LECTURES ON
FEVSLER GEOMETRY Zhongmin Shen
World Scientific
LECTURES ON
FEVSLER GEOMETRY
LECTURES ON
FEVSLER GEOMETRY Zhongmin Shen Indiana University-Purdue University Indianapodis, USA
m World Scientific ll
Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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LECTURES ON FINSLER GEOMETRY Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-4530-0 ISBN 981-02-4531-9 (pbk)
Printed in Singapore by World Scientific Printers
to my daughter, Alice
Preface
In 1854, B. Riemann introduced the notion of curvature for spaces with a family of inner products (Riemannian metrics). There was no significant progress in the general case until 1918, when P. Finsler studied the variation problem for spaces with a family of norms (Finsler metrics). Meanwhile, A. Einstein used Riemannian geometry to present his general relativity. By that times, however, the geometry of Finsler spaces was still at its infant stage. Until 1926, L. Berwald extended Riemann's notion of curvature to Finsler spaces and discovered a new non-Riemannian quantity using his connection. In his Paris address in 1900, D. Hilbert formulated 23 problems, the 4th and 23rd problems being in Finsler's category. Finsler geometry has broader applications in many areas of natural science [AIM] and will continue to develop through the efforts of many geometers around the world [BCS3]. This book comes out of a series of lecture notes based on my work at IHES (Curvature, distance and volume in Finsler geometry, preprint, 1997). Viewing Finsler spaces as regular metric spaces, we discuss the problems from the modern geometric point of view. The book is intended to provide basic materials for Finsler geometry including Riemannian geometry. The presentation is aimed at a reader who has completed a one-year graduate course in differential geometry, and who knows the elements of homotopy theory in topology. The first four chapters cover some basic theories on regular metrics (Finsler metrics) and regular measures (volume forms) on manifolds. In Chapter 5, we introduce the notion of geodesies via the calculus of variation. Using the geodesies, we define the Chern connection. The Chern vii
Vlll
Preface
connection is an important tool to investigate the geometric structure of Finsler spaces. In Chapter 6, we introduce the notion of Riemann curvature via geodesic variation. This is different from Riemann's and Berwald's approaches. We discuss the dependence of the Riemann curvature on geodesies. In Chapter 7, we introduce various non-Riemannian quantities (e.g., the Chern curvature, the Landsberg curvature and the S-curvature, etc) and their geometric meanings. In Chapters 8 and 9, we use Cartan's exterior differential method to discuss various curvatures and relationships. In particular, we discuss Finsler spaces of constant curvature. Several important examples are studied. These examples make Finsler geometry more fruitful. Starting with Chapter 10, we study the metric properties of Finsler spaces and metric-measure properties of Finsler m spaces. We first derive the second variation formula of length and show the geometric meaning of the T-curvature. Then in Chapters 11 and 12, we discuss the index form, Jacobi fields, exponential maps and their relationship. The discussion leads to some basic comparison theorems that are discussed in Chapter 13. In Chapters 14, 15 and 16, we use a distance function to study the global geometric structure of Finsler spaces and Finsler m spaces. We obtain several estimates on the geometry of level sets of distance functions. The geometric meaning of the S-curvature lies in the volume comparison theorem. In Chapters 17 and 18, we apply the Morse theory to the canonical energy functional on the loop space. In particular, we prove a vanishing theorem for the homotopy groups of Finsler spaces with pinched curvature. In the last chapter, we view a Finsler space as a point of a space of Finsler spaces equipped with the Gromov-Hausdorff distance. We briefly discuss the precompactness and finiteness of certain classes of Finsler spaces. I would like to take this opportunity to thank several people in my personal life and academic life. First, I would like to thank my thesis advisor Detlef Gromoll for his help and advice in Riemannian geometry during my graduate study in SUNY at Stony Brook, and thank another advisor after the thesis, S.S. Chern for bringing me to a wider field — Finsler geometry. I thank Bart Ng for providing me with a good research environment at my current institution. Finally, I thank my wife, Tianping, for her consistent support and understanding. Without them, I would not
Preface
have written this book, for sure. Zhongmin Shen Indianapolis, USA
ix
Contents
Preface
vii
Chapter 1 Finsler Spaces 1.1 Metric Spaces 1.2 Minkowski Spaces 1.3 Finsler Spaces
1 1 6 12
Chapter 2 Finsler m Spaces 2.1 Measure Spaces 2.2 Volume on a Finsler Space 2.3 Hyperplanes in a Minkowski m Space 2.4 Hypersurfaces in a Finsler m Space
19 19 21 27 30
Chapter 3 Co-Area Formula 3.1 Legendre Transformations 3.2 Gradients of Functions 3.3 Co-Area Formula
35 35 41 46
Chapter 4 Isoperirnetric Inequalities 4.1 Isoperirnetric Profiles 4.2 Sobolev Constants and First Eigenvalue 4.3 Concentration of Finsler m Spaces 4.4 Observable Diameter
55 56 58 64 69
Chapter 5 Geodesies and Connection 5.1 Geodesies
75 75
xi
xii
5.2 5.3 5.4
Contents
Chern Connection Covariant Derivatives Geodesic Flow
80 88 90
Chapter 6 Riemann Curvature 6.1 Birth of the Riemann Curvature 6.2 Geodesic Fields 6.3 Projectively Related Finsler Metrics
95 95 99 101
Chapter 7 Non-Riemannian Curvatures 7.1 Cartan Torsion 7.2 Chern Curvature 7.3 S-Curvature
107 107 112 117
Chapter 8 Structure Equations 8.1 Structure Equations of Finsler Spaces 8.2 Structure Equations of Riemannian Metrics 8.3 Riemann Curvature of Randers Metrics
125 125 130 134
Chapter 9 Finsler Spaces of Constant Curvature 9.1 Finsler Metrics of Constant Curvature 9.2 Examples 9.3 Randers Metrics of Constant Curvature
137 137 141 145
Chapter 10 Second Variation Formula 10.1 T-Curvature 10.2 Second Variation of Length 10.3 Synge Theorem
153 153 156 162
Chapter 11 Geodesies and Exponential Map 11.1 Exponential Map 11.2 Jacobi Fields 11.3 Minimality of Geodesies 11.4 Completeness of Finsler Spaces
165 165 167 171 175
Chapter 12 Conjugate Radius and Injectivity Radius 12.1 Conjugate Radius 12.2 Injectivity Radius 12.3 Geodesic Loops and Closed Geodesies
179 179 183 189
Contents
xiii
Chapter 13 Basic Comparison Theorems 13.1 Flag Curvature Bounded Above 13.2 Positive Flag Curvature 13.3 Ricci Curvature Bounded Below 13.4 Green-Dazord Theorem
193 193 198 201 203
Chapter 14 Geometry of Hypersurfaces 14.1 Hessian and Laplacian 14.2 Normal Curvature 14.3 Mean Curvature 14.4 Shape Operator
207 207 213 218 221
Chapter 15 Geometry of Metric Spheres 15.1 Estimates on the Normal Curvature 15.2 Convexity of Metric Balls 15.3 Estimates on the Mean Curvature 15.4 Metric Spheres in a Convex Domain
229 229 234 236 241
Chapter 16 Volume Comparison Theorems 16.1 Volume of Metric Balls 16.2 Volume of Tubular Neighborhoods 16.3 Gromov Simplicial Norms 16.4 Estimates on the Expansion Distance
245 245 252 256 259
Chapter 17 Morse Theory of Loop Spaces 17.1 A Review on the Morse Theory 17.2 Indexes of Geodesic Loops 17.3 Energy Functional on a Loop Space 17.4 Approximation of Loop Spaces
265 265 268 271 273
Chapter 18 Vanishing Theorems for Homotopy Groups 18.1 Intermediate Curvatures 18.2 Vanishing Theorem for Homotopy Groups 18.3 Finsler Spaces of Positive Constant Curvature
279 279 280 286
Chapter 19 Spaces of Finsler Spaces 19.1 Gromov-Hausdorff Distance 19.2 Precompactness Theorem
291 291 293
xiv
Contents
Bibliography
299
Index
305
Chapter 1
Finsler Spaces
1.1
Metric Spaces
Metric spaces exist everywhere in our life. A metric space is a set of points equipped with a metric (distance function). With a metric, we measure the distance from a point to another point in the set. More precisely, a metric on a set M is a function d : M x M —> R which has the following properties
(i) for any p,q g M, d(p,q)>0, equality holds if and only if p = q; (ii) for any p,q,r G M, d{p,q) < d{p,r) + d(r,q). In addition, if d has the following reversibility, (iii) for any two points p,q € M, d(p,q) = d{q,p), then d is called a reversible metric. Let R n — I(xl)
= (x\---,xn)
xi 0 ;
(iii) ||u + v|| [0, oo) by d(u,v) := \\v — u\\. d is a "flat" metric on V.
|j
Example 1.1.2 (Funk) Let ft be a bounded domain in E " = (R", | • |). Assume that Q, is strictly convex, i.e., for any line segment L in R n , if the endpoints of L are contained in fi, then the whole L is contained in il. For an ordered pair of points p, q € Cl, let L pg denote the ray issuing from p and passing through q. Since Q is strictly convex, there is a unique intersection point zpq := Lpq n d£l. Define
d( P , g ):=lnjj^j.
(1.1)
We will show that d is a non-reversible metric on 0. d is called the Funk metric. It suffices to prove the triangle inequality: d(p,q) q ~ ^ 2 2 A- 1 y/\p-q\ - (\p\ \q\2 - (p,q)2) ~(q,qp)
Metric Spaces
5
Note that lim d(Q, g) = oo,
lim dip, 0 ) = l n 2 .
H
Example 1.1.3 (Klein) Let d denote the Funk metric on a strongly convex domain flcR . Define dx '• fi x ^ —> [0,oo) by dK{p,q)-=
2{d{p,q)+d(q,p)},
(1.2)
Observe that for p,q,r £ Q, dK(p,q)
=
^{d(p,q)
+ d(q,p)j
< 2fa'r) + d(r ' 5 ^} + 2 I d(9 ' r) + d(r'P)} =
^{d{P,r)
=
dK(p,r)
+ d{r,p)} + -{d(q,r) +
+ d(r,q)}
dK(r,q).
T h u s dx is a metric too. We call dx the Klein metric.
Let ( M , d) be a metric space. [a, 6] ->• M , define
)j
For a Lipschitz continuous curve c :
fe-i
Ld(c) := sup ] P . From the definition, d 0 and yeV: (M3) For any y £ V\{0}, the symmetric bilinear form g y on V is positive definite, where Sy(u, v) := - ^ - ^ \F2{y + su + to)] | s = t = 0 . The pair (V, F) is called a Minkowski space We will show that any Minkowski norm F on a vector space V satisfies F(u + v) < F(u) + F(v),
u,veV.
(1.4)
See Lemma 1.2.2 below. Hence it induces a metric d on V by d(u,v) := F(v— u),
u,v£V.
We call a metric d on V a Minkowski metric if it is induced by a Minkowski norm F on V. There are many Minkowski norms on a vector space.
Minkowski Spaces
7
Example 1.2.1 (Randers norm) Let a an Euclidean norm and (3 a linear form on an n-dimensional vector space V. Define F(y) := a(y) + p(y).
(1.5)
Clearly, F satisfies (Ml) and (M2) in Definition 1.2.1. We are going to show that (M3) holds if and only if \\(3\\ < 1. Fix a basis {b,}™=1 for V and express a(y) = \Ja.ijyiy:> and 0(y) = &*y\
y = yl\n € V,
where f a^ 1 is a positive definite symmetric matrix. We have ||/3||:= sup i p(y) = iJaVbibi «(v)=i where Ia t J ) = (aij)
. Let
9ij(v) •= g„(bi,bj) =
-[F\iyj(y).
An easy computation yields
*. [0, oo) with $(0) = 0 such that the following limit exists for any x G fi and y £ TxVl = R™ lim
i[8(,,,^)-*(x,»)]
s->0+
S
We obtain a nonnegative function F = F{x, y) on TO = flxR". We call F the Fechner function and $ a psychometric transformation associated with \I>. Clearly, F satisfies F(x,Xy) = XF{x,y),
A > 0.
Finsler
Spaces
15
A psychometric function is said to be Finslerian if the associated Fechner function is a Finsler metric. From the definition, given any Finsler metric F on a domain ft C R n and any fi > 0, the function
*(x,y):=[F(x,y)f is a psychometric function on
ft.
jj
Example 1.3.4 Let d be the Funk metric on a strictly convex domain ft in R™ (see Exercise 1.1.2). Assume that ft is strongly convex. By definition, there is a Minkowski norm ip on R™ and a point p £ CI, ft - M =
{2/ G R" I -O+ e
U-a;| \z — x — ey
(y, z - x)
y£TxCl
= Rn.
Clearly, the above equation is equivalent to the following equation: x+-^— F (y)
= zedCl.
(1.19)
Thus if d is induced by a Finsler metric, then F is determined by (1.19). We can easily show that F determined by (1.19) is indeed a Finsler metric and this Finsler metric F induces the Funk metric d. We shall also call F the Funk metric on ft.
Finsler Spaces
16
The Klein metric dx on ft is defined by dK{p,q) •=
^{d(p,q)+d(q,p)J.
We can show that dx is induced by the following Finsler metric FK(v):=\(F(-y) + F(yj),
(1.20)
where F is the Funk metric on 0 . We shall also call FK the Klein metric on fi. Take a look at the special case when Q, — B is the unit ball in the Euclidean space M. . A direct computation gives an explicit formula for the Funk metric,
F(y) := V l g l a - ( N a M a - < » , l / ) a ) + (»,g) 1 — \x\z
G r B » = R„
where | • | and {,) denote the Euclidean norm and inner product in M. , respectively. The Funk metric on B is a Randers metric. The Klein metric FK is given by i — \xy Thus the Klein metric on B is a Riemannian metric.
it
The following lemma is due to Okada [Ok]. Lemma 1.3.2 satisfies
(Okada) The Funk metric F : TO. = Q, x R n -> [0, oo) Fxk=FFyk.
(1.21)
Finsler Spaces
17
Proof. By assumption, CI is strongly convex, i.e., there is a Minkowski norm
,
(2.2)
Volume on a Finsler Space
21
We consider an oriented manifold M equipped with a volume form dfx. We can view dfj, as an n-form on M. Let X a vector field on M. Define an (n - l)-form X\d/j, on M by X\dn{X2r--,Xn):=dn(X,X2,---,Xn). Define d(X\dfi) = div(X)dfi.
(2.3)
We call div(X) the divergence of X. Clearly, div(X) depends only on the volume form dp,. In a local coordinate system (xl), express d/i = a(x)dx1 •••dxn. Then for a vector field I = I ' | r o n M , ,.,.., I d , \ 8Xl X{ da div X) = - — • \aXvl)i = — - H -—.
(2.4)
Applying the Stokes theorem to r\ = X\ dji, we obtain
I div(X)dn
=
j d(X\d/j,)=0,
f dW(X)dfi
=
I d(X\dfi)=
[
JM
JdM
JM
2.2
if dM = 0 (2.5) X\dfi,
if
