), 639-652. [48] Y. Ichihyo, On special Finsler connections with vanishing hv-curvature tensor, Tensor, N. S. 32(1978), 146-155. [49] M. Ji and Z. Shen, On strongly convex graphs in Minkowski geometry, Can. Math. Bull. 45(2) (2002), 232-246. [50] A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR 37(1973), [Russian]; Math. USSR-Izv. 7(1973), 535-571. [51] M. Kitayama, M. Azuma and M. Matsumoto, On Finsler spaces with (a, P)metric. Regularity, geodesies and main scalars, J. Hokkaido Univ. Education (sect. II A), 46(1995), 1-10. [52] S. Kikuchi, On the condition that a space with (a, 0)-metric be locally Minkowskian, Tensor, N.S. 33 (1979), 242-246. [53] C.-W. Kim and J.-W. Yim, Finsler manifolds with positive constant flag curvature, Geom. Dedicata (to appear). [54] W. Klingenberg, Riemannian geometry, de Gruyter Studies Math. 1, 2nd rev. ed., de Gruyter Berlin New York 1995. [55] S. Kobayashi, Theorem of Busemann-Mayer on Finsler metrics, Hokkaido
188
Bibliography
Math. J. 20 (1991), no. 2, 205-211. [56] S. Kobayashi and K. Nomizu, Foundations of differential geometry I, II. Interscience Publishers (1963-69). [57] D. Kosambi, Parallelism and path-spaces, Math. Z. 37(1933), 608-618. [58] D. Kosambi, Systems of differential equations of second order, Quart. J. Math., Oxford Ser. 6(1935), 1-12. [59] L. Kozma, On Landsberg spaces and holonomy of Finsler manifolds. Contemporary Mathematics, 196(1996), 177-185. [60] L. Kozma, On holonomy groups of Landsberg manifolds, Tensor, N. S. 62(2000), 87-90. [61] G. Landsberg, Uber die Totalkriimmung, Jahresberichte der deut Math. Ver. 16(1907), 36-46. [62] G. Landsberg, Uber die Krummung in der Variationsrechnung, Math. Ann. 65(1908), 313-349. [63] C. LeBrun and L. J. Mason, Zoll manifolds and complex surfaces, J. Diff. Geom. 61(2002), 453-535. [64] M. Matsumoto, On C-reducible Finsler spaces, Tensor, N.S. 24(1972), 2937. [65] M. Matsumoto, On Finsler spaces with Randers metric and special forms of important tensors, J. Math. Kyoto Univ. 14 (1974), 477-498. [66] M. Matsumoto, Foundations of Finsler Geometry and special Finsler Spaces, Kaiseisha Press, Japan, 1986. [67] M. Matsumoto, Randers spaces of constant curvature, Rep. Math. Phys. 28(1989), 249-261. [68] M. Matsumoto, The Berwald connection of a Finsler space with an [a, (3)metric, Tensor, N.S. 50(1991), 18-21. [69] M. Matsumoto, Theory of Finsler spaces with (a, ft)-metric, Rep. Math. Phys. 31(1992), 43-83. [70] M. Matsumoto and S. Hojo, A conclusive theorem on C-reducible Finsler spaces, Tensor, N. S. 32(1978), 225-230. [71] M. Matsumoto and H. Shimada, The corrected fundamental theorem on Randers spaces of constant curvature, Tensor, N. S. 63(2002), 43-47. [72] X. Mo, The flag curvature tensor on a closed Finsler space, Res. Math. 36(1999), 149-159. [73] X. Mo, On the flag curvature of a Finsler space with constant S-curvature, Houston J. Math, (to appear). [74] X. Mo and Z. Shen, On negatively curved Finsler manifolds of scalar curvature, Can. Math. Bull., (to appear). [75] X. Mo, Z. Shen and C.Yang, Some constructions of protectively flat Finsler metrics, preprint, 2004. [76] S. Numata, On Landsberg spaces of scalar curvature, J. Korea Math. Soc. 12(1975), 97-100. [77] T. Okada, On models of protectively flat Finsler spaces of constant negative curvature, Tensor, N. S. 40(1983), 117-123.
Bibliography
189
[78] H. B. Rademacher, A sphere theorem for non-reversible Finsler metrics, In "A Sampler of Finsler Geometry" MSRI series, Cambridge University Press, 2004. [79] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59(1941), 195-199. [80] A. Rapcsak, Uber die bahntreuen Abbildungen metrischer Rdume, Publ. Math. Debrecen, 8(1961), 285-290. [81] C. Robles, Einstein metrics of Randers type, Ph.D. thesis, University of British Columbia, Canada, 2003. [82] C. Robles, Geodesies in Randers spaces of constant curvature, Trans. Amer. Math. Soc. (to appear). [83] V. S. Sabau and H. Shimada, Classes of Finsler spaces with (a, f3) -metrics, Rep. Math. Phy. 47 (2001), 31-48. [84] Z. Shen, Finsler spaces of constant positive curvature, In: Finsler Geometry, Contemporary Math. 196(1996), 83-92. [85] Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. 128(1997), 306-328. [86] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, 2001. [87] Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001. [88] Z. Shen, Projectively flat Randers metrics of constant flag curvature, Math. Ann. 325(2003), 19-30. [89] Z. Shen, Projectively flat Finsler metrics of constant flag curvature, Trans. Amer. Math. Soc. 355(4) (2003), 1713-1728. [90] Z. Shen, Finsler metrics with K=0 and S=0, Can. J. Math. 55(2003), no. 1, 112-132. [91] Z. Shen, Two-dimensional Finsler metrics of constant flag curvature, Manuscripta Math. 109(3) (2002), 349-366. [92] Z. Shen, Nonpositively curved Finsler manifolds with constant S-curvature, Math. Z., (to appear). [93] Z. Shen, Landsberg curvature, S-curvature and Riemann curvature, In "A Sampler of Finsler Geometry" MSRI series, Cambridge University Press, 2004. [94] Z. Shen and H. Xing, On Randers metrics with isotropic S-curvature, preprint, 2004. [95] C. Shibata, H. Shimada, M. Azuma and H. Yosuda, On Finsler spaces with Randers' metric, Tensor, N. S. 31(1977), 219-226. [96] J. Simons, On transitivity of holonomy systems, Ann. Math. 76(1962), 213234. [97] Z. I. Szabo, Positive definite Berwald spaces (Structure theorems on Berwald spaces), Tensor, N. S. 35(1981), 25-39. [98] H. C. Wang, On Finsler space, with completely integrable equations of Killing, J. London Math. Soc. 22(1947), 5-9. [99] J. H. C. Whitehead, Convex regions in the geometry of paths, Quart. J.
190
Bibliography
Math. Oxford Ser. 3(1932), 33-42. [100] H. Xing, The geometric meaning of Randers metrics vrith isotropic Scurvature, Advances in Mathematics (China), (to appear). [101] H. Yasuda and H. Shimada, On Randers spaces of scalar curvature, Rep. Math. Phys. 11(1977), 347-360. [102] E. Zermelo, Uber das Navigationsproblem bei ruhender oder verdnderlicher Windverteilung, Z. Argrew. Math. Mech. 11(1931), 114-124. [103] W. Ziller, Geometry of the Katok examples, Ergod. Th & Dynam. Sys. 3(1982), 135-157.
Index
(a, /3)-metric, 11 (a,/?)-norm, 5
Finsler metric, 9 Finsler volume form, 17, 22 flag curvature, 108 flat Finsler metric, 45 fundamental form, 2 fundamental tensor, 34 Funk metric, 12, 23, 25, 46, 64-66, 91, 167
almost isotropic S-curvature, 91, 166 Berwald metric, 38, 54, 79, 80 Bianchi identity, 40, 48 Bryant metric, 165 C-reducible, 27 Cartan tensor, 34 Cartan torsion, 26 Chern connection, 36 color pattern, 26 connection forms, 33 constant flag curvature, 44, 109 constant S-curvature, 91 covariant derivative, 73 curvature forms, 33
Gauss curvature, 108, 109 geodesic variation, 117 Hilbert metric, 65, 115, 159 holonomy group, 78 indicatrix, 2, 80 integral curve, 51 irreversible Minkowski norm, 2 isotropic flag curvature, 109, 136 isotropic S-curvature, 91, 152
distortion, 88 E-curvature, 90 E-tensor, 90 Einstein metric, 109 Euclidean norm, 4 Euclidean space, 4 exponential map, 59
Jacobi field, 118 .
Klein metric, 10, 66 Klein model, 10 Landsberg metric, 38, 54 landsberg metric, 81 length structure, 15 Levi-Civita connection, 39, 80
Finsler manifold, 9 191
192
Index
linear covariant derivative, 71 linearly parallel vector field, 71 locally Minkowski metric, 45 locally Minkowskian, 45 locally projectively flat, 63, 110
shortest time problem, 20 slit tangent bundle, 9, 33 spray, 37, 51 spray coefficients, 37 structure equation, 31
Matsumoto torsion, 26, 128, 132 mean Cartan tensor, 88 mean Cartan torsion, 26 mean Landsberg tensor, 38 Minkowski metric, 10, 75 Minkowski norm, 2 Minkowski space, 2
tangent bundle, 9 tangent space, 9
nonpositive flag curvature, 120 parallel translation, 77 parallel vector field, 74 positively complete, 59 product Finsler manifold, 13 product Finsler metric, 13 projective factor, 63 projective hyperbolic metric, 10 projective spherical model, 11 projectively equivalent, 60 projectively flat, 63 projectively flat Finsler metric, 110 Randers metric, 12 Randers norm, 4 reversibility, 119 reversible Finsler metric, 17 reversible Minkowski norm, 2 Ricci curvature, 108 Ricci identity, 104 Ricci scalar, 108 Riemann curvature, 107 Riemannian metric, 9, 38 S-curvature, 89 scalar flag curvature, 109 Schur Lemma, 129 sectional curvature, 109 shortest path, 56
Zermelo's navigation problem, 20
