Riemann-Finsler Geometry

RIEMANN-FINSLER GEOMETRY

NANKAI TRACTS IN MATHEMATICS Series Editors: Yiming Long and Weiping Zhang Nankai Institute of Mathematics

Published Vol. 1

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Scissors Congruences, Group Homology and Characteristic Classes by J. L. Dupont The Index Theorem and the Heat Equation Method

by Y. L Yu

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Least Action Principle of Crystal Formation of Dense Packing Type and Kepler's Conjecture by W. Y. Hsiang

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Lectures on Chern-Weil Theory and Witten Deformations by W. P. Zhang

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Contemporary Trends in Algebraic Geometry and Algebraic Topology edited by Shiing-Shen Chern, Lei Fu & Richard Hain

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Riemann-FinslerGeometry by Shiing-Shen Chern & Zhongmin Shen

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Iterated Integrals and Cycles on Algebraic Manifolds by Bruno Harris

Vol. 8 ' Minimal Submanifolds and Related Topics by Yuanlong Xin

Nankai Tracts in Mathematics - Vol. 6

RIEMANN-FINSLER GEOMETRY

Shiing-Shen Chern Nankai Institute of Mathematics P. Ft. China

Zhongmin Shen Indiana University Purdue University Indianapolis USA

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Library of Congress Cataloging-in-Publication Data Chern, Shiing-Shen 1911-2004 Riemann-Finsler geometry / S.S. Chern, Zhongmin Shen. p. cm. -- (Nankai tracts in mathematics ; v. 6) Includes bibliographical references and index. ISBN 981-238-357-3 (alk. paper) - ISBN 981-238-358-1 (pbk. : alk. paper) 1. Finsler spaces. 2. Geometry, Riemannian. I. Shen, Zhongmin, 1963II. Title. III. Series. QA689.C48 2005 516.3'75-dc22 2005040818

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Preface

Two years ago David Bao, Zhongmin Shen and I published a book on Riemann-Finsler geometry through Springer Verlag. Riemann-Finsler geometry is not a generalization of Riemannian geometry. Riemann knew and began with the general case. He saw the main features of Riemannian geometry and remarked that the general case does not involve new ideas. In this assessment he was only partially correct. It certainly cannot include global problems. In local problems these are also subtleties, which need manipulation. The aim of this book is to provide an elementary account of Finsler geometry to show that Finsler geometry is essentially not more difficult. Such an account is desirable, as Finsler metrics have come up in many applications. I am ashamed to say that Shen actually wrote the whole book, although the idea of the book originated from me. The manuscript was written at least five times and I am impressed by its clarity and simplicity. I went through it in a seminar but do not wish to evade any responsibility if mistakes are found.

S.S. Chern January 2003

V

vi

Preface

At the end of 2001, I visited S.S. Chern at Nankai Institute of Mathematics in Tianjin, P.R. China. During my visit, Chern told me that he wished to have a concise book written for graduate students and young geometers who are interested in Riemann-Finsler geometry. The primary goal is to introduce some basic concepts, examples, theorems and to bring the readers to the most current research areas in Finsler geometry. We had a thorough discussion on topics and I started to collect some materials for the manuscript. Soon I realized that it is very difficult to write such a book. There is no simple proof for some important examples and theorems. Often times, the computation has to be carried out using a computer program such as MAPLE. Since the winter of 2001, I have been meeting Chern twice a year at Nankai Institute of Mathematics to work on our book project. The first draft of the manuscript was completed in the summer of 2002. However, I continued to make changes based on my discussion with Chern and comments from our colleagues. Chern wanted to check all the details by himself. Thus the submission of the manuscript was postponed for several times. On December 3,1 was stricken with the saddest news from my colleagues that S.S. Chern was no longer with us. Personally, I lost a great advisor. Without Chern's support and encouragement throughout the last decade, I would not have done any work in Finsler geometry. I would like to take this opportunity to thank X. Chen, L. Kozma, X. Mo, C. Robles, H. Shimada, and G. C. Yildirim for their valuable comments.

Zhongmin Shen December 2004

Contents

Chapter 1 Finsler Metrics 1.1 Minkowski Norms 1.2 Finsler Metrics 1.3 Length Structure and Volume Form 1.4 Navigation Problem 1.5 Cartan Torsion

1 2 9 15 20 25

Chapter 2 Structure Equations 2.1 Chern Connection 2.2 Structure Equations 2.3 Finsler Metrics of Constant Flag Curvature 2.4 Bianchi Identities

31 31 40 43 47

Chapter 3 Geodesies 3.1 Sprays 3.2 Shortest Paths 3.3 Projectively Equivalent Finsler Metrics 3.4 Projectively Flat Metrics

51 51 56 60 63

Chapter 4 Parallel Translations 4.1 Parallel Vector Fields 4.2 Parallel Translations 4.3 Berwald Metrics 4.4 Landsberg Metrics

71 71 76 79 81

vii

viii

Contents

Chapter 5 S-Curvature 5.1 Distortion and S-Curvature 5.2 Randers Metrics of Isotropic S-Curvature 5.3 An Equation on the S-Curvature

87 87 92 104

Chapter 6 Riemann Curvature 6.1 Riemann Curvature 6.2 Second Variation of a Geodesic 6.3 Nonpositive Flag Curvature

107 107 115 120

Chapter 7 Finsler Metrics of Scalar Flag Curvature 7.1 Some Basic Properties 7.2 Global Rigidity Theorems 7.3 Randers Metrics of Scalar Flag Curvature

127 127 131 139

C h a p t e r 8 P r o j e c t i v e l y Flat Finsler Metrics 8.1 Projectively Flat Randers Metrics 8.2 Projectively Flat Metrics with Constant Flag Curvature . . . . 8.3 Projectively Flat Metrics with Almost Isotropic S-Curvature .

149 149 155 166

Appendix A Maple Programs A.I Spray Coefficients of Two-dimensional Finsler Metrics A.2 Gauss Curvature A.3 Spray Coefficients of (a, /3)-Metrics

171 171 176 178

Bibliography

185

Index

191

RIEMANN-FINSLER GEOMETRY

Chapter 1

Finsler Metrics

To measure the length of a smooth curve C parametrized by a map c = c(t), a < t < b, in a manifold M, it suffices to define a nonnegative scalar function F(x, •) on every tangent space TXM. Then the length of C is denned by

CF(C) = J

F(c(t),c(t))dt.

It is required that £f(C) be independent of parametrization. F must be positively homogeneous with degree one, F(x,\y) = XF(x,y),

A > 0.

The length structure induces a nonnegative function d : M x M — > [0,oo) by d(p,q):=ia£L(C), where the infimum is taken over all smooth curves C from ptoq. In general, d is irreversible, i.e., d(p,q) ^ d(q,p) for some pairs of points {p, q). It is required that F uniquely determine dp. We impose a convexity condition on F, that is, F(x,yi + y2) 0 for any yeV, and F(y) = 0 if and only if y = 0; (b) F(Xy) = XF{y) for any y e V and A > 0; (c) F is C°° on V \ {0} such that for any y £ V, the following bilinear symmetric functional gy on V is an inner product,

The inner product g^ is called the fundamental form in the direction y. The pair (V, F) is called a Minkowski space. A Minkowski norm F is said to be reversible if F(—y) = F(y) for y 6 V. Given a Minkowski space (V, F), let SF:={yeY\F(y)

= l).

Sp is a closed hypersurface around the origin, which is diffeomorphic to the standard sphere S n - 1 CR". S,p is called the indicatrix of F.

3

Minkowski Norms

Figure 1.1 Let u,v e V \ {0} and u := u/F(u), v := v/F(v). Assume that u / ±v. Consider V?(t) := F2(tu + (1 - *)«)• We have that ') + 4>4>',

spiazaj:

6

Finsler Metrics

where the functions are evaluated on s := /3/a with \s\ = (s)>0,

((s)-s 0.

(1.7)

Consider the following family of functions, ) C R": F =

W\V? + M(M%12 - (x, y)2) + yfjjjx, y))2 (l + /iM 2 )Vl!/l 2 +MW 2 |y| 2 -(a:,l/> 2 ) '

where /i < 0 and rM := l/y^/I. The reader should try to find a and /? so that F = (a + (3)2/a. The metric when ^ = —1 is of particular interest. F._

(V\y\2 + -(M 2 M 2 - {x, y)2) + (x, y))2 (l-|a:| 2 )Vl2/l 2 -(N%l 2 -{^2/) 2 ) "

The metric in (1.17) was constructed by L. Berwald [17]. It has many special geometric properties. We will discuss it later in the book.

.

13

Finsler Metrics

One may construct a product Finsler metrics from a pair of Finsler manifolds. Let (Mi,F{) and (M2,F2) be Finsler manifolds. A Finsler metric F on M := M\ x M2 is called a product Finsler metric of Ft and F2 if at any point x = {xi,x2) 6 M, F( {

, = (F^xuyx) ' \F2{x2,y2) V)

ify = Uy =

yi®0eTxM

0®y2eTxM

where TXM S T Xl Mi © T I2 M 2 . In this case, (M,F) is called a product Finsler manifold of (Mi, Fi) and (M2, i7^)For a pair of Finsler manifolds, there is no canonical way to define a product Finsler metrics on the product manifold. When the Finsler metrics are Riemannian, we can define the product Finsler metrics in the following way. Example 1.2.5 Let a\ and a 2 be Euclidean norms on vector spaces Vi and V2 respectively. Let / : [0, oo) x [0, oo) —> [0, oo) be a C°° function satisfying VAX),

f(Xs,\t)=Xf{s,t),

and

f(s,t)>0,

V(s,i) ^ (0,0). (1.18)

Define a function F : V := Vx © V2 -> [0, oo) by

F(y) := ^/([a^yi)]2,

Mrf),

where y = y\ ® y2 6 V = Vi © V2. i71 = ^(y) has the following properties (a) F(y) > 0 for any y eV, and F(y) = 0 if and only if y = 0; (b) F(Ay) = AF(y) for any y € V and A > 0; (c) F(y) isC°° on V\{0}. Let n\ = dim Vi, n2 = dimV2 and n = n\ + n2 = dimV. We shall assume the following ranges of indices: 1 < a, b, c < n i ,

ni + 1 < a,/3,7 < n,

l(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,){y) be a Minkowski norm on Rn and

U:=[yeRn\{y) 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)(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
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)