Variational Problems in Geometry (Translations of Mathematical Monographs)

Translations of

MATHEMATICAL

MONOGRAPHS Volume 205

Variational Problems in Geometry Se'' Nishikawa

k

American Mathematical society

Variational Problems in Geometry

Translations of

MATHEMATICAL MONOGRAPHS Volume 205

Variational Problems in Geometry Seiki Nishikawa Translated by Kinetsu Abe

Amerl=n Mathematical 8ocisty Providence. Rhode Island

Editorial Board Shoshichi Kobayashi (Chair) Masamichi Takesaki

6 fol 0: FL16 M** 9 3 KIKAGAKUTEKI HENBUN MONDAI by Seiki Nishikawa Originally published in Japanese by Iwanami Shoten, Publishers, Tokyo, 1998 T anslated from the Japanese by Kinetsu Abe

2000 Mathematics Subject Ciasaication. Primary 53-01, 53C21, 53C43, 58E20, 58J25.

Library of Congress Cataloging-In-Publication Data Nishikawa, Seiki. [Kikigakuteki henbun mondai. English]

Variational problems in geometry / Seild Nishikawa ; translated by Kinetsu Abe

p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 ; v. 205) (Iwanami series in modem mathematics) Includes bibliographical references and index. ISBN 0-8218-1356-0 (acid-free paper) 1. Harmonic maps. 2. Variational inequalities (Mathematics). 3. Riemannian manifolds. I. Title. It. Series. III. Series: Iwanami series in modern mathematics QA614.73.N5713 2001 514'.74-dc21 2001046350

© 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.

® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Information on copying and reprinting can be found in the back of this volume. Visit the AMS home page at URL: http://vvv.ams.org/

10987654321

070605040302

Contents Preface to the English Edition

ix

Preface

Outlines and Objectives of the Theory Chapter 1. Are Length of Curves and Geodesics 1.1. Are length and energy of curves 1.2. Euler's equation 1.3. Connections and covariant differentiation

1

1

9 16

Geodesics Minimal length property of geodesics Summary Exercises

28 38 43 44

Chapter 2. First and Second Variation Formulas 2.1. The first variation formula

47 47 54 65 69 77

1.4. 1.5.

2.2.

Curvature tensor

The second variation formula Existence of minimal geodesics Applications to Riemannian geometry Summary Exercises 2.3. 2.4. 2.5.

Chapter 3. Energy of Maps and Harmonic Maps 3.1. Energy of maps 3.2. 3.3. 3.4. 3.5.

Tension fields

The first variation formula Harmonic maps The second variation formula Summary Exercise

82 82 85

85 90 99 103

110 114 114

v

CONTENTS

vi

Chapter 4. Existence of Harmonic Maps 4.1. The heat flow method 4.2. Existence of local time-dependent solutions 4.3. Existence of global time-dependent solutions Existence and uniqueness of harmonic maps 4.4. 4.5. Applications to R.iemannian geometry Summary Exercises Fundamentals of the Theory of Manifolds and Functional Analysis A. I. Fundamentals of manifolds A_2. Fundamentals of functional analysis

Appendix A.

Prospects for Contemporary Mathematics Solutions to Exercise Problems Bibliography Index

Preface to the English Edition This book, published originally in Japanese, is an outgrowth of lectures given at Tohoku University and at the 1995 Summer Graduate Program of the Institute for Mathematics and Its Applications, University of Minnesota. In these lectures, through a discussion on variational problems of the length and energy of curves and the energy of maps, I intended to guide the audience to the threshold of the field of geometric variational problems, that is, the study of nonlinear problems arising in geometry and topology from the point of view of global analysis. It is my pleasure and privilege to express my deepest gratitude to Professor Kinetsu Abe who generously devoted considerable time and

effort to the translation. I would also like to take this opportunity to express my deep appreciation to Professor Phillipe Tondeur who invited me to join the 1995 Summer Graduate Program, and to my friend Andrej Treibergs for making his notes [261 available to the organization of the last chapter. Seiki Nishikawa April 2001

vH

Preface It is said that techniques for surveying were developed from the need to restore lands after frequent floods of the Nile River in ancient Egypt. Geometry is the area of mathematics whose name originates from this method of surveying; namely, "to measure lands" (geo = lands, metry = measure). As such, it is an ancient practice to study figures from the view of practical applications. It is also said that the ancient Greeks already knew of the method of indirect surveying using the congruence conditions of triangles. A minimal length curve joining two points in a surface is called a geodesic. One may trace the origin of the problem of finding geodesics back to the birth of calculus. Many contemporary mathematical problems, as in the case of geodesics, may be formulated as variationl problems in surfaces or in the more generalized form of manifolds. One may characterize the geometric variational problems as a field of mathematics that studies the global aspects of variational problems relevant in the geometry and toplogy of manifolds. For example, the problem of finding a surface of minimal area spanning a given frame of wire originally appeared as a mathematical model for soap films. It has also been actively investigated as a geometric variational problem. With recent developments in computer graphics, totally new aspects of the study on the subject have begun to emerge. This book is intended to be an introduction to some of the fundamental questions and results in geometric variational problems, studying the variational problems on the length of curves and the energy of maps. The first two chapters approach variational problems of length and energy of curves in Riemannian manifolds with an in-depth discussion of the existence and properties of geodesics viewed as the solution to variational problems. In addition, a special emphasis is ix

PREFACE

x

placed on the fact that the concepts of connection and covariant differentiation are naturally induced from the first variation formula of this variational problem, and that the notion of curvature is obtained from the second variational formula. The last two chapters treat the variational problem on the energy of maps between two Riemannian manifolds and its solutions, namely harmonic maps. The concept of harmonic maps includes geodesics and minimal submanifolds as examples. Its existence and properties have successfully been applied to various problems in geometry and topology. This book takes up the existence theorem of Eells-Sampson, which is considered to be the most fundamental among existence theorems for harmonic maps. The proof uses the inverse function theorem for Banach spaces. It is presented to be as self-contained as possible for easy reading. Each chapter of this book may be read independently with minimal preparation for covariant differentiation and curvature on manifolds. The first two chapters, through the discussion of connections and covariant differentiation, are designed to provide the reader with a basic knowledge of Riemannian manifolds. As prerequisites for reading this book, the author assumes a few elementary facts in the theory of manifolds and functional analysis. They are included in the form of appendices at the end of the book. Details in functional analysis may

be skipped. The reader, however, is encouraged to do the exercise problems at the end of each chapter by himself or herself first. The solutions may be consulted if necessary, since many of the exercise problems complement the contents of the book. This book is an outgrowth of lectures delivered at Tohoku Univer-

sity and the 1995 Summer Graduate Programs held at The Institute for Mathematics and Its Applications, University of Minnesota. The first half of the book aims at a junior and senior level, and the last half at a first and second year graduate level. Each half roughly consists of the amount of topics that may be covered in one semester. In the actual lectures, the author also discusses the harmonic maps between Riemann surfaces. This portion is not included in this book due to the limited space. The reader who is interested in the study of harmonic maps is advised to first study the harmonic maps between Riemann surfaces.

It would be this author's wish as well as pleasure if this book could interest many readers in variational problems in geometry.

PREFACE

xi

Last but not least, the author expresses his sincere gratitude to the editorial staff of Iwanarni Shoten for their valuable help in the publication of this book. Seiki Nishikawa

December 1997

Outlines and Objectives of the Theory Among geometric variational problems, the extreme value problem regarding the length of curves is as old as those in calculus. Chapter 1 of this book is devoted to discussions of variational problems of curves in manifolds. As is well known, the length of a curve joining two points in a plane is given by integrating the magnitude of tangent vectors. Similarly, one can define the length and energy for curves in more general Riemannian manifolds by measuring the magnitude of the tangent vectors using Riemannian metrics. In Chapter 1, Euler's equation is calculated. It characterizes the critical points of the length and energy of curves when they are considered as functionals defined in the space of curves. Consequently, the equation of geodesics is obtained. The concepts of connections and covariant differentiation are naturally induced from the equation of geodesics in a manifold. Covariant differentiation, an essential tool for studying variational problems in manifolds, is an operation that defines the derivative of a vector field by a vector field in a manifold. The most fundamental connection, called the Levi-Civita connection, is uniquely determined in a manifold equipped with a Riemannian metric, i.e., a Riemannian manifold. The notion of parallel transport is induced from this connection. The discovery of the notion of parallel transport in Riemannian manifolds (1917) and Einstein's use of geometry based on a four-dimensional indefinite metric for his general relativity (1916) greatly advanced the study of Riemannian geometry.

Geodesics in Riemannian manifolds correspond to straight lines in the plane and they are locally characterized as the curves of minimal length between points. One can construct a special local coordinate system, called a normal coordinate system, using these minimal geodesics about each point in a Riemannian manifold. Parallel transport and normal coordinate systems are the most basic tools in Xi;i

xiv

OUTLINES AND OBJECTIVES OF THE THEORY

comparing the geometry of a Riemannian manifold with the geometry of a model space (for example, Euclidean space).

In Chapter 2, using covariant differentiation, the first variation formula (Euler's equation) for the variational problem regarding the energy of curves in Riemannian manifolds is computed in the general case where the image of a curve is not always contained in a local coordinate neighborhood. The second variation formula is subsequently computed. Just as the notion of connections is derived from the first variation formula, it is seen that the second variation formula possesses an intimate relationship to the notion of curvature in R.iemannian manifolds. In other words, the notions of curvature tensor

and the curvature of a Riemannian manifold are naturally induced from the second variation formula for the energy of curves. Given two points in a Riemannian manifold, the distance between these two points is given by the least upper bound of the lengths of piecewise smooth curves connecting them. Whether a Riemannian manifold becomes a complete metric space with respect to this distance is an important question. It was relatively recently (1931) that Hopf-Rinow gave necessary and sufficient conditions for the question. The results by Hopf and Rinow are significant not only in making the notion of completeness succinct, but also in showing that this completeness is the condition that guarantees the existence of a minimal geodesic joining two given points. As stated above, the second variation formula for the energy of curves is closely related to the curvature of Riemannian manifolds. Using this, one can study the effects of the curvature of a Riemann ian manifold on its topological structure. Myers' theorem and Synge's theorem are discussed as typical examples of such applications. The

former states that the fundamental group of a compact and connected, Riemannian manifold of positive curvature is a finite group, and the latter states that an even-dimensional compact, connected and orientable Riemannian manifold of positive curvature is simply connected. Research on Riemannian manifolds using existence and properties of geodesics is being actively pursued. In Chapter 3, harmonic maps and the energy of maps are discussed. They generalize the variational problem of the energy of curves in Riemannian manifolds. Namely, a functional called the energy of maps is defined in the mapping space consisting of smooth maps between Riemannian manifolds, and harmonic maps given as its

OUTLINES AND OBJECTIVES OF THE THEORY

xv

critical points are investigated. The energy of maps is a natural generalization of the energy of curves. Examples of harmonic maps appear in various aspects of differential geometry. Harmonic functions, geodesics, minimal submanifolds, isometric maps, and holomorphic maps are a few typical examples. The first variation formula, which characterizes the critical points

of the energy functional, can be obtained by essentially the same approach as in the case of geodesics. However, the computations become unnecessarily complicated and only yield results of a local nature without use of the covariant differentiation that is naturally induced from the Levi-Civita connection of Riemannian manifolds. To alleviate these difficulties, it is designed in this chapter to derive, through discoveries in the process, the computational rules for the covariant differentiation that is induced from the Levi-Civita connection in tangent bundles and their tensor products over Riemannian manifolds. This route may not be the most direct one, but the author believes that it is more effective in familiarizing the reader with the definition and the rules of computations for covariant differentiation than the axiomatic approach. At first, the reader may feel uneasy, especially about the portion of the induced connections. Nonetheless, actual computations help promote understanding of the notion. The fastest way to grasp the rules of computation involving covariant differentiation is actually to engage in the computations. The computations of the first variation formula for the energy functional of maps yield a vector field called the tension field. It is given as the trace of the second fundamental form of the maps. A harmonic map is then characterized as a map whose tension field is identically 0. Chapter 4 is devoted to the existence problem of harmonic maps

between compact Riemannian manifolds. Whether or not a given map is homotopically deformable to a harmonic map is one of the most fundamental questions among geometric variational problems. It may be regarded as a generalization of the existence problem of closed geodesics. To this end, the "heat flow method" is first introduced. This is an effective technique for deforming a given map to a harmonic map. Then, using this technique, it is proved that any map from a compact Riemannian manifold M into a compact Riemannian manifold N of nonpositive curvature is free homotopically deformable to a harmonic map. This theorem was first proved by Eells-Sampson in 1964.

xvi

OUTLINES AND OBJECTIVES OF THE THEORY

The proof of this theorem using the heat flow method first requires the existence of a time-dependent solution to an initial value problem with any initial map of the parabolic equation for harmonic maps. The original proof uses successive approximations to construct a solution after converting the problem to a problem of integral equations via the fundamental solution of the heat equation. In this book, the solution is constructed through use of the inverse function theorem in Banach spaces in an effort to minimize the amount of preparation. The existence of time-dependent local solutions is always guar-

anteed, but the existence of global time-dependent solutions is not self-evident, since the parabolic equation for harmonic maps is nonlinear. In fact, proving the existence of global time-dependent solutions

entails some estimates of the growth rate of solutions in time. The curvature of the Rieinannian manifold N plays a crucial role in est.imating the influence of nonlinear terms. An estimation formula that guarantees the existence and convergence of time-dependent global solutions is obtained using the Weitzenbock formula for the heat operator under the condition that N is of nonpositive curvature. The Weizenbock formula, in general, gives the relationship between second order partial differential operators naturally acting on tensor fields on Riemannian manifolds and the Laplace or heat operator acting on functions. It is revealed that the Riemann curvature and its Ricci identity play essential roles for existence of solutions to those differential operators. In this chapter, an a priori estimate regarding the growth rate of solutions is obtained using the Weizenbock formula for the energy density of solutions to the parabolic equation for harmonic maps and the heat operator. This idea is originally due

to Bochner. It has become an effective and fundamental technique for the proofs of theorems such as the Kodaira vanishing theorem and more recently in gauge theory.

As in the case of geodesics, one can also investigate the structures of Riemannian manifolds using the existence and properties of harmonic maps. The theorem of Preissman, one of the typical applications of harmonic maps, is discussed. The theorem states that a nontrivial, Abelian subgroup of the fundamental group of a compact manifold of negative curvature is infinitely cyclic. The research of Riemannian manifolds using the existence and properties of barmonic maps seems to possess a promising future. For example, new proofs from a more analytical point of view for the topological sphere theorem and the Frankel conjecture were recently given by exploiting

OUTLINES AND OBJECTIVES OF THE THEORY

xvii

the existence theorem of harmonic spheres due to Sacks and Uhlenbeck. A strong rigidity theorem regarding complex structures in Kahler manifolds of negative curvature was also obtained using the existence theorem of Eells and Sampson.

CHAPTER 1

Arc Length of Curves and Geodesics "Given two points in a surface, find a curve joining them of the minimum arc length." A solution to this question is called a geodesic. Finding geodesics is a typical problem in the calculus of variation. Its origin could be traced back to the birth of calculus.

In this chapter, the variational problem of arc length and the energy of curves in Riemannian manifolds is discussed as an introduction to geometric variational problems. The critical points in this variational problem satisfy a differential equation called the Euler

equation. The concept of covariant differentiation is naturally induced from this equation. The first variational formula of the energy of curves is obtained. Geodesics are characterized as the critical points of this variational problem.

1.1. Are length and energy of curves The reader, who has already learned the theory of surfaces, knows how the arc length of a curve drawn on a surface is defined. We begin by reviewing definitions. Let M be a surface in the Euclidean space E3 of dimension threeLet c be a smooth curve in M. Using the coordinates in E3, we denote the parametric representation of the curve c = c(t) by c(t) = (x(t), pi(t), z(t)),

a < t < b.

The tangent vector c'(t) of the curve c at the point c(t) is given by

c(t) = (x'(t), y'(t), z'(t)), a < t < b. If we consider the curve c as the trace of a moving particle c(t), the

tangent vector c/(t) is nothing but the velocity vector of the particle at time t. Since c(t) is a vector in E3, its magnitude Ic'(t)I may be measured in terms of the Euclidean inner product (, ). In fact, the

i

2

1. ARC LENGTH OF CURVES AND GEODESICS

magnitude lc'(t)I is given by XI(t)2

(c'(t) _ (C(t),c(t))1'2 =

+ y'(t)2 + x'(t)2,

and it represents the speed of the particle at time t. Then length L(c) of the curve c is given by integrating the magnitude of the tangent vector c(t) in t as b

L (c) =

j

Ic'(t) Idt.

The arc length L(c) is the distance traveled by the moving particle along the curve c(t) from time a to time b. Similarly, we can define the are length of a smooth curve in a more general manifold by measuring the magnitude of the tangent vectors

and by integrating it along the curve. In the case of a curve in a surface, the magnitude of the tangent vectors is measured by using the inner product of the Euclidean space E3 that contains the surface. In a general c°O manifold, a tensor field called a Riemannian metric plays the role of the inner product in E. This entails the introduction of Riemannian metrics. Let M be a c°° manifold of dimension m. Given a point x in M, denote by (U, 0) a local coordinate neighborhood system around x. Here, U is an open subset of M and 0 is a homeomorphism from U onto an open subset i(U) in the mrdimensional Cartesian space Rm. Using the coordinates of Rm, O(x) can be expressed as '0(x) = (x1(x), ... , xm(x))

E Rtm.

We call (x1(x), x2 (x), ... , xm (x)) the local coordinates of x and the m-tuple (x1, x2, ... , xm) of functions the local coordinate system

with respect to (U, i). Since 0 is a homeomorphism from U onto ,O(U), we identify a point x E U and 0(x) E R' via ¢. Hereafter, we denote the local coordinates of x by

x = (x1, x2, ... , xm) = (xs),

1 < i < M.

As far as the local discussion of manifolds is concerned, identifying U and O(U) is more convenient than mapping points of U into Rm under 46 each time; consequently, it helps the reader reach the essence of the arguments more promptly. With this convention understood, for example, given a cO° function f : M - R, we may simply write its local coordinate representation f o 0-1 : 0(U) -+ R by F

1

2

m

1.1. ARC LENGTH AND ENERGY OF CURVES

3

instead of

f (x) = f o O-' (x' (x), x2(x), ... , xm(x)),

x E U.

Let TxM be the tangent space of M at x. Given a c°O function f and local coordinate functions x' near x, the directional derivative of f in the direction x' is expressed as

()(f)= Lf

(x),

15 i<m.

x

/ (

From the definition of the tangent space (see §A.1(b), Appendix),

ft6

is a tangent vector to M at x.

/z

l1

(1.1)

, ... ,

8x2

,

10

x

&m 01 )XI

forms a basis for the tangent space TxM; hence, any vector v E TxM is given uniquely as a linear combination of the basis elements in

(1.2)

V= i-1 DO (ax=

x

(C ', C2'. .. , gym) are called the components of C with respect to

the local coordinate system (x', x2,... , x"'). By applying v to each coordinate function x', we see that C' = v(x`),

1 < i < M.

On the other hand, applying the differential (dxi)x of each coordinate function x', we get

(r / lx/l ( ( axi x )( ldx)x\\a _

x)=

1 0 for any v E ;.t. The inner product 9x is not arbitrarily given at each tangent space T.M. In particular, if gz is smoothly related to x E M in the following sense, gx is said to define a Riemannian metric in M. DEFnvITION 1.1. Suppose that an inner product gx is given in the tangent space TxM at each point x of a cO° manifold M. Furthermore,

we assume that given an arbitrary coordinate neighborhood U in M and its local coordinate system (2I, x2, ... , x'), a function defined by

(1.4)

9i; (x) = 9x t

(ai)

, i axi

1 < i, j < m

is c°°O in U for all i, j. Then we call the family of the inner products g = {gx}ZEM a Riemannian metric on M. When a Riemannian metric g is given to a c°0 manifold M, we call the pair (M, g) of M and g a Riemannian manifold The function gs f is called the (i, j) component of g with respect to the local coordinate system (xi, x2, ... , x"'). The m x m matrix (ge.1) is a positive definite symmetric matrix at each point x E M. In fact, the symmetry of gx implies that

()) = (()

/l

=Fur

9x being positive definite is nothing but the matrix (9ij (x)) being positive definite.

As is readily seen, the set of all bilinear forms gives rise to a vector space over R under naturally induced addition and the scalar product. We denote this vector space by TxM` 0 TxM', that is, TYM a TxM' = { f :TxM 0 T.M -+ IR J f is a bilinear map}. TxM` OTxM` is called the tensor product of TxM` and TxM". Given the dual base (1.3) for TM*, we define a bilinear form (dx' )x g (dam )x by

(dx`)x ® (dx')x(v, w) = (d r')x(v)(dXj)x('w),

Then we can easily verify that

{(dx')x0(dxj)xI I N be a c°° immersion of a cO° man-

ifold M into another cO° manifold N, where (N, h) is a Riemannian manifold. At each x E M, we set gx(v.w) = h9,(z)(dco (v), dPo (w)),

v, w E TIM.

Then the family g = {9z}xEM defines a Riemannian metric on M. This g is called the induced metric from h through 'p, and it is denoted by g = W*h. As for a general c°° manifold M, not necessarily a submanifold, we can define the Euclidean inner product in each coordinate neighborhood by identifying it with an open subset of the

1. ARC LENGTH OF CURVES AND GEODESICS

6

Euclidean space. Using the partition of unity, we can glue them together to obtain at least one (in fact, infinitely many) Riemannian metric on M. Consequently, we can regard a c°O manifold with the second countability axiom as a Riemannian manifold. When there is no fear of confusion, we simply denote by M a Riemannian manifold (M, g).

We now consider curves in M. Let (a, b) (-oc < a < b < oo) be an open interval in real line R. We also regard R and (a, b) as a one-dimensional c°° manifold and its open submanifold, respectively. A cO° map c : (a, b) -+ M from (a, b) into M is called a c°° curve or a smooth curve in M defined over (a, b). The coordinate function t in (a, b) is called the parameter of the curve c. If c(t) = x, we say "the

curve c passes through point x in M at t." Since we have defined a curve c as a map from (a, b) into M, two such curves are considered distinct if they are distinct as maps, even if their images are identically

the same as sets in M. Let [a, b] (-oo < a < b < oo) be a closed interval, and let (a - e, b + E) ( > O) be an open interval containing [a, b]. We call a curve c : [a, b] -- M a c"O curve or a smooth curve in M defined in [a, b], if it is the restriction of a c°O curve c : (a - e, b + e) --+ M to [a, b]. Also given a continuous curve c : [a, bJ -- M, if there is a partition a = ao < a1 < . < a,. = b of the interval [a, b] such that c is a C0° curve in each interval [aa_1, a,] (1 < i < r), we call c a piecewise smooth curve in M defined over [a, b]. Let c : (a, b) -+ M be a c0O curve defined in an open interval (a, b)

and let to be a point in (a, b). Regarding (a, b) as a c°O manifold, the directional derivative with respect to the coordinate function t determines a tangent vector

l (! ET.(a,b) 1 to

to (a, b) at to. The image of this vector under the derivative map d,

: Tto (a, b) - TO(E) M is a tangent vector

(1.5)

dcen ((f)) E T4to)M

to M at c(to). We call this vector the tangent vector of c at t = to, and denote it by c' (to). Let (xl, x2, ... , x"') be a local coordinate system defined near c(to) and let c' = x` o c (1 < i < m). Then the curve c is expressed

I.I. ARC LENGTH AND ENERGY OF CURVES

7

about c(to) as c(t) = (cl(t),... ,cm(t))-

The tangent vector c'(to) is nothing but the vector given as c'(to) =

( )

)

\ / (to) l a1 ) C(w

If c is a c' curve defined in a closed interval (a, b], noting that c is the restriction to [a, b) of a cd0 curve defined in an open interval containing [a, b], the tangent vector c'(to) of c at each point to E [a, b] can be defined as above. If c' (to) 0 0 at every to E [a, b], we call c a regular c'°0 curve.

Since M is a Riemannian manifold, we can measure the magnitude (the norm) Ic'(t)I of the velocity vector c(t) using the inner product g,,(t) defined in each tangent space Tc(t)M. In fact, ]c'(t)I is given by

Ic'(t)I =

ge(t) (c'(t), c'(t)), t E [a, b],

and is a continuous function of t. DEFINITION 1.3. Given a c°O curve c : [a, b] - M defined in a closed interval [a, b] (-oo < a < b < oo), the integral of Ic'(t) I

L(c) =

rb

Ja

Ic'(t)Idt

is called the length of the curve c.

For a piecewise smooth curve, since it is composed of smooth curves defined in closed intervals, we can define its are length as the sum of the arc lengths of those smooth curves. We have defined a curve as a map; hence, even though the image is fixed, we could arbitrarily choose the domain and the parameter. We see here that the arc length of a curve is a "geometric quantity" which is independent of domains and parameters. LEMMA 1.4. Let c : [a, b] M be a curve in M defined in the closed interval [a, b]. Let 9 : [a, ,(3] - [a, b] be a diffeomorphi mn. Then the curves c and c o 9 : [a, fl] -+ M have the same arc length; namely, the following holds: L(c) = L(c o 0).

I. ARC LENGTH OF CURVES AND GEODESICS

8

PRooF. By (1.5), (c o 0)'(t) = c'(0(t)) (d0/dt)(t) holds; hence, 13

L(c o 0)

J«

(c o 0)'(t) I dt = J«

) (t) dt.

1 c'(0(t)) I I

On the other hand, since 0 is a diffeomorphism, we may alway assume dO

dO >Oor

>0.

In the former case, noting 0(a) = a and 0(f3) = b, we have dO

L(c o O) _

lc'(0(t)) I

(t)dt =

6

IC'(t) I dt = L(c).

Ja In the latter, since 0(a) = b and 0(0) = a, we get L(c o 0)

dO

Ja

Ic'(O(t)) I

b

(t)dt JQ

1c'(t) ldt = L(c).

0 Given a c°° curve c : [a, b] - M defined in the closed interval [a, b], we set t

s(t) = j Ic (u)jdu. The function s; [a, b] - [0, L(c)] is called the are length of the curve c. In particular, if c is a regular curve, c' (u) 0 0 always holds. Then s is a monotone increasing function of t; therefore, the inverse function s-I : [0, L(c)] -, [a, b] exists. We define a c°° function by c o s-1 : [0, L(c)] -t M and call it the arcwise parametrization of c. When c is parametrized by are length, we have 1(co s-')'(s)J = (c'(t)IIG (t)I-1 = I. This implies that the magnitude of the tangent vector is always 1. DEFINITION 1.5. Given a C°Q curve c : [a, b] -, M defined in a closed interval [a,6] (-oo < a < b < oo), the integral of le,(t)12/2 Ic'(t)I2dt

E(c) = 1 a

is defined to be the energy or the action integral of the curve c.

For a piecewise smooth curve c, as in the case of the length, we first decompose c as a union of C°° curves, and then define the energy E(c) of the curve c as the sum of the energies of the COQ components.

1.2. EULER'S EQUATION

9

Although the length of a curve is independent of the parametriza-

tion, the same does not hold true for the energy of a curve. In fact, under the same conditions as in Lemma 1.4, it can readily be seen from the definition that E(c) j4 E(c o 0) in general. Nonetheless, as we shall see in the following sections, it is often more convenient to investigate the functional E(c) rather than to deal directly with the functional L(c) when studying variational problems (critical point problems) concerning length of curves. We will take up the relationship of L(c) and E(c) in §1.2.

1.2. Euler's equation Let (M, g) be an m-dimensional Riemannian manifold, and let c : [a, b] --; M be a C°° curve in M defined in a closed interval [a, b]. For the velocity vector c'(t) of the curve c, we defined, using the Riemannian metric g, its magnitude as Ic'(t)I =

9c(t) (c'(t), c'(t))

t E [a, b].

The length L(c) and energy E(c) were, respectively, given by b

L(c) =

E(c) = 2

I c'(t)I dt,

Jn

f

(c'(t) 12dt.

Let (x) = (xi, ... , xm) be a local coordinate system of M defined in a neighborhood of c(t). By setting ct = X' o c, c(t) is expressed as

c(t) = (C' M' ... , cm(t))As was seen in (1.6), c(t) is then given by i

(1.7)

c'(t) _ a-1

(s-)

(t)

(_) (t) E T°it)M. C

Denoting the components of g by gt.j, c' (t) local coordinate system (x), given by I

m I

c'' (t)

I = , 11:

I

is, in terms of the

(t) d d (t)

Now let (x) = (x',... , xm) and (Y) = (fi, ... , Y) be two local coordinate systems of M around c(t). The base of the tangent space

1. ARC LENGTH OF CURVES AND GEODESICS

10

T.,M

isfies the transformation formula

a

m

i

-x

axe (x) j,l

a axi

x

Hence, from (1.4), the components gig and gii of g relative to (x`) and (2i), respectively, satisfy the transformation formula m

gij(x) = F fti )I k,t=1

(1.8)

(x)

j

(x)gkt(x).

00

On other hand, from (1.7), we see that the components of the velocity vector c'(t) undergo the transformation: i

}(t)=

(1.9)

(c(t))(d(t).

1=1

From (1.8) and (1.9) it follows that the local coordinate representations of g (c' (t), &(t)) relative to the coordinates (x') and (xi) satisfy the equation ( 175' -&q (X)

t,,y=1

(t)

r ddt0

(t) =

gkt(x)

(1) (t)

(t).

k,t.1

As is seen from the definition, the above equation tells that the local coordinate representations of the inner product g,, (c' (t), c'(t)) given

by the Rican metric does not depend on the choice of local. coordinate systems. Therefore, we often define the length L(c) and the energy E(c) of a curve c, using a local coordinate system as b

L(c) a

m

dcdcJ

E gij dt dt 1j=1

,

1 lb ",

E(c) _

-2

> g#jdedt-dtdc)-dt.

Now we consider the critical value problem of L(c) and E(c) regarding the CIO curves in a Riemannian manifold M. Namely, we invesigate which c gives rise to a critical point among all the C°°

curves joining two points p = c(a) and q = c(b) in M. For the sake of simplicity, we assume that a curve c; [a, b] , M is a regular Coo curve whose image lies in a local coordinate neighborhood

1.2. EULER'S EQUATION

11

U. Let (x1, ... , xm) denote the local coordinate system in U. We define 2m independent variables x1,... , x"`, C1, ... , Cm as follows. Let

x= (x11 ...,xm, l,... 'Cm) E U,

ER. Weset for

L(c) Jr

(x, rr )

= .f (x1, ...

,xm ,1 , .. .

,

gijsiv,

Sm) _ s,j =1

and for E(c)

f(z,

gij s J.

)=J(21,...,x

i,j=1

Under these conventions, we can express both L(c) and E(c) as (1.10)

F(c) =

b f (c'(t)... Ja

,

cm(t),

11 (t)'...

dem ,

dt

(t} dt.

We find conditions for the integral F(c) in (1.10) in general to assume

a critical value. To that end, we arbitrarily choose a C°° function t defined in an open interval that contains [a, b] and which satisfies ry(a) = i7(b) = 0.

(1.11)

Using this 77, we set, for each i = 1,... , m and any sufficiently small real number c, c1(t; E)

= c (t) + Eij(t),

cj(t; E)

=

c? (t)

(i 54j)-

Then c(i; E)(t) = (C' (t; E), ... , Cm(t; E)),

t E [a, b],

gives a family {c(i; E)} of C°O curves in U. From the definition, for sufficiently small c, each c(i; E) gives rise to a regular curve joining p = c(a) and q = c(b). c(i; 0) gives c itself. For these curves, F(c(i; E)) is given by

F(c(i; E)) =

f

b

. , c+ e (t), ... , Cm (t), dc'

(t), ... ,

_

(t} + E dg (t),

... ,

d

(t) dt.

1. ARC LENGTH OF CURVES AND GEODESICS

12

This is a differentiable function of c in a neighborhood of 0. Hence, a necessary and sufficient condition for F(c(i; e)) to assume a critical

value at =0 is

d

(1.12)

0;

namely, {

axi

(c(t),c'(t))t7(t) + (j,)

(c(t)>

c'(t))

(t) dt = 0.

FIGURE 1.1. Family of cA0 curves {c(i; E)}

Here, if we apply integration by parts to the second term on the left hand side of the equation, noting (1.11), we get

Lb

(L' ) (c(t), c'(t))

I

b

L') (c(t), c(t))i7(t) In

(t)dt =

jab d a

d

--

l of (c(t), c'(t))q(t)dt (c(t), c'(t))j(t)dt.

a

a

(}

(c(t), (1(t)) } 9(t)dt = 0.

The above equation is equivalent to (1.13)

Jb 1 Of )

`

(c(t), fi(t))

-

d

1.2. EULER'S EQUATION

13

Noting that (1.13) holds for any q satisfying (1.11), we see that the following equation

(/)

(1.14)

e

(c(t), c'(t)) -

l

\

dj) (c(t), c (t)) = II

must hold. In fact, suppose that the left side of the equation (1.14) does not equal 0, say > 0, at some point to E [a, b]. Then, by continuity, the left side is positive in a neighborhood to C [a, b] of to. If we choose a C°° function 77 which takes positive values in 1o but 0 identically outside Io, the left side of (1.13) becomes positive, contradicting

the assumption (Exercise 1.3 at the end of Chapter 1).

From the above discussion, we have learned that if a curve c assumes a critical value of F(c) among the C°° curves joining p = c(a) and q = c(b), then (1.14) holds for i = 1, ... , m. We call the equation (1.14) Euler's equation for (1.11). We may consider F(c) as a function defined in the family of curve {c(i; e)}. More generally, a function

defined in a space consisting of functions or mappings is called a functional, indicating a function of functions. When F(c) is regarded as a functional, (1.12) is often called the first variation Jomtda of the variational problem concerning the functional F(c). Solutions to the first variational formula or to Euler's equation give extreme values of the corresponding functional; however, they may necessarily be neither relative maximal nor minimal in general. Therefore we call them critical points of the functional, and the values of the functional at those points are called critical values.

Now we actually compute Euler's equation (1.14). In order to distinguish the functionals L(c) and E(c), we will denote f for L(c) and E(c) by f 1(x, ) and f2 (x, ), respectively. By the assumption, the curve c is regular and C°°. Hence, the arc length parametrization of c gives, as was seen in § 1. 1,

Ic`(t)I = fi(c(t),c'(t)) - 1.

(1.15)

By the definition of f, and f2, we have f,(x,

) =

2f2(x, C)

Therefore, equation (1.14) for L(c) is given as

%22

(c(t),c'(t)) _

I(

-a

2

2) (c(t), c'(t)) = 0. 2

1. ARC LENGTH OF CURVES AND GEODESICS

14

From (1.15), this is equivalent to 12

ai } (c(t), e(t)) = 0. C

(c(t), c'(t))

(1.16)

This implies that under the assumption of (1.15), Euler's equations for L(c) and E(c) are both reduced to computing (1.16) for 21

f2 (X,

M

xm

I

,

m) = 2

E 9ij(xxiti.

i.j=1

We compute (1.16) for each i = 1,... , m. From the symmetric property of (g13), we get m

M

9%j We + 2 >9js(xW _

2 9=1

a9jk t

j,k=1

9ij(x) 'j, j=1

3=1

m

2

m

(X)Vk.

We now see that (1.16) is equivalent to m

d

dt

7=1

m

dcj 9ij (c(t)) dt (t)

_

d(t) = 0.

dc

(c(t)) ) dt (t) dt 2J.,k(Igjk ax= =1 1

This equation can be rewritten as in

m

j=2

9ij (C(t)) d2 2 (t) + d

E j,k=1

dck

ax } (CM) dt (t) at (t)

M

-2

dc

9ij

'9

j,k=1

ax' l) (C(t))

dcj (

dd-`tk

(t) = 0.

Hence, noting the second term being symmetric, we obtain (CM) (1.17) 1

m

da

13 (t)

a9,j

19ik

+2 j,k=1 (axk+axj

I

dck axi)(c(t)) d(t)d(t)=0.

19j k

1.2. EULER'S EQUATION

15

The positive definite matrix (gq., (x)) has its inverse matrix at each point x in U. Namely, we define the inverse matrix (gU(x)) by m 9:k

gik (x)9k1(x) = 8e ,

(x)9kj (x) = S9,

1 < i, j < m.

k=1

k=1

Then we get an m x m symmetric matrix (g'') whose components

tare C' functions in U. Using g' J, we define in U a family

jk

(1 < i, j, k < m) of C°° functions by

{i}_1gu(o91i+a91ko9ik

(1.18)

{

2

a-1

8XI

jk 's are called Christo f'el symbols, and they occupy an important

position in Riemannian geometry. We point out here that

jk }'s are

determined solely by the components gt, of the Riemannian metric

g in the local coordinate system (x'), and that they are independent of the curve c. Using the Christoffel symbols, we can rewrite (1.17) in a simpler form. In fact, if we replace the script i by 1, and add all the equations obtained by cross-multiplying g`l to (1.17)

over I=1,...,m,weget i

(1.19)

m

J-(t)+ E {;k}(C(t))(t)(t)=o1

1

i

m.

This is what we have obtained by actually computing Euler's equation for L(c) and E(c). In general, the curves satisfying (1.19) are called geodesics in a Riemannian manifold. DEFINITION 1.6. Let C : [a, b] --; M be a C°° curve in a Riemann-

ian manifold M. In a coordinate neighborhood U of each point in c, we denote by ` C(t) = (C1(t), ... , Cm(t))

the coordinate representation of c near the point. Then the curve c is said to be a geodesic if each c' (t) (1 < i < m) satisfies equation (1.19).

From the above observations, we have realized that among the

C°O curves joining two points p = c(a) and q = c(b) in U, the

1. ARC LENGTH OF CURVES AND GEODESICS

16

geodesics, when parametrized by arc length, are the extreme points of L(c) and E(c). We point out here the following fact. Let c : [a, b] - M be a C°O curve in M. Suppose there are two local coordinate systems

(xi) = (.T',. - , x') and (t) = (ct1, ... , x") in an open subset U, through which c passes. Then the Christoffel symbols { 3' } and 1JJ

jk

satisfy the following relationship (see Exercise 1.4 at the end

of this chapter): M

7k

5P=I

ati

m

v-xp

J

p

k

+ 9E1

07x9 01xT Q7'

' 52k

On the other hand, when c is represented as c(t) = (c, (t),

.

en (t))

{e1(t), . , . , c""(t)), in these coordinate systems, respectively, (1.9)

holds between the components of the tangent vector ca(t). If we pay attention to these transformation formulas and the coordinate transformation in U 0xi dxk axk 1

_

°

a simple computation yields the following relationship: m

dt2

(t) +

tc1

k

(OXIdt

(t)

(t)

(t) + j,k=1

{}(c(t))(t)(t)). jk dt dt

From this relationship follows that if the curve c in U satisfies (1.19) in the local coordinate system (x1), (1.19) also holds in (if). This fact suggests that the geodesics in Riemannian manifolds can be defined geometrically, independent of local coordinate systems.

In the next section, we will take up this problem from a more general point of view.

1.3. Connections and covariant differentiation Let (M, g) be an m-dimensional Riemannian manifold and let [a, b] --y M be a C°° curve in M defined in a closed interval [a, b]. In the last section, we derived the equation of geodesics from

c:

1.3. CONNECTIONS AND COVARIANT DIFFERENTIATION

17

the first variation formula of length and energy, when the image of c is contained in a coordinate neighborhood of M. In this section, as part of preparation for a general first variation formula, we discuss differentiation for vector fields. For the time being, we assume M to be simply a C°O manifold and U an open subset of M. A vector field X = {X (x)}iEU in U is defined to be a correspondence that assigns to each x E U, a tangent vector

X (x) E T=M. In particular, if U = M, X is simply called a vector field in M. Let (xe) = (x1, ... , xm) be the local coordinate system of an arbitrary local coordinate neighborhood V which intersects U. At each point x E U n V, X (x) is uniquely expressed as T (1.20)

X(x) _

(x)

(axi)Them

functions l;' (1 < i < m) are called the components of X. When the components are C°° functions, X is said to be a C°° vector field in U.

For example, assuming U to be a local coordinate neighborhood, set

ax=) (x) - a_i)

x E U.

Then we obtain a C°° vector field in U :

l

x(JJJ ZEU

1

The family of these vector fields {

ural frame in U.

ll

i

I 1 < i < M } is called the natJJ

Given an open subset U in M, denote by 1(U) the set of all C°° vector fields in U. Also denote by C°° (U) the set of all C°° realvalued functions defined in U. C°°(U) is a commutative ring under the ordinary sum and product of functions. On the other hand, given any X, Y E 1(U) and f E C°° (U), we define X + Y and f X, respectively, by

(X +Y)(x) = X(x) +Y(x),

(fX)(x) = f(x)X(x), x E U.

As can be seen readily, then X + Y, f X E X(M). Fbr any X, Y E 1(M) and for any f, h E C°°(U), we have f (X + Y) = f X + f Y, (f + h)X = f X + hX, (f h)X = f (hX ). Hence, we see that 1(U) is a C°° (U) module.

1. ARC LENGTH OF CURVES AND GEODESICS

18

Now given X E X(U) and f E C°° (U), we can define a function

Xf in U by

(Xf)(x)=X(x)f, xEU. By (1.20), we have, in the local coordinates (xi), m

(X f)(x) = >Ei(x)af (x),

(1.21)

x E u n V.

i=1

Hence, X f E C°°(U). We call X f the derivative of f by a vector field X or in the direction of a vector field X. From (1.21), we see that if X(x) = 0, then (Xf)(x) = 0. We see dearly that the following properties of the differentiation hold: (i)

(ii)

(iii)

X(f + h) = X f + Xh, (X + Y)f = X f +Yf, X (f h) _ (X f )h + f (X h).

Given X, Y E X(U) and f E COO (U), we set

[X, YJ(x)(f) = X(x)(Yf)

- Y(x)(X f ),

x E U.

Then we can readily see that

[X,Y](x)(fh) = ([X, Y](x)f)h(x) + f(x)([X, Y](x)(h)) This implies [X, YJ E T,,M; hence, [X, YJ = { [X, YJ (x) }1EU defines a vector field in U. Let V and 77i denote the components of X and Y

in the local coordinate system (xi), respectively. Then noting

a

i' axe

(x)f = axaa_ -

i)I (x) = 0,

1 < 2,. 0) be an open interval that contains [a, b]. The restriction of a CO° vector field X = {X (t)}tE(a.. ,..f) along a C°° curve c : (a - e, b + e) --+ M is called a CO° vector field along the C°° curve c : [a, b] - M. For example, the tangent vectors of a C°° curve c give rise to a C°° vector field d = {c' (t) } t et,,,bl along the C°° curve c. This is called the tangent vector field of c. Also given a C°° vector field X in M, a vector field along c defined by X o c = {X (c(t)) } is a C00 vector field along c. This is called the restriction of a C°° vector field X to c. The set of all C°'° vector fields along c will be denoted by X(c). Given X, Y E X (c) and f E C°° ([a, b) ), we define vector fields X + Y and f X, respectively, by

(X + Y)(t) = X(t) + Y(t),

(fX)(t) = f(t)X(t), t E [a,bJ.

Then they are C°° vector fields along c. Under these operations X(c) becomes a C°'° ([a, bJ) module.

PROPOSITION 1.10. Let V be a linear connection on a CO° manifald M. Then there is a uniquely determined linear map

D : X(c) -+ X(c) satisfying the following: (i) given X, Y E X(e) and f E C°° ([a, bJ),

D (ii)

(X+Y)= aX +

dDt(fX)=

x+f DX

;

if X E X(c) is the restriction of X E X(M) to c,

DX = Ve(t)X, att

tE

[a, b].

PROOF. First, we verify uniqueness. As before, let to E (a, b] and let (a{) denote a local coordinate system of M about c(to). We respectively represent c in (x*) by c(t) = (c1(t), ... , cm(t)) and X E X (c) by in

X (t) _ i=1

fi(t) 49 s

) c(t)

1.3. CONNECTIONS AND COVARIANT DIFFERENTIATION

23

From (i), (ii) and (1.7), we get

DX

dt - j=1

(1.24)

= k=I

a

a

dt axj +

axj

(c+

Mr 13

a

(c)d

axk

#,j=1

DX

(to) is uniquely determined. Conversely, if we define DX /dt by (1.24) in a coordinate neighborhood intersecting with c([a, b]), we can readily verify that the expression satisfies (i), (ii). It follows, therefore, that if UnV flc([a, b]) # 0, DX /dt' defined in U and V as above coincides in U n V due to the Hence,

uniqueness. This implies that Dt : X (c) - X (c) is well defined.

0

Given X E X(c), D/dt E X(c) determined in Proposition 1.10 is called the covariant derivative along c of X. In particular, when D/dt = 0, X E X (c) is called a parallel vector field along c. The following holds.

PROPOSITION 1.11. Denote by V a linear connection in a C°° manifold M, and let c : [a, b] -p M be a Cl* curve in M. Given an arbitrary tangent vector v E Tc(a), there is a unique vector field X E X(c) parallel along c with X (a) = v.

PROOF. First, we prove the proposition when c([a, b]) is contained in a coordinate neighborhood U. Denote by (xi) the coordinate system in U. In terms of (x'), we express c and v by c(t) _ (c1(t), ... , cm(t)),

a

v= s=1

axe

l

) c(t)

respectively. We set the desired vector field X E X(c) as X (t) El n, V (t)((a/(9xi),(t)). Then, from (1.24),

DX

m

(t) _

m

k

(t) + i=1

i,?_I

i

F z.(c(t)) 4dt (t)tj(t)

axa

x

()()

_

1. ARC LENGTH OF CURVES AND GEODESICS

24

Hence, in order for DX/dt = 0 to hold, it suffices that each Ek(t) satisfies the following system of differential equations: to

(1.25)

£(t) + 1: I ;j (c(t))

o,

1 < k < M.

This is a system of first order linear ordinary differential equations. Hence, given an initial condition (v1, ... , v), there is a unique soluv' (1 < i < m) tion (41, ... , ) to (1.25) defined in [a, b] with due to the fundamental theorem regarding the existence and uniqueness of the systems of linear ordinary differential equations (see Rikigaku to Jyobibun Hoteishiki, Gendai Sugaku e no Nyumon, Iwanami Koza). Next, in the general case, noting that c([a, b]) is a compact subset, it can be covered by a finite number of coordinate neighborhoods, in which the proposition holds. By uniqueness, the solutions coincide in

the intersections of the neighborhoods to produce a solution defined 0 in the entire [a, b]. This completes the proof.

In Proposition 1.11, we call the tangent vector X(b) to M the vector obtained by parallelly displacing v = X(a) along c. By oorresponding X (b) to X (a), we get a map P,, : TT(Q) 4 Ta(b) between the tangent spaces. We call Pc the parallel displacement along c.

It is easily seen from the uniqueness and existence of solutions to the system of linear ordinary differential equations (1.25) that the parallel displacement P, : TT(0) -> T,(b) is a linear isomorphism form Te(a) onto TT(b).

Now we go back to the Riemannian manifold (M.9). The following

result, often called the fundamental lemma in Rieannian geometry, is a theorem, which is to be the starting point of Riemannian geometry.

THEOREM 1.12 (Levi-Civita). Let (M, g) be a Riernannian manifold.

Then among the linear connections on M, there is a unique

linear connection V such that for any X, Y, Z E X (M), (i) (ii)

X g(Y, Z) = g(V XY, Z) + 9(Y, V X Z),

VXY - VYX = [X, Y].

1.3. CONNECTIONS AND COVARIANT DIFFERENTIATION

25

PROOF. We first show uniqueness. Assuming that such a V exists, from (i) follows

X9(Y, Z) = 9(VxY, Z) +9(Y: VxZ), Yg(X, Z) = g(VYX, Z) + g(X, VYZ), -Zg(X, Y) = -g(V ZX, Y) + g(X, VZY). Adding the left hand sides together and the right hand sides together, respectively, and using (ii) yields

Xg(Y, Z) + Yg(X, Z) - Zg(X, Y) =g(V xY, Z) + g(V YX, Z)

+g(X,VYZ - VZY) +g(Y,VXZ - VZX) =2g(VxY, Z) +g([Y,X], Z) + g(X, [Y, Z]) + g(Y, [X, ZI). Namely, we get (126)

2g(VxY, Z) = Xg(Y, Z) + Yg(X, Z) - Zg(X, Y)

- g(X, [Y, Z}) - g(Y, [X, ZJ) + g(Z, [Y, XD.

Noting that the Riemannian metric g defines a nondegenerate inner product gx in each tangent T. ,M, we see from (1.26) that V XY can be uniquely expressed in terms of g and the commutators of vector fields. Hence, if V exists, it is unique. Next, we show existence. Given X, Y E X(M), define V xY by (1.26). It is easily verified that VxY defines a covariant derivative from its definition and the properties of the commutator product. We immediately see from (1.26) that VxY satisfies the conditions (i),(ii) 0 in the theorem. DEFINITION 1.13. Given a Riemannian manifold (M, g), the linear connection V in Theorem 1.12 is called the Levi-Civita connection or the Riemannian connection of (M, g). In general, a linear connec-

tion V is said to be compatible with the Riemannian metric g if it satisfies the condition (i), and symmetric if it satisfies the condition (ii), respectively.

From now on, we will consider the Levi-Civita connection in a Riemannian manifold (M, g) unless otherwise mentioned. Let (x) be a local coordinate system in M and let I' denote the 9 connection coefficients in (x_) of the Levi-Civita connection. From

1. ARC LENGTH OF CURVES AND GEODESICS

20

(1.22) and (1.26) follows

V.

1 (89j,

k

rji Stet - 2 k-= I

OX'

499;1

+ ax.

_ a9s;) ax!

/

If we change the index k to h, multiply both sides by the components (gk1) of the inverse matrix (g$) of (g j ), and add them over 1, we get (1.27)

1

t=i

a a9-a, ag;t gk1\Ox +Ox - Oz1 !II

Hence, it follows that the connection coefficients 1' 's of the LeviCivita connection are nothing but the Christoffel symbols

g3

's.

ExAMPLE 1.14. If M is Euclidean space Em, we have

1'ij = d, 1 < i, 3, k < m, since gE, = &,f in the natural frame relative to the coordinate functions in R'. Hence, from (1.23), we see that the Levi-Civita connection of

E"' is precisely the standard connection. If X E X(c) is a parallel vector field along a curve c, (1.24) implies that each component of the vector becomes a constant; hence, X E X(c) is parallel in the sense of Euclidean geometry.

As is readily seen from (1.22) and (ii) of Theorem 1.12, a linear connection that is symmetric is equivalent to the connection coefficients 14jj's that are symmetric in the indices i, j; namely,

I

=F

,

1Z=1

(x)2 = JJJ

1.4. GEODESICS

33

As a Riemannian metric in S, we consider the induced metric. If we set

U={(x',...,xm+1)ES"xm+1>01 and define a map 0: U 0(X1

lRm by m+1

(

m)

1

(U, 0) gives rise to a coordinate neighborhood of S. Noting that

¢-1(x1,...,x"`)= xi,...,x"=,

1-

m

(xi)s

gives in U the embedding of S' into Eit+i, the components gij of g, gl) and the connection coefficients n-'s are, through simple computations, given, respectively, as X)

gi1 = di - xi a

9;j = 6;J + (X,,+1)2' X,

r

1

= gil, 1 < i, j, k < m. j Assuming that c is a normalized geodesic, the equations of a geodesic d2ck

dt

x1`

m

+ i,7=1

i

ckgs;

dt d

=0,

1 < k < m,

are reduced to a system of second order ordinary differential equations of constant coefficients (1.34)

1x'(q)eiE V, qEU, s=1

with (xl (q), ... , xm (q) ), we can define a local coordinate system (zt ).

We call it a normal coordinate system about p (relative to {ej). Theorem 1.24 can be refined as follows. THEOREM 1.25. At each point p in M, there exist a neighborhood

W and a positive number 6 > 0 such that given a point q E W, the exponential map expgat the point q is a C°° difeotnorphism from B (0) C TqM onto expq(Bs(0)) D W. Hence, W may be regarded as a normal Coordinate neighborhood about every point in W

PROOF. Given p E M, take e > 0 and the neighborhood V of p in Theorem 1.20. In the neighborhood U={(q,v)jgEV,vETTM, IvI<E} of (p, 0) in the tangent bundle TM, define a map F : U -+ M x M by F(q, v) = (q, expq v),

(q, v) EU.

Note that F(p, 0) = (p, p). As was seen in the proof of Theorem 1.24, d(expp)o = I; namely, the derivative d(expp)o at 0 E TM is the identity map. Hence, the derivative dF(p,o) : T(p,o)TM - TpM x TpM of F at (p, 0) E TM is an isomorphism. In fact, by the definition of F, it can be readily seen that the matrix representation of dF(p,o) is given by I

0

I I

By the Inverse Mapping Theorem, there exist a neighborhood U' C U of (p, 0) In TM and a neighborhood W' of (p, p) in M x M such that F is a C°° diffeomorphism from U' onto W'. We may assume that U' is given in the form

U'={(q,v) (qE V',

VETqM, Ivl Ir'(t)12.

Hence, (1.39)

I

1

I w'(t) I dt > j 1 Ir'(t)Idt >

f ' r'(t)dt = r(1) - r(e)

holds. Since r(1) = r(c), we get L(w) > L(c) by letting c -- 0.

1. ARC LENGTH OF CURVES AND GEODESICS

42

If L(w) = L(c), the equality in (1.38) and (1.39) holds. Hence,

for any t, 18f / ti{r(t), t) = 0 and fir, (t)I = r'(t) > 0 must hold. This implies that v(t) = 0, that is, v(t) is a constant vector, and w is nothing but a simple reparametriztion of c. Consequently, we get w([0,11)=c([0,IDWhen w([O, 11) is not contained in B, there is a parameter value

tj E (0,1) at which w intersects with the boundary of B. Thus, we have the inequality L(w) > L(wi[0,t11) > p > L(c), where p is the radius of B.

0

We note here that Theorem 1.28 is a local result, and that a geodesic may not always be length minimizing when its length gets larger. In fact, in the case of the geodesics in the unit sphere S'r` of Example 1.22, it is clear that a geodesic c emanating from a point p is no longer length minimizing when it goes beyond the antipodal point of p resulting in L(c) > ir. On the other hand, we see that a length minimizing piecewise smooth curve is a geodesic, as stated below. THE0R.EM 1.29. Let c : [a, b] - M be a piecewise smooth curve with a parametrization proportional to arc length. For any piecewise smooth curve w joining c(a) and c(b), if L(c) < L(w) holds, then c is a geodesic.

PROOF. Given t E [a, b), let W be the neighborhood of c(t) given in Theorem 1.25. We can choose a sufficiently small interval I C [a, b] with t E I so that c(I) C W and cII : I W represents a piecewise smooth curve joining two points in some geodesic ball. Then, by the assumption and the first half of Theorem 1.28, the length of cit equals the length of the geodesic joining these two points. Hence, we see that cuI is a geodesic from the fact that the parameter of clI is proportional to are length, combined with the second half of Theorem 1.28. Since 0 t E [a, b] is chosen arbitrarily, the entire c is a geodesic.

Given two points p, q in a Riemannian manifold M, the infimum of the lengths L(tv)'s of all piecewise smooth curves w's joining p and

q is expressed by d(p,q) and called the distance between p and q. Namely, the distance between p and q is defined to be d(p, q) = inf{L(w) I w is a piecewise smooth curve joining p and q}.

SUMMARY

43

If M is connected, the distance d(p, q) is well defined, since there are piecewise smooth curves joining p and q (Exercise 1.9 at the end of

this chapter). In this case, the function d : M x M -- JR, which assigns d(p, q) to (p, q) E M x M, gives rise to a distance function on M. In fact, by the definition, d(p, q) > 0 holds, and d(p, q) = d(q, p) is evident. Given p, q, r E M, the triangle inequality d(p,r) 0. Now assume that p # q and take the geodesic ball B,(p) about p. If q E B, (p), d(p, q) > 0 from Theorem 1.28. If q is not in B, (p), Theorem 1.28 also implies d(p, q) > E > 0, since any piecewise smooth curve joining p and q intersects with the geodesic sphere S, (p). In either case, d(p, q) > 0, if p # q. From the above arguments, we also see that for any p E M and a sufficiently small E, the geodesic ball BE (p) and the geodesic sphere SE (p) of radius E, respectively, are the sets defined, in terms of the distance d, as

B,(p)={gEMI d(p,q)<E}, S., (p) = {q E M I d(p,q) = e}. From these observations it follows that the metric topology defined

in M by the distance function d coincides with the topology of the differentiable manifold M (Exercise 1.10 at the end of this chapter).

Summary 1.1 The definitions of length L(c) and energy E(c) of a curve in a Riemannian manifold M. 1.2 The first variation formula and the Eider equation giving the critical points of variational problems. 1.3 The definition of the Levi-Civita connection V, i.e., a unique symmetric linear connection compatible with the Riemannian metric on a Riemannian manifold M. V defines the parallel transport of tangent vectors along curves in M. 1.4 The geodesics in a Riemannian manifold M give rise to the exponential map expp, which defines the normal coordinate system about p. 1.5 The geodesics are locally length minimizing curves in a Riemannian manifold M.

1. ARC LENGTH OF CURVES AND GEODESICS

44

Exercises 1.1 Given local coordinate systems (xi) = (xl,... , xm) and , rn) about a point x in an r-dimensional C°'° mani(X') = fold M, show the following: (1) Between the bases of the tangent space TM, transformation equations a (;-}

g

p-1 n

8 oxi

a

8 z

hold.

(2) Given the components (') and (x) of a tangent vector v E TAM with respect to (x1) and (x ), respectively, transformation equations

ax

_

(x)"

j=1

9=1

hold.

1.2 Prove that a C°° manifold M satisfying the second axiom of countability possesses Raemannian metrics. 1.3 (The fundamental lemma in the theory of variation) Let f : [a, b] -, R be a continuous function defined in a closed interval [a, b]. Assume that

f.

f (t)q(t)dt = 0

for any C°0 function i, : [a, b] - R. Prove, then, that f - 0. 1.4 Regarding a linear connection V on a C°° manifold M, prove the following.

(1)Let (x`)_(xl,...,xm)and (z)_(r1,...,Y') be two local coordinate systems about a point x E M. The connection coefficients {Pjk } and {I z, } with respect to the the coordinate systems satisfy the t ransorrmmation rule in

{rik}=E P=1

q.r=1

r''

OX9 ax,

Itt

t

dt (t)

(t)

tt>ti

alt

W.

PROOF. Using the definition (2.1), we compute

dE

_

1

of Of

d

k

w 2 - =ads j

(rat ' 8t )

9

.

In each domain of the definition [t,.4+13 x (-E,E), Of 10t and Of/Os define C°° vector fields along the curves f,,, ft, respectively. By Proposition 1.15 and Lemma 1.26, we have d da9

8f 8f rat

' at

-

2g

Dof If

D Of _ D Of

as 8t '

as et

at)

cat as

Hence, in each [tj.t,+1], we get d

te+l

ds it"

(,

of

at 8t }

rt'+,

=J 2

2g

tj+1

D

Of of

a N, 5i) dt D

8f Of 50

Noting that Proposition 1.15 implies d

D Of

dt9(a'cat) -9(atof' cat)+gas'3t5t)'

2. FIRM AND SECOND VARIATION FORMULAS

52

integration by parts gives us, in each [t;, 4. .F1],

(D Of Of dt

atas'at) of of u+

9

as' at

j

Consequently, we get

(2.3)

=

eI of k

g f Daf

c,+1

-

g i

as at at } dt.

nof Daf , J9ru)dt. ^

Setting s = 0 in (2.3) and noting c(t) = f(t, 0), V(t) = 8s (t, 0) and V(0) = V (a) = 0, we get (2.2) from the definitions ofd

(tt), (t ).

From the first variation formula in Theorem 2.4 follows COROLLARY 2.5. A necessary and sufficient condition for a piece-

wise smooth cvroe c : [0, a] -+ M to be a geodesic is E'(0) = 0 for any pieoewise smooth variation f of c.

PRooF. If c is a geodesic in M, from the definition, de = 0 and

(tt) =

(t{ ), 1 < i < k; hence, necessity is obvious. In order to see sufllciency, we assume that E'(0) = 0 holds for any piecewise smooth variation f . Choose a piecewise smooth function

dt

h:[0,a]-'JRin each tt(0 0 fort 96 to hold. Let Dc' dt be a piecewise smooth vector field along c. Given a piecewise smooth variation with V as its variation vector field (see Proposition 2.2), we have from (2.2)

V(t) = h(t)

E'(0)=-

12dt

ah(t) I Jo

= 0.

Hence, we see that Dc'/dt = 0 in each open interval (t{, tf+i ). Since c is a piecewise smooth curve, Dc'/dt = 0 in [ti, ti+i]. Consequently, c I [ti, ti+i] is a geodesic.

2.1. THE FIRST VARIATION FORMULA

53

Next, consider a variation V such that V (O) = V (a) = 0 and in

each ti (1 Si fk(x)T (Ek, ... , Xs)(x) = 0. k=1

Under the preparations above, we define the curvature tensor of a Riemannian manifold. In what follows, we assume that M is the Riemannian manifold (M, g) of dimension m and V denote the LeviCivita connection of M.

2. FIRST AND SECOND VARIATION FORMULAS

58

DEFINITION 2.8. Given X, Y E X(M), we set

R(X, Y) = V xV y - V yV x - Vjx,vx].

We define a map R:X(M)xX(M)xX(M)-+ X(M) by R(X, Y, Z) = R(X, Y)Z,

X, Y, Z E X(M).

This R is called the curvature tensor of M or (M, g).

For example, if M is the rn-dimensional Euclidean space El, R(X, Y, Z) = 0 for any X, Y, Z E X(M) - In fact, when Z E X (M) is regarded as an Rm valued C°° function Z = (f 1, ... , f'n) in M, we have

V VyZ = (XYf',... ,XYr). Hence from the definition of [X, Y], we get

R(X, Y) = VxVy - VyVX - V[x,yJ = 0. Therefore, we may consider the curvature tensor R representing a quantity that measures the degrees of deviation of M from the Euclidean space El. As we can also see from the following lemma and Proposition 2.6,

R is a type (1,3) C°O tensor field over M. This is the reason why R is called the curvature tensor of M.

LEMMA 2.9. R : X(M) x X(M) x X(M) -+ X(M) is a trilinear map, and for any X, Y, Z E X(M) and fl, f2, f3 E C' (M), we. have R(f1X, f2 Y, f3 Z) = f1f2f3R(X, Y, Z)

PROOF. It is obvious from the definition that R is linear with respect to the sum and the scalar product in each variable. As for the property regarding the product with a CO° function, say f3s we have

R(X,Y,f3Z) = R(X,Y)(f3Z) = VX VY(.f3Z) - DYVX(f3Z) - VIX,YJ(f3Z)

= f3(VXVYZ - DyV Z - D[X.y]Z) + (XYf3)Z - (YX f3)Z - ([X, Y]f3)Z = f3R(X, Y, Z).

Similarly, we can verify the above for fl, f2

0

The curvature tensor R satisfies the first Bianchi identity given as follows:

2.2. CURVATURE TENSOR

59

PROPOSITION 2.10. For any X, Y, Z E 1(M), R(X, Y, Z) + R(Y, Z, X) + R(Z, X, Y) = 0.

PROOF. From the symmetric property of the Levi-Civita connection, the left hand side equals

VXVyZ - VyVXZ - V{X,y)Z +VyVZX - VzVYX - v(y,z]X + VzVXY - VXVZY - V[Z,X1Y = Vx[Y, ZJ - V1y,z] X + Vy[Z, X] - V 1Z,xjY +VZIX,YJ - V[X,y1Z

= [X,fY,Z]]+[Y,[Z,X]]+[Z,[X,Y]]. Hence, we obtain the desired conclusion from the Jacobi identity for the vector fields.

R is a C°° tensor field of type (1,3). Set

R(X,Y,Z,W) =g(R(X,Y)Z,W), X , Y,Z,W E 1(M). This defines a map

R : 1(M) x 1(M) x 1(M) x 1(M)

C°°(M). We readily see from Proposition 2.6 that R is a C°° tensor field of type (4,0). This R is called the Riemannian curvature tensor of M or (M, g). We have the following with respect to R. PROPOSITION 2.11. For any X, Y, Z, W E 1(M), we have (i)

R(X,Y,Z,W) = -R(Y,X,Z,W),

(ii)

R(X, Y, Z, W) = -R(X, Y, W, Z), R(X, Y, Z, W) = R(Z, W, X, Y).

(iii)

PROOF. (i) follows readily from R(X, Y) = -R(Y, X). (ii) is equivalent to R(X, Y, V, V) = 0, where V = Z + W. Since the LeviCivita connection is compatible with V and g, we see R(X,Y, V, V) = g(VxVYV,V) - g(VyV V, V) - g(V(x,YJV V) = 2 {XYg(V, V) - YXg(V, V) - [X, Y]g(V, V)} = 0.

As for (iii), the first Bianchi identity implies that R(X, Y, Z, W) + R(Y, Z, X, W) + (Z, X, Y, W) = 0.

2_ FIRST AND SECOND VARIATION FORMULAS

60

By replacing X, Y, Z, W in this equation in the order X -., Y --, Z W -+ X, we obtain a set of four equations. Adding those equations together with (i) and (ii), we get

2R(Z,X,Y,W)+2R(W,Y,Z,X) = 0. Hence, R(Z,X,Y,W) = R(Y,W,Z,X) holds. Given a local coordinate system (z') in a local coordinate neigh-

borhood U in M, the curvature tensor of M and the Riemannian curvature tensor, in terms of the components in (x'), are expressed, respectively, as follows: in Rajkdx` (& d. T'

R

dal

dx 0dx'` 0dzl.

R= In U, we have

R

8

a

a

'"

E Milk !=1

11a a

a

8

Rtjkt = 9 R 8xs , axj } axk 8xl

jia

_ E X,011 . r=1

Flurthennore, it is readily verified from the definition (see Exercise 2.4 at the e n d of t h i s c h a p t e r ) that the components R , 's of the curvature tensor are given by

a

m

8

irrjk - Tiyr ik) r-1 The equations in Propositions 2.10 and 2.11 are, respectively, given as (2.7)

jk -

Mjk = axi U

I'iik +

Rjk+Rpk,+Rij=O, Rijkt = -RjkU, Ri,ki = -Rijlk, Rijkt = Rktij. RJ 1A.nx. Since some authors define the curvature tensor R as

R(X, Y)Z = VyVxZ - VxVYZ + V(x,y1Z, or express a

s

R (axi' x, }

a axk =

rn

1 l=1

a

2.2. CURVATURE TENSOR

61

it is imperative to pay attention to the positions of the subscripts and superscripts in symbols and component representations.

For a given x E M, denote by a a two-dimensional subspace of the tangent space TIM. Then given an orthonormal basis {v, w} for or with respect to gI, the value of K(v, w) = R(x)(v, w, w, v) = 9x(R(x)(v, w) w, v)

is determined, independent of the choice of {v, w}, only by a. In fact, let {v', w'} be another orthonormal basis for a. Then {v', w'}, in terms of {v, w}, can be expressed as v` = cos OV + sin 9w,

w' _

sin 9v ± cos 9w.

From the property of R in Proposition 2.11, we readily see that R(x)(v, w, w, v) = R(x) (v', w', w', v'). We denote this value by K(a) and call it the sectional curvatum of or. From the definition, the sectional curvature K(a) is determined by the curvature tensor R. Conversely, the curvature tensor R(x) at a point x E M is completely determined, if the sectional curvatures of all two-dimensional subspaces a C Tx M are given (see Exercise 2.6 at the end of this chapter).

EXAMPLE 2.12. Let M be the in-dimensional Euclidean space Em. Since the curvature tensor R - 0, all the sectional curvatures are 0.

EXAMPLE 2.13. Let M be the unit sphere Sm C E'+1 with the induced metric g from E11+1. In terms of the local coordinate system

defined in Example 1.22, the components g,, and r t of g and the connection V are, respectively, given as

gcl=aci+(

xixj +1)2,

rks,=xk gij,

1 M from an open subset O of R2 into M and a C°° vector field V along u, we have

DV=R(5,i) 5; Z V-D 5i FX

2. FIRST AND SECOND VARIATION FORMULAS

66

PROOF. If we express V in the coordinate functions (x, y) of R2 and a local coordinate system (.Ti) in M as m f

v'(x,y) t ; ; ) 10

V(x,y) =

'

by the definition we get

D V-

D

8

m

vi

1

8Bl

D DV

m 8vi

axi

8+ 8x

e,1

v,

D8

t ax

8v' D a 8v= D 0 axay 8xi + ay ax 8x' + az 5i azi } dI a2v4 8

8x

DD a t

Hence, if we subtract the equation obtained from the second equation by interchanging x and y from the second equation itself, we get

a OyV - Oy a v

_ c"` *(D D

DD 8 ey8x}

=Lv {ax8y

8Xi.

On the other hand, if we express u(x, y) = (u=(x, y)) in the local coordinate system (a') of M, we get

au

8x -

l m

8u& a 8z 8x1 '

rn

8n

ay - j=L

8u1 a 8y axe

Noting Preposition 1.10, we get

D8

a

TY 8xi

axi ,

1=1

DDa

D

axv 8xi = 8x

mau3

a

°

axi

M auk auJ

".r

1=I

-f- >

Jk-i

ay

a

"` au'

ax 61Y

E'"a2u1 axay °

1=1

8 axe

a S axi

2.3. THE SECOND VARIATION FORMULA

B7

Hence, if we interchange x and y in the second equation and subtract the resulting equation from the original equation, we have

DD

ft0y

DD d 8yax 8:r, 8uk

' (V ea V 3.3

8y

9.k=1

since

8

(0

R

8

8r' _ (V,!. V i

,

holds from the definition of R, noting 1-0

DD_DD -by

8r) V

_

m

l

- V,

'0

a ] = 0, we finally get

v iau kaw

8r

8u u = R ax Oy

)

V

a R (8xk '

a

8

8x`

V.

For a geodesic c : 10, a] - M in M, the first variation E'(0) of the energy functional E for an arbitrary piecewise smooth variation f : 10, a] x (-e, e) -. M is always 0 as seen in §2.1. Namely, geodesics are the curves that give the critical points of the energy functional. As in the extremal problem of calculus, it is needed to investigate the second variation E"(0) of the energy functional at the critical point c, in order to study the behavior of the energy functional E about a geodesic c. In what follows, we calculate the second variation formula.

THEOREM 2.16 (The second variation formula). Let c : [0, a] -

M be a geodesic in M and let f : [0, a] x (-E, E) -} M be a piecee smooth variation of c. Reganiing the energy functional E : (-c, E) --s R, the following holds: q (V, 2 E"(O) (2.8)

jTt2

k s=1

(

di-

dt

2. FIRST AND SECOND VARIATION FORMULAS

68

Here, V and R are, resp , the variation vector field of f and DV (t=) are the cunmftm tensor of M. Furthermore, DV (t;) and given, resper.vely, by

(t;) = t

(t;) _

(t),

(t).

tdti

t>t{

PROOF. If we differentiate the equation (2.3) in the proof of Theorem 2.4 in s, we get

D 8f 8f

d2E _k_

ds2 inn L 8s 8s' 8t

(2.9)

t

+t

+ ,=O

14

(Of D Of ti+1 as, as at ) it.

D 8f D 8j)dt(8f D D Of

-a9

9(8s 8s' at 8t

8s' 8s 8t 8t l

dt.

If we set s = 0, since c is a geodesic, we have D do .- 0. dt dt From this follows that the third term of the right hand side equals 0. Noting that the variation f leaves the end points c(0), c(a) fixed and

that 8f /8t(t, 0) = c'(t) is smooth, we see that the first term of the right hand side also equals 0. On the other hand, from Lemmas 1.26 and 2.15, we get D D Of _ D D B,f D Of _ D Of 8f 8f 8,f R( as, as 8t at 8t as at + 8s ' 8t) at 8s 8t 8t From this readily follows that the second term on the right hand side becomes

o

8fD8f'--g V(t;),DV(t`)_DV(t:) 8s F. at }

it-

i=I

(

at

8t

Furthermore, we get at s = 0,

D D8f _ D D as & &

V + R(V,

dc dc dt )

dt;

dt dt hence, the forth term of the right hand side can be rewritten as

f

9

j

8s' 8 8t at j

2

tit

tR

/ V, dt [

Ja 9 \V (t)' dt2 Henee, we have obtained the desired equation (2.8).

dt.

0

2.4. EXISTENCE OF MINIMAL GEODESICS

69

COROLLARY 2.17. Let c : [0, a] -+ M be a geodesic in M, and let The energy functional E : (-E, E) -i R satisfies

f : [0, a] x (-E, E) -p M be a piecewise smooth variation of c. p

E"(0) = J {g(v', V') - g(R(V, c')c', V) }dt.

(2.10)

0

Here, V and R are, respectively, the variational vector field of f and the curvature tensor of M, and V' = DV/dt. PROOF. Noting, as in the proof of Theorem 2.4, that d (V, DV dt dt g

l _- g

DV DV D2V + g V' &-2 dt ' dt

holds holds with the variational vector field in each interval [ti, tt+I], we readily get (2.11) g

(V, D2V

+ R(V,

dt

o

k

g E {-o

V,

DV ) dt

{9(V, , V) - g(R(V, c')c', V)}dt. ti

We substitute this into (2.8) to get (2.10).

0

Equation (2.10) implies that the second variation E"(0) of the energy functional E depends only on the variational vector field V along the geodesic c and the curvature tensor R of the R.iemannian manifold M.

REMARK. As was stated in §2.1, if we regard the set fl(M; p, q) of all piecewise smooth curves between two given points p and q in a Riemannian manifold M as a "manifold", and the energy functional E : 1I(M; p, q) - R as a "function" defined in this manifold, we may consider the second variation formula (2.10) as the "He sian"of the function E at the critical point c. The "index" of this Hessian and the topology of Q(M; p, q) are closely related to each other, for example, see Milnor [12].

2.4. Existence of minimal geodesics Let (M, g) be a connected yn-d mensional R.iemannian manifiold.

As seen in § 1.5, the distance between two given points p, q E M is

2. FIRST AND SECOND VARIATION FORMULAS

70

defined as the infimum of the lengths L(w) of piecewise smooth curves w joining p and q; namely,

d(p, q) = inf {L(w) I w is a piecewise smooth curve joining p and q }.

The topology in M induced from the distance function d coincides with the topology of M as a differentiable manifold. Theorems 1.25, 1.28 and 1.29 together yield that there exists a unique minimal geodesic c, i.e., L(c) = d(p, q), joining any two points p and q in a sufficiently small geodesic sphere W of M.

Given two points p, q E M, there may not always exist a minimal geodesic joining p and q. For example, consider the Riemannian manifold obtained by removing the origin from the Euclidian space E'11. Even if such a minimal geodesic exists, it may not be unique, as can be seen readily in the case of an antipodal pair in the unit sphere SR`.

In this section, we investigate the existence problem of minimal geodesics. We begin with conditions for any two points p, q E M of a Riemannian manifold M to have a minimal geodesic joining them. DEFtNITION 2.18. Let M be a R.iemannian manifold. If the exponential map expp at p is defined for any tangent vector v E TpM at any point p E M; namely, any geodesic c(t) emanating from p is defined for every t E R, M is called geodes°icaldy complete. For example, as seen in Examples 1.21, 1.22, 1.23, the rn-dimen-

sional Euclidean space E'", the unit sphere (STI, g) in E"'+', and the m-dimensvonal real hyperbolic space (H, g) are all geodesically complete. They are also complete as a metric space (M, d) with the distance d induced from the Riemannian metric; namely, any Cauchy sequence with respect to d converges. The following important result called the Hopf-Rinow theorem shows that these two notions of completeness are not only equivalent, but also the exact condition which assures the existence of minimal geodesics.

TxE OREM 2.19 (Hopf-Rinow). Let (M, g) be a connected Rie-

mannian manifold and let p be a point in M. Then the following conditions are equivalent: (i) The exponeniialznap exp, at p is defined in the entire tangent Spam

Tit.

(ii) Any bounded dosed subset of (M, d) is compact.

2.4. EXISTENCE OF MINIMAL GEODESICS

71

(iii) (M, d) is a complete metric space. (iv) (M, g) is geodesically complete. In what follows, an (M, g) satisfying one of the above conditions, therefore, all of them, will simply be called a complete Riemannian manifold.

THEOREM 2.20 (Hopf-Rinow). If (M, g) is a complete and connected Riemannian manifold, for given two points p, q E M, there is a geodesic c joining p and q such that L(c) = d(p, q); namely, there is a minimum length geodesic joining p and q. We first prove Theorem 2.20.

PROOF. Set d(p, q) = r. We only need to treat the case r > 0. Given a point p, under the condition (i) of Theorem 2.19, we show that for any q E M, there is a minimum length geodesic joining p and q.

FIGURE 2.2

Let W be a normal coordinate neighborhood of the point p as given in Theorem 1.25. For sufficiently small 6 > 0, let Ba(p) C W denote the geodesic ball of radius 6 with p as its center. Since the boundary S = SS (p) = {x E W1 d(p, x) = S} of Ba (p) is a compact set and since the distance d(q, x) from a point q is a continuous function in S, there is a point xo E S such that d(q, xo) assumes the minimum. Using the exponential function expp, xo can be expressed as xo = expp(dv),

v E TpM, IvJ = 1.

From the assumption, the normal geodesic c(t) = expp(ty) is defined at all t E JR for this v. The proof will be done if we can show c(r) = q.

2. FIRST AND SECOND VARIATION FORMULAS

72

In fact, since L(c) = r, c will be a minimum length geodesic joining p and q. In order to show c(r) = q, we see that the equation

d(c(t), q) = r - t

(2.12)

holds for each t E [0, r]. Setting t = r in (2.12) we see that c(r) = q since d(c(r), q) = 0. Now set I = It E [0,r]1 (2.12) holds}.

Clearly, 136 0 since 0 E I. Also from the definition, I is a closed subset of [0, r]. Hence, assuming to E I, it suffices to verify that if to < r, there is a sufficiently small 8' > 0 such that (2.12) holds for to + b'. From this, we get sup I = r. Since I is a closed subset, r E I; consequently, we get I = [0, r]. This completes the proof. In order to see that (2.12) holds at to+8', we consider the geodesic

ball B (c(to)) of radius b' and center c(to) and its boundary S' _ S',g(e(to)). As before, there is a point xo' E S' such that the distance d(x, q) between q and x E S' is minimized at xfl. It suffices, then, to see xo' = c(to + 6') for this xfl'. In fact, from d(c(to), q) = r - to due to to E I and from d(c(to), q) = 8'

+ DES' d(x, q) = 8' + d(xo', q).

we see that r - to = 5' + d(xo', q) = 6' + d(c(to + b'), q).

(2.13)

From this, we get

d(c(to + 8'), q) = r - (to + 8'); consequently, we see that (2.12) also holds at to + 8. Next, we show c(to + b') = xo'. From the first equality in (2.13) and the triangle inequality of the distance function, we have d(p, xo') >d(p, q) - d(xo', q) = r - (r - to - 6') = to + 6'. We finally see that d(p, xo') = to +6', since the length of the piecewise

smooth curve obtained by connecting the portion of the geodesic c between p and c(to) and the minimum geodesic c' between the points xo' and c(to) is to+b'. Hence, by Theorem 1.29, this piecewise smooth

curve is nothing but a geodesic joining p and xo'. In particular,

0

c(to + 8') = xo', which was what needed to be shown. We next prove Theorem 2.19 in the order of (i) (iv)

(i).

(ii)

(iii)

2.4. EXISTENCE OF MINIMAL GEODESICS

73

Let A C M be a bounded closed set. (ii). PRooF. (i) Since A is a bounded subset of (M, d), the diameter p = diam(A) = sup{d(x, y)lx, y E Al is finite. Hence, there is a geodesic ball B centered at a point p such that A C B. From the condition (i) and Theorem 2.20 follows

AC BCexppBr(0) for the r neighborhood B,.(0) C TpM of the origin 0 in the tangent space T.M. Since the closure B,.(0) of B,.(0) is a compact subset of T,pM, the image expp B,.(0) of the exponential map exp, is a compact subset of M; hence, its dosed subset A is also compact.

Let {pk} be a Cauchy sequence in (M,d). Since from the definition [p&) is a bounded subset of M, we see from (ii) that {pk} is relatively compact. Therefore, {pk} has a convergent (ii)

(iii).

subsequuence. Since (Pk} is a Cauchy sequence, it also converges. (iii) (iv). Let p E M be an arbitrary point of M, and let c be a normal geodesic emanating from p. For c, we set

I+ = it > 01 c is defined in [0, t] }.

Then we show t+ = sup I+ = +oa. Assume t+. < +oo, and let {tk } c I+ be a sequence converging to t+. From the assumption, given any E > 0, for sufficiently large k, ! with I tk - ti l < e, we have d(c(tk), C(tt)) < L(c I [tk, t1]) = Itk - tj I
k2 > 0, M is compact, diam M < sr/k, and the fundamental group ir(M) is finite.

so

2. FIRST AND SECOND VARIATION FORMULAS

In Corollary 2.25, condition K > k2 > 0, regarding the sectional curvature, cannot be weakened simply to K > 0. In fact, the parabolic surface of revolution in the three-dimensional Euclidean space E3 given by

M={(z,y,z)EE'31z=z2+y2} is complete and the sectional curvature K > 0 everywhere, but not compact. We discuss another application of the second variation formula. THEOREM 2.26 (Synge) - Let (M, g) be a compact, connected, and

orientabk Riemannian manifold of even dimension. If the sectional curvature K of M is always positive, then M is simply connected PROOF. First of all, we note the following. For the piecewise smooth variation f : [0, a] x (-E, c) -f M of a geodesic c : [0, a] -M given in Corollary 2.17, if we do not assume condition (iii) of Definition 2.1; namely, if the deformation obtained from the family { f, } of curves in the variation f does not necessarily keep the end points c(0), c(a) fixed, the variation vector field V (t) of the variation f may not satisfy V(O) = V (a) = 0. Consequently, the first term on the right hand side of the equation (2.9) in the proof of Theorem 2.16

is not necessarily equals 0. Also the terms at t = 0 and t = a in the second term on the same right hand side remain nonzero. These

terms correspond to the cases at t = 0 and t = a in the first term on the right hand side of equation (2.11), as one can see in proof of Corollary 2-17.

Keeping these in mind, if we repeat the arguments in the proofs of Theorem 2.16 and Corollary 2.17, we see readily that the second variation formula of the enemy function E fur the general variation f, which does not necessarily keep the terminal points c(0), c(a) fixed, is given by (2.16)

E"(0) _

10

f{g(V, V') -g(R(V, c')c',V)}dt

D

- g (Ts , f (0, 0), c"(0)) + g

(D ,

f (0, a), c'(0)

Here, V` = DV/dt and R is the curvature tensor of M. Now we assume that there is a nontrivial free homotopy class a E CI (M) of a loop in M. By Theorem 2.23, there is a minimum closed geodesic c in a. Let t be the arc length parameter and let e : [0,1] -- M

2.5. APPLICATIONS TO RIEMANNIAN GEOMETRY

81

represent the arc length parametrization of c. As was seen in Proposition 1.15, the parallel transport Pc : TT(o)M TC(i)M = T'(o)M defines a linear isometry of TT(o)M onto itself. Since c is a dosed geodesic, Pc(c'(0)) = c'(l) = c'(0) holds. From the assumption, M is orientable; therefore, Pc defines an orientation preserving orthogonal transformation I cl(0)i -, c'(0)1 PC

in the orthogonal complement c'(0)-i C Tc(o)M of c'(0) in TCo)M. On the other hand, M is even dimensional from the assumption. Hence c'(0)1 is odd dimensional. As is well known in linear algebra, the orthogonal transformation PP I c'(0)1 has 1 as an eigenvalue, and it leaves the corresponding eigenvector v # 0 fixed. Namely, we have

P°(v) = v. Consequently, by transporting v parallelly along c, we obtain a vector field V(t) along c which is perpendicular to c at each point c(t). If we consider the variation of c given by f (t, s) = exp0t) sV(t),

(t, s) E [0, a] x (-e, c),

as treated in Proposition 2.2, noting V' - 0 and v(O) = V(a), the second variation formula (2.26) combined with the assumption K > 0 on the curvature readily yields

E"(0)

j

g(R(V, c')c', V )dt < 0.

Of course, E'(0) = 0 for the first variation formula of E. Hence, we see that E(fe) < E(c) for sufficiently small 0 < s < c. With the length of f8, this implies that L(fs)2 < 2lE(f$) < 21E(c) = L(c)2. Since it is clear from the definition that f, and c are free homptopic C°° curves, this yields the contradiction that the length L(c) of c is the minimum in a. Consequently, we see that M possesses no nontrivial 0 five homotopy class; hence, M is simply connected. We point out the following regarding the assumption in Theorem 2.26. The three-dimensional real projective space P3(R) is a compact orientable C°° manifold, and it possesses a natural R iemannian metric g so that the universal covering map tii : S3 -> P3(R) gives rise to an isometric and locally diffeomorphic map. With respect to this g, the sectional curvature K of P3(R) is always positive, but P3(R) is not simply connected. Hence, the assumption that the manifolds

2. FIRST AND SECOND VARIATION FORMULAS

Sa

are even dimensional cannot be removed. The assumption that they are orientable cannot also be removed. Indeed, two-dimensional real projective space P2(R) is a nonorientable C'0a manifold, and it possesses a Riemannian metric g such that the sectional curvature K is always positive as in the case of P3(R). However, p2(R) is not simply connected (see Exercise 2.10 at the end of this chapter).

Summary 2.1 The first variation formula regarding the energy E(c) of a curve c and characterization of the geodesics as the critical points of the energy functional E. 2.2 The de initions, the l3iema nian curvature tensor, and various curvatures on a Riemannian manifold. 2.3 The second variation formula of the energy E(c) of curves. 2.4 Existence of a minimal geodesic joining two arbitrary points p, q E M in a complete Riemannian manifold M, and existence of a al closed geodesic within each nontrivial free hoinotopy class of n a closed curve in a compact Riemannian manifold. 2.5 Theorems of Myers and Synge regarding the topology of complete Riemanniau manifolds of positive curvature.

Exercises Let (xi) and (J{) be local coordinate systems about a point x in a C°O manifold M. Show the following:

2.1

(1) Between the components 7'' andi...i, of a tensor field T E T$(M) of type (r, s), a transfiamiation identity &k3 ftk1 . fr 1

1

klt ,*. LI,.

Itr

holds.

(2) Given two local coordinate systems (xi), (z) in an open sub-

set U, that the components of T E ;(M) with respect to (x{) are C'° implies that the components with respect to (r) are C°° . 2.2 Show that the space T.1 (x) of all tensors of type (1, s) over the tangent space Tz to a C0° manifold M is canonically isomorphic

to the vector space Hom (TM x ... x TM, .,x) of all s linear maps from the direct product TxM x ... x TxM into T.M.

EXERCISES

2.3

83

A linear connection

V : X(M) x X(M) - X(M) in a C°° manifold M does not define a C°° tensor field of type (1,2) over M. Why not? 2.4 Let (xi) denote a local coordinate system in an m-dimensional Riemannian manifold M. Show that a=

'jk = ax rjk - aaxe r4k +

m

(rirr,k - l1 rr k) r=1

holds regarding the components of the curvature tensor R2.5 Let o C TxM be a two-dimensional subspace of the tangent space TaM of M at a point x E M. Show that the sectional curvature K(a) of a is given by

K(cr) =

gx (R(v, w)w, v) ga(v,v)gx(w,w) - 9x(v,W)

with respect to any base {v, w} for o. 2.6 Show that the curvature tensor at a point in a Riemannian manifold is completely determined by the sectional curvatures of all possible two-dimensional subspaces in the tangent space at the point. 2.7 Suppose that the parallel transport between two arbitrary points in a Riemannian manifold M is determined independently of the choice of piecewise smooth curves between the points. Then show that the curvature tensor R of M is identically 0; namely, R(X, Y)Z = 0,VX,Y, Z E X(M).

2.8 Let co : M -+ N be a C°° diffeomorphism from a Riemannian manifold (M, g) into another Riemannian manifold (N, h). If gx(v,w) = hp(x)(dPx(v),&Px(w)), Vv,w E TxM, holds at each point x E M, cP is said to be isometric. Given an isometric diffeomorphism cp : M -+ N, show the following:

(1) Let V and V' be the Levi-Civita connections of M and N, respectively. Then d&o(V xY) = v4(x)dco(Y),

X, Y E X(M).

Also let R and R' be the curvature tensors of M and N, respectively.

2. FIRST AND SECOND VARIATION FORMULAS

84

Then dsp(R(X, Y)Z) = R'(dco(X ), d(p(Y))d4p(Z),

X, Y, Z E X(M),

holds. (2) If c is a geodesic of M, V (c) is a geodesic of N. 2.9 Let (M, g) be a Riemannian manifold and let c:,1 : M ---; M be a COO covering of M. Prove the following:

(1) M has a Riemannian metric g so that w becomes a locally isometric diffeomorphism. (2) (M, g) is complete if and only if (M, g) is complete.

2.10 Let M be a compact and connected even-dimensional Riemannian manifold. Assume that all the sectional curvatures of M are positive. Prove that M is either simply connected or the fundamental group irl(M) = Z2.

CHAPTER 3

Energy of Maps and Harmonic Maps In this chapter, we define a functional called the energy of maps in the spaces of smooth maps between Rieman.n.ian manifolds and discuss harmonics maps which are the critical points of the functional. The energy of a map is a spontaneous generalization of the energy of a curve as seen in Chapter 1. The harmonic maps include, for examples, harmonic functions, geodesics, minimal submanifolds, isometry, holomorphic maps, etc.

3.1. Energy of maps Let (M, g) be an mrdimensional Riemannian manifold. Denote

by (xl, ... , x') a local coordinate system in an open subset U in M. The local coordinates of a point x E U in M are expressed as (x')(1 < i < m). At each point x E U,

forms a base for the tangent space TIM, and

f(dxl)z,... ,(dxm)r} forms the dual base for the dual space TIM*. Namely,

W)x((ax'/zl

=o,

1 g=i dxidxi

.

ij=1

Here the components g1j of g in (x') are C° functions in U defined by (a

al

gs'

g

axs' ax? 85

3. ENERGY OF MAPS AND HARMONIC MAPS

86

and (g`j) forms a positive definite m x m symmetric matrix at each point in U. Let (gij) denote the inverse matrix of (g1 j ). Namely, we set (3.1)

E9'k9kj = b`,

94k9kj _ Sj

1 < i, j < M.

k=1

k=1

The R.iemannian metric g, induces a natural linear isomorphism between the tangent space T,,M and the dual space TXM* defined by

5:T,,M-+TM*,

(3.2)

0:TxM* _,TTM.

In fact, for given X. E TM, wx E TM', we may set XT(Y3)

°

9x(wz1Yz) = w1(YY), YY E T1 M.

91(X1,Yx),

If we, using the local coordinate system, express m m a X1 _ X'(x)(; )x, wx =

X , ws are, respectively, given by

m m XX = E(Eg;j(z)X3(x))(dx`)., i=1 j=1 m in w!

_ E E 9ii (x)wj (x) i=1

j=1

Using this linear isomorphism, an inner product gx in TZM* dual to the inner product gx in TxM is defined, for given wx, es E T,,M*, to be 99(Wx, 8x) = 9, (WI' 61).

From this definition combined with (3.1) and (3.3), an easy computation yields 9x((dx{)a, (dx')x) = g1 (x).

Namely, the inverse matrix (gij(x)) of (gj(x)) is precisely the matrix representing the components of the inner product g.*, in TxM*. Let (N, h) be an n-dimensional Riemannian manifold and let u : N be a C° map from M into N. Denote by (y1, ... , y") a M local coordinate system in an open subset V in N. Express the local

coordinates of a point y E V in N by y = (y") (1 < a < n). Then

3.1. ENERGY OF MAPS

87

for a given x E U with u(x) E V, the local coordinates of u(x) are, in terms of C°° functions Ua = yk o u in U, expressed as (3.4)

u(s) = (ul (xl, ... , 2m), ... , un (x1, ... , xm)).

The Riemannian metric h of N is also expressed in V as n

h = E h.,fldyadyI3. a,0=1

Let us discuss the differential of u at a point x denoted by

du= : TM -+ duz is a linear map from the tangent space TTM into Tu(,,)N. If we represent u as (3.4) using the local coordinates, we get n

du, (()) =

(8x=) (x) ( ay ) UkX) ,

I



< 97 b.

In other words, due is the linear map represented by the n x m matrix

((8ua/8x')(x))

A.s is well known, the linear space Hom(T.M, Tu(s) N) consist-

ing of all linear maps from TsM into Tu(,,)N is naturally linearly isomorphic to the tensor product ,,s* ® Tu(s)N. In fact, it is obtained by assigning to f E Hom(TxM, Tti(s)N) a bilinear map f t in TsM* ®TT(s)N defined by f I (v, w) = w(f (v)),

v E TM, w E Tu(s)N*.

Hence, the differential dux of u can be regarded as an element of TJ M* 0 T,,(s)N. Namely, we have (3.5)

du,, E Hom(TsM,T,,(s)N)

TJM* OTT(z) N.

Since a base for TXM* ®Tu(,,)N is given by (3.6)

{(dxt)s ®(8/ a)L(s) I < i < m, 1 < a < n},

due is represented by sn

(3.7)

n

l i)// (x)(dx`)z ®(

du,, _ m .L ( i=1 a_1

)// (xy

On the other hand, the inner products gi in T=M* and hu(x) in Tu(s)N naturally induce an inner product (, ) in the tensor product

3. ENERGY OF MAPS AND HARMONIC MAPS

88

TXM* 0 T,,(,,)N. In fact, for the base in (3.6), we set

(dx')x u(x)

®(

g'3ha#(u(x))

u(x)

x

and extend it bilinearly to arbitrary elements. In other words, we define the components of the inner product (, ) as the tensor products (g'"(x))®(ha,9(u(x))) = (gaj(x)(ha0(u(x))) of the matrices representing the components of gx and hu(x). We denote by IduxIx the norm

of du with respect to this inner product. As seen readily from the definition of the inner product and (3.7), 1dti

is given by

j

Idu, I = (du,, duz)= n M

a

E g''(x)hao(u(x))

l (x)

(x)

From these facts, we see the following. Given a C°° map u : M -> N from a R.iemannian manifold (M, g) into another Riemannian manifold (N, h), we consider the induced vector bundle u-ITN by u over M from the tangent bundle TN of N. u-ITN is the vector bundle over M whose fiber over x E M is the tangent space Tu(g) N of N at u(x). Next denote by T*M the cotangent bundle of M, and

consider the tensor product T*M ® u-'TN of T*M and u-'TN. T*M®u-iTN is nothing but the vector bundle whose fiber at x E M

is TTM 0 Tu(z)N. Denote by I'(T*M 0 u-'TN) the space of all C°° cross sections of the vector bundle T*M 0 u-'TN. Namely, r(T*M®u-iTN) is the set of all CIO maps a : M --> TM* ®u-ITN such that a(x) E TXM* 0 Tu(.,)N holds at every x E M. Define a map du : M -+ TM* ®u-iTN by setting du(x) = du, for the differential du,, of u at each point x in M. From (3.5) and the local representation (3.7) of dux, we see readily that du determines a C°O cross section of the vector bundle T*M ® u-iTN. Namely, we have

du E r(T*M (g u-'TN).

On the other hand, a fiber metric (

,

) is naturally defined in

the vector bundle T*M 0 u-'TN from the inner product (, )z in the tensor product TZM* 0 Tu(x)N. In fact, given a,' a E I'(T*M 0 u-'TN), we simply define (a,' a) (x)

= (a{x),' a{x)) z, x E M.

89

3.1. ENERGY OF MAPS

Given du E r(T*M®u-1TN), we can speak of the norm Idul with respect to this fiber metric (, ) . From the definition, ldul (x) = Idu,, Iz, and Idul is given by (3.8)

auQ

n

m

Idu12

= E E 9 haA(u) (axi) i,j=1 a,3=1

(2)

Under the above observations, we defined the energy density of a map u as follows. DEFINITION 3.1. Given a C°° map u : M -p N from a R.iemannian manifold (M, g) into another Riemannian manifold (N, h), a C°° function e(u) E C°°(M) defined by

e(u)(x) = 2 Idul2(x),

x E M,

is called the energy density function of u or simply the energy density. It is clear from (3.8) that the energy density e(u) is a C°° function. Let {eI, ... , em}, {e'1, ... , e',b} be, respectively, orthonormal bases

for the tangent spaces TM, ,,,(,)N with respect to gz and hu(,). We express du,, in these bases as n

dux(ei) _

E

'M.

or=1

Then it is readily verified, from its definition, that Idul(x) is given by m 1du12(x)

i=1 a=1

Consequently, we may regard Idul2 (z) as representing the square sum of the "rate of expansion" for the differential du,, : TTM -+ T, (z)N of u in the mutually orthogonal directions. This is the reason why we call e(u) the energy density of u. DEFINITION 3.2. Let (M, g) be a compact Riemannian manifold. Then the integral of e(u) given by (3.9)

E(u) =

e(u) d1ts

Jet is called the energy or the action integral of the map u. Here, p9 represents the standard measure induced in M from the Riemannian metric g (see Exercise 3.1 at the end of this chapter).

90

3. ENERGY OF MAPS AND HARMONIC MAPS

Let C°D(M, N) denote the space consisting of all the CO0 maps from the Riemannia manifold M into the Riemannian manifold N. .

If M is compact, the energy E(u) E R is defined by (3.9) for each u E C°°(M, N). Hence, the energy of maps is regarded as defining a functional

E:C(M,N)-+R. Our objective in the following is to find maps which are critical points of this functional E.

3.2. Tension fields As preparation for obtaining the first variation formula charaaoterizing the critical points of the energy functional E, we consider the second fundamental form of a map and its tension field. Let (M, g) and (N, h) be Riemannian manifolds of dimensions m

and n, respectively, and let u E C'0 (M, N) be a C°° map from M into N. Let TM` 0 u`ITN denote the tensor product of the cotangent bundle TM' of M and the bundle u-1TN induced over M by u from the tangent bundle TN of N. As seen in the last section, TM* 0 u ITN has the fiber metric L , ) induced naturally from the metrics g, h. First, we verify that there is a naturally induced connection compatible with this fiber metric (, ); in other words, (, ) is parallel with respect to the connection. The Riemannian metric g induces the Levi-Civita connection V in the tangent bundle TM of M. Namely, as in Theorem 1.12, given CO° vector fields, or C°° cross sections X, Y, Z E r(TM) of TM, them exists a unique connection V which satisfies the following conditions: (3.10)

X9(Y, Z) = 9(VxY, Z) + 9(1', VxZ),

(3.11)

VXY - VYX = [X, 11.

In fact, given a local coordinate system (x') of M, we define the connection coefficients {r!,) of the Levi-Civita connection V in (a-') by in

(3.12)

V.

&zs

From (L27), r k is given by rn

ag;!

r;; = 21 E gki !=1

C

a9;!

ax, + axe

- '09ij a24

3.2. TENSION FIELDS

91

Here, gi3's are the component of g in (xi) and gii's are the components of the inverse matrix of (gi,) For a given Y E P(TM), if we define

VY(X) = VXY, X E r(TM), we obtain the following tensor field of type (1, 1): VY E r(TM* (& TM)!---' Hom(TM,TM).

Consequently, the Levi-Civita connection V of M defines a map

V : r(TM) _ r(TM* (9 TM), which assigns a CO° tensor field VY E r(TM* (& TM) of type (1,1)

to a C°° tensor field Y E F(TM) of type (1, 0). We call VY the covariant differential of Y. We can define, from the Levi-Civita connection V in TM, a con-

nection V* in TM* as follows. First, we note that from the linear isomorphism in (3.2), we obtain bundle isomorphisms

b:TM -,TM*, between the tangent bundle TM and the cotangent bundle TM*. Using these bundle isomorphism, given w E r(TM*) and X E r(TM), we define V*Xw E r(TM*) by V*Xw(Y) _ (Vxwo)b (Y),

(3.13)

Y E r(TM).

It is easily verified that V*Xw satisfies the same computational rules as the covariant differentiation VXY in the tangent bundle TM. Call V*Xw the covariant derivative of w by X. For given w E r(TM*), define

V*w(X) = V*xw, X E r(TM). Then we obtain a type (0, 2) tensor field V*w E r(TM® (9 TM*)

Hom(TM (9 TM*).

The map

V*: r(TM*) -. F(TM* (& TM*) that assigns the type (0, 2) C°° tensor field V*w to the type (0,1) Co' tensor field w is called the connection in TM* determined by V. From (3.13) and the definitions of b and #, we see that V *Xw is given through a simple calculation by (3.14)

V*XW(Y) = Xw(Y) - w (VXY) .

3. ENERGY OF MAPS AND HARMONIC MAPS

92

Consequently, we may accept (3.14) as a definition for V *Xw. Rewriting (3.14), we obtain

Xw(Y) = V*Xw(Y) + w(VXY).

This equation explains that the connection V in TM and the connection V* in TM* are in a mutually dual relationship. From the definitions of g* and V* and from (3.10), we can readily verify that, given X E r(TM) and w, 0 E r(TM*), the equation (3.15)

Xg*(w, 0) = g* (V* XW, 0) + g* (w, V*X0)

holds. Namely, we see that V* is a connection compatible with the fiber metric in TM*. Let U be a coordinate neighborhood in M and let (x') denote a local coordinate system in U. Then we see that (3.14) yields in

(3.16)

V * .-

dxk

= - E r 3 dx1,

1 < i, k < m.

j=1

We note that the connection coefficients of the connection V* induced in TM* from V are given as negative the connection coefficients of V. Let K E r(TM® ® TM*) be a tensor field of type (0, 2) and let L E I'(TM ®TM) be a tensor field of type (2, 0). As a generalization of the above, we can define, by covariantly differentiating L and K,

a tensor field V*K E r(TM* 0 TM* 0 TM*) of type (0,3) and a tensor field VL E r(TM* 9 TM (9 TM), respectively, by (317)

V*K(X,Y, Z) = XK(Y, Z) - K (VXY, Z) - K (Y, V Z) , VL(X, w, 0) = XL(w, 0) - L (V*xw, 0) - L (w,V*X9) .

Using these definitions, if we define K and L, respectively, by

V*xK(X,Y) =V*K(X,Y,Z),

VxL(w,0) =VL(X,w,0),

we can easily verify that the covariant derivatives V * x K and V x L

of K and L in X satisfy the same rules of calculation as V xY. In particular, if V * K = 0 and V L = 0, K and L, respectively, are called parallel tensor fields with respect to the connections V* and V. For example, g and g* are tensor fields of type (0, 2) and (2, 0), respectively. Following (3.17), a calculation of their covariant differentials V*g and Vg*, from (3.10) and (3.15), yields

V*g = 0, Vg*.

3.2. TENSION FIELDS

93

Namely, g is a parallel tensor field of type (0, 2) with respect to V*, and g* is a parallel tensor field of type (2, 0) with respect to V. In other words, that the Levi-Civita connection and g are compatible is nothing but that g and g* are parallel tensor fields with respect to V* and V, respectively. Given a local coordinate system (x') in a local coordinate neighborhood U of M, we express the components of the type (0, 3) for field V*g and the type (2, 1) tensor field Vg' by °igjk and Vigjk, respectively. Namely, V i9Jk and Dig1 k, respectively, are C°° functions in U defined by

(k) a

°igjk=V`y` Vi9'jk = Vg'

g j,b

xk =V

Ole

= V 8m' g! (dxi, dxk).

From (3.12), (3.16) and (3.17), simple computations yield that V,gjk and Vigjk are given, respectively, by V1gjk =

gjk

axi

m

-

m

i jglk 1=1

(3.18)

°igjk = axk + J

m

- E tlk t

1=1

m

j+ilgik +

1=11

I'ikl911. 1L=1

Consequently, we may regard (3.18) as definitions for Vigjk and

Vigjk. That g and g' are parallel with respect to V' and V, respectively, means (3.19)

Vigjk = 0,

1 < i, j, k < m.

Vigjk = 0,

We note that from (3.18), these equations are nothing but the componentwise expressions of (3.10) and (3.15) in the local coordinate system.

We can uniquely define a connection 'V in the vector bundle u_ I TN induced from the tangent bundle TN of N by a C° map

u : M -+ N from M into N (see Exercise 3.2 at the end of this chapter). In fact, let U and V be coordinate neighborhoods of M and N such that u(U) C V, and (x') and (y*) denote local coordinate systems in U and V, respectively. For each I < a < n, (3.20)

(&Yaoul(x)=

}

'6y" UW

,

3. ENERGY OF MAPS AND HARMONIC MAPS

94

defines a C°° cross section of u-1TN over U. At each point z E U,

(r° u}) (x)' ... ,

(ayn o u

) (x) }

gives rise to a base for the fiber TT()N of u-'TN over x. Then we can define a covariant differentiation ' V as such satisfying, for each

1

0) an open interval in the real line R. Given a C°° function u E C°0 (M, N), a CO0 map F : M x I - N is called a C°° variation or a smooth variation of F if

F(x, 0) = u(x),

(3.29)

x E M,

is satisfied. Given a variation as defined above, set

x E M, t E I.

Ut(x) = F(x, t),

Then ut gives rise to a CO° map ut : M -> N, and uo = u holds from (3.29). Furthermore, since F(x, t) is of C°° in t, the family of maps jut I t E I} C C°°(M, N) obtained as above defines a smooth variation of the given map u = uo in the space of maps C°° (M, N). In what follows, we denote by F = jut } t E I a C°° variation of u E C°° (M, N)

for the sake of simplicity. When a C°° variation F = {ut}tEI of N u E C°° (M, N) is given, at each x E M, ut (x) = F(x, t) : I defines a C°° curve in N, passing through u(x) at t = 0. Consequently, the set of the tangent vectors to these curves at t = 0, denoted by

ut(x) =

V (t) = t=o

&

(x, 0) E Tu(s)N,

x E M,

defines a C°C section V E r(u-'TN) of the induced bundle u -'TN. In other words, V (x) defines a C°° vector field in N along the map u. The tangent space T(x,t) M x I of the product manifold M x I at a point (x, t) is naturally identified with the direct sum TxM ED TtI of the tangent spaces TM and TtI. In this setting, V is nothing but the vector field given by V (X) = dF(,,o)

0,

j

0

j,

(x, 0) E M X I.

This vector field V E r(u-'TN) is called the variational vector field of the map u.

Given a C°° map u E C°°(M, N), let V E r(u ITN) denote a C°° vector field in the induced bundle u-'TN. Then, for a sufficiently

3. ENERGY OF MAPS AND HARMONIC MAPS

100

F(x. t)

FIGURE 3.1. A variation of a map and its variational vector field

small e > 0, a C'°° variation F = {ut}tEj of u can be defined by F(x, t) = exp,3(.) (tV (x)), (x, t) E M x I. With this variation, dI

V (X) _

ut (x),

x E M,

t=0

holds. Hence, the set r(u-'TN) of the C°° sections of the induced bundle u ITN is nothing but the space of the variational vector fields along U.

Given a C°° variation F = {ut}tEl, we investigate the change of the energy functional E. Since F is a C°° variation, the energy

E(ut)

2

Jr I dut I2dpg

becomes a CO° filnction in t for each ut E C°Q(M,N). Regarding the first variation of E(ut), we have

THEoiu M 3.8 (The first variation formula). Let F = {ut}tEl be a CO° variation of a C°° map u E C°° (M, N). Then

5E(ut)I

(3.30)

where V = d

r

t_o

M = -J (V,T(u))dp9,

ut is a variation vector field of u, and T(u) is the

I

too

tension field of u. (,) is the natural fiber metric in the induced bundle

u-'TN.

3.3. THE FIRST VARIATION FORMULA

101

PROOF. Let F(x,t) = ut(x) : M x I -+ N be a COO map defining the C°° variation of u. Consider now the vector bundle T(M x I)* ®F-1TN over M x I. As was seen in the previous section, T(M x I)* 0 F-'TN admits a natural fiber metric (, ) and the standard connection V compatible with the metric. Under the Tx M®Tt I, we denote the covari-

natural identification T(x,t) (M x I)

ant differentiation with respect to the connection V in the directions (8/8x4, 0) E T(x,t) (M x I) and (0, d/dt) E Tx,t (M x I), respectively, by

vt = V(o,d/dt).

Vi = v(a/ax+,0),

As before, denote by (xi ), (ya) the local coordinate systems in M and N, and by gij, h,,.,g the components of the Riemannian metrics g, h, respectively. Since V is a connection compatible with the fiber metric (,) , for each 1 < i, j, k < m, 1 < a, < n, from (3.24) follows (3.31)

vigjkha#(ut) = 0,

Vtg3khafl(ut) = 0. Since from the definition, E(ut) is given by

E(ut) =

2

f

M

;dµ9 exi

L is=1 a,Q=1

noting (3.31),

dtE(ut) t=0 =

2

f

M

n

a

9s'ha18(ut) a 4 8zj

bl

(ij=1 a,4=1

m

dp9

t=0

a

n

&i (7xj ) 1t=0

M

On the other hand, applying Lemma 3.7 to the C°° map F and the vector fields (8/8x4, 0), (0, d/dt) over M x I, by noting that [(0, d/dt), (8/8x4, 0)] = 0, we get V(o,d/dt)dut

\ \a '' 0)) -- V(a/ax+,o)dut `(0' 4))

=dut(

(0

dt)'(

=0.

i'0))

Consequently, for each 1 < i < m, 1 < a < n, we get (3.32)

vt Our = 8xt

via

8t

3. ENERGY OF MAPS AND HARMONIC MAPS

102

If we write a variational vector field of u as n

V=1:

a=1 Va

ou,

=a

noting V,

we get It=01

m

d'IE(ut)

=

a

IEE

94'h(u)©i

M j j_1

= fm

M

n-+

E

L

g'jha0(u)ViVQ

i,j=1 a,$=1

att t _ o

dig

(quo W

d1A9

= J(VVdu)d9. Here (, ) on the right hand side denote the natural fiber metric in the vector bundle TM* 0 u-'TN. Hence, from the following lemma, we obtain (3.30) as the first variation formula of E(u ).

0

LEMMA 3.9. Given u E COO (M, N) and V E r(u-'TN), we have fm

(VV, du)djas =

" fkt

(V,

r(u))dµ9.

Here, the symbol (, ) on the left and right hand sides, respectively, represent the natural fiber retrace in T M* 0 u-1 TN and u-1 TN. PROOF. Let X be a C°O vector field over M given by m

m

X=E i=1

n

E E E 9'hag(u)V i=1 j=1 0,6=1

'

Denote the covariant differentiation of X by in

VX = E ViXj dxt ®tax ij=1 i VEX' (see Exercise 3.3 The divergence of X is given by divX = at the end of this chapter). Then noting the identity

V(V0du)=V 0du+V0Vdu

3.4. HARMONIC MAPS

103

together with (3.24), (3.26), (3.27), and (3.28), we get from a simple calculation m n auls

E ij=1 a,3=1

divX = E

M

9'3h«p(u)ViyVa

a

.i

n

+ E E giiha0(u)VaV,Vjuu3 i,j=1 a,J=1

=(VV, du) + (V, r(u)).

Green's theorem (see Exercise 3.4 at the end of this chapter) implies JM

divX dµ9 = 0;

hence, we get the desired result.

From the first variation formula of Theorem 3.8, we get the following-COROLLARY

3.10. Given u E C1 (M, N), a necessary and sufficient condition for the first variation of E(ut) of an arbitrary COQ

variation F = tvt}tEJ to satisfy dtE(ut)t_o = 0 is for the tension field of u to identically vanish, i.e., r(u) L 0. PROOF. It suffices to note that we can take any CO° section V E

I'u-'TN as the variation vector field in the first variation formula (3.30).

Corollary 3.10 indicates that a C°° map u E COO (M, N) with r(u) = 0 is a critical point of the energy functional E : C°°(M, N) -R.

As seen from the proofs of Theorem 3.8 and Lemma 3.9, we used

the connection V in the vector bundle TM* 0 u-1TN and Green's theorem regarding the divergence of a vector field. However, the basic

idea for the proof of the first variation formula for the energy E(u) of a map is essentially the same as in the case of the first variation formula for the energy E(c) of a curve.

3.4. Harmonic maps From the first variation formula for the energy of a map obtained in §3.3, we have seen that the critical points of the energy functional are given by COO functions whose tension fields vanish identically.

3. ENERGY OF MAPS AND HARMONIC MAPS

104

Such maps are generally called harmonic maps. In this section, we discuss the definition of harmonic maps and some examples. In what follows, (M, g) and (N, h), respectively, denote connected Riemannian manifolds of dimensions m and n, and denote by u : M N a C°D map from M into N. We begin with a rigorous definition of a harmonic map.

DEFINITION 3.11. A C°° map u E C°°(M, N) is called a harmonic map if its tension field r(u) is identically 0; namely,

r(u) = trace Vdu - 0

(3.33)

holds in M. (3.33) is called the equation for harmonic maps.

When M is compact, a harmonic map u is nothing but a map which is a critical point of the energy function E : C°° (M, N) - R. In fact, as was seen in Corollary 3.10, the map u E COO (M, N) being harmonic means that

=0 dtE(ut) holds for an arbitrary C°° variation F = {ut}tEl of u. From Lemma 3.4 and (3.28), we see the following, regarding the equation for harmonic maps. Let (xs) and (y') denote local coordinate systems in M and N, respectively. With these local coordinate systems, we express the map u by

u(x) = (u1(x1,... ,xm),... ,u"(x1,... ,2m)) = (U12 (x')), and denote the tension field r(u) of u by n

r(u)°i / o u E r(u 1TN).

r(u) _ a=1

Then r(u)° is given by

r(u)

m

02ua °C

=1 9 =,j

= Lu°1 +

=j 8xk

8x'8xi

Lr L.

i,.1=1 Q,'T=1

8U.*

r

k

k=1

=jrja

9

Q-f (u)

+Q,y=1

1a ply

(u

)

8U,0 OU7

ax' 8xj

OuQ ou7

8x' axj

Here, r'k and r'Qy, respectively, represent the connection coefficients of the Levi-Civita connections in M and N, and A is the Laplace

3.4. HARMONIC MAPS

105

operator in M (see Exercise 3.3 at the end of this chapter). Consequently, equation (3.33) for harmonic maps is given, in the local coordinate systems in M and N, as m

(3.34)

A0 +

n

7 gjjr'G

(u)

5T Xi

=0, 1 < a < n.

From (3.34), using the local coordinate systems, we see that equar tion (3.33) for harmonic maps is a system of second order quasi-linear elliptic differential equations. The nonlinear parts are second degree polynomials of the first order partial derivatives of ua. If we set 9aJrn

$'r (u) 1,J=1 0,'Y= I.

ou auk axs 92;j '

and decompose (3.33) formally as

au' + r(u)(du, du)a = 0,

1 < a < n,

we understand the characteristics of the equation for harmonic maps better. Note, however, that the above decomposition is invariant only in coordinate transformations of M, but not in those of N. Examples of harmonic maps appear in various problems in differential geometry. In what follows, we take up some of these examples. EXAMPLE 3.12 (Constant maps and identity maps). The

sun-

pleat example of a harmonic leap is a constant map. In fact, let u : M -* N be a constant map. Then there is a point q E N such that

u(x)=q, xEM.

The derivative du of u is identically 0; hence, the tension field of u is also 0. Especially, if M is compact, the energy E(u) of a constant map u becomes 0. The converse also holds true. Namely, the constant maps are nothing but the maps that give the absolute minimum value of the energy functional E : C°° (M, N) -; R. Now, if we assume u : M - M to be the identity map

u(x) = x, x E M, then dug : TAM -i TM is the identity map for each x E M. Hence, 7(u) = 0, and u is a harmonic map. EXAMPLE 3.13 (Harmonic functions). Assume (N, h) to be the one-dimensional Euclidean space E. Namely, E is the real line R with the Euclidean inner product. A C°° map from M into N is nothing

100

3. ENERGY OF MAPS AND HARMONIC MAPS

but a COO function u : M -p R. As was seen in Example 1.14, r= 0 in the equation (3.34) for harmonic maps. Thus, u being harmonic is equivalent to Au = 0 (Laplace equation) regarding the Laplace operator of Al. Consequently, a harmonic map

u : M R is nothing but a harmonic funchon in Al. Fbr a given function u : M -i R, the energy E(u) defined by (3.9) is called the Dirichlet integral of u. In general, it is well known as the Diriclrlet principle in analysis that the Laplace equation Lu = 0 is the Euler equation, namely, the first variation formula for the Dirichlet integral. In this context, we may consider the equation r(u) = 0 for harmonic maps as a generalization of the Laplace equation for functions in the case of maps between Riemannian manifolds. This is the reason why a map u with r(u) = 0 is called a harmonic map.

EXAMPLE 3.14 (Geodesics). Next, we assume (M, g) to be the one-dimensional Euclidean space E. In this setting, a C0° map fircm Al into N defines a C°'° curve u : J --s N. Considering t as a coor-

dinate function in R, we get i Jk = 0, as in Example 3.13. Hence, the equation (3.34) for harmonic maps reduces to the equation for geodesics in N

Consequently, a harmonic u : R -+ N is a geodesic in N, and t is nothing but an affine parameter for u. On the other hand, if we assume (M, g) to be the one-dimensional

unit sphere S1 C E2, a COO map u : St - N determines a smooth loop in N. A harmonic map u : Sl -+ N becomes a closed geodesic in N. The energy E(u) of u is precisely the energy of curves defined in §1.1. The equation for harmonic maps r(u) = 0, therefore, can be regarded as a generalization of the equation for geodesics in the case of maps between Riemannian manifolds. The next example is also best understood if it is viewed from the same aspect.

EXAMPLE 3.15 (Minimal submanifolds). Let u : M - N be a C' °° immersion of a Riemannian manifold (M,g) into another (N, h). As was seen in Example 1.2, theme is an induced metric u`h in M.

3.4. HARMONIC MAPS

107

When especially g = u'h, namely, at each x E M, gx(v, w) = hu(x)(dux(V), dux(w)),

v, w E TIM,

holds, we call u an isometric immersion from M into N. Given a C°° immersion u : Al -* N, we may define the normal vector bundle of M

TM-i = U dux(T2M)1 xEM

using an orthogonal decomposition

Tu(x)N = duI(TTM) @ duI(TM)l, x E M, of the tangent space Tu(I)N of N at u(x) with respect to k(x). Then, as seen easily from Lemma 3.3 and the definition (3.31) of the induced connection, we see that the second fundamental form

©duEI'(TM`®TM'(& u-'TN) of u defined in §3.2 coincides with the ordinary second fundamental form

AEF(TM*9, TM`0TM1) when M is regarded as a Riemannian submanifold of N. In fact, since

u is a local imbedding, by identifying X E r(TM) with du(X) E r(u-'TN) about each point x E M, (3.25) represents the identity (©'XY)(x) _ (V xY)(x) + A(X, Y)(x), x E M, expressing the decomposition of the covariant derivative V'XY into

the TM and TM' components. We see that the tension field r(u) of u coincides, up to a constant, with the mean curvature vector field H = trace A/m of the Riemannian submanifold M of N.

In general, when the mean curvature is identically 0, we call the Riemannian submanifold M a minimal submanifold of N. Consequently, an isometric immersion u : M -+ N being harmonic is equivalent to M being a minimal submanifold of N. In particular, if u; M -+ N is an isometric diffeomorphism, Vdu = 0; hence, u is a harmonic map (see Exercise 2.8 in Chapter 2). EXAMPLE 3.16 (Riemannian submersions). We consider a some-

what opposite situation to Example 3.15. In general, a C°° map u : M -+ N is called a submersion if the derivative du, : TxM --+ Ta(e) N is surjective at each x E M. It is clear from the definition that

m < n holds between the dimensions of M and N if there is a submersion from M onto N. Also the inverse function theorem implies

3. ENERGY OF MAPS AND HARMONIC MAPS

108

that u-1(u(x)) is an (m - n)-dimensional C°° submanifold of M for each x E M. We call u-i (u(x)) a fiber passing through x E M. Let u: M --+ N be a submersion from a Riemannian manifold (M, g) onto another (N, h). At each x E M, denote by V,: the tangent space to the fiber u-1(u(x)) at x. If we orthogonally decompose the tangent space T=M at x with respect gx, we have

T1M=V, (DHH, xEM. VS and H. are called, respectively, the vertical component and the horizontal component at x. In particular, if the restriction duz IHx : H. - Tu(r) N of the derivative du,, of u to H= at each x gives rise to an isometric isomorphism, u is called a Riemannian submersion. Given a vector field x E r(TN) over N, a vector field X E I'(TM) in M satisfying X E H=,

dux(. (x)) = X(u(x)), x E M

is called the horizontal lift of X.

Now let u : M -- N be a Riemannian submersion. Then the following proposition implies that u being harmonic is equivalent to each fiber u-1(u(x)) being a minimal submanifold of M.

PRoPoSmoN 3.17. Let u : M - N be a Riemannian submersion from a Riemannian manifold (M, g) onto another Riemannian manifold (N, h). Then a necessary and sufficient condition for u to be a harmonic map is that the fiber u-1(u (x)) passing through x for each x E M is minimal as a Riemannian submanifold of N. PROOF. Given x E M, denote by (yn) a local coordinate system

of N about u(x). Applying the Schmidt orthonormalization to the natural frame {8/8yP } determined by the coordinate system at each

point x E M, we get a set lei

I

1 < i < n} of C° orthonormal

vector fields around y. At each point y, {e'i(y) I 1 < i < n} forms an orthonormal base for T.N. Such a set {e'i} as defined above is called an orthonormal frame field. Denote by {e1i... , e,,} the horizontal lift of {e'1, ... , e,,), and e x t e n d it to a n orthonormal {ei , ... , en, eni1, ... , c. } around x. Then, from the definition and Lemma 3.3, the tension field r(u) of u is given by m

T(u)

M

Vd21(ei, ei) i=1

(T.,du(ez)

_ i,=1

- du(Ve1ei)}.

109

3.4. HARMONIC MAPS

Noting the definition of the induced connection 'V and Exercise 3.7 at the end of this chapter, we see that V.,du(ej) = V'e',e'i = du(Vepei)

holds for each 1 < i < n. For n + 1 < i < m, e -j

to the

vertical component at each point. Hence, 'Ve,du(ei) = 0 follows. Consequently, we obtain

t(u) _ - > du(Veiet) = -du

Veiei)

.

s=n+1

From this follows that a necessary and sufficient condition for r(u) = 0 is given by m

V,e, ei E Vz = (du=)-' (0),

x E M.

i=n+1

On the other hand, if we regard the fiber u-1(u(x)) passing through x as a Riemannian submanifold in M, we see that m

trace A(x) = the H. component of E Ve; ei (x) in+1 holds from the definition of the second fundamental form A. Consequently, we see that each fiber u' 1(u(0)) must be a minimal subman0 ifold of N in order for u to be a harmonic map.

Finally, we look into the case of complex manifolds. Here, we presume that the reader is acquainted with the definition and fundamental properties of Kahlerian manifolds (see Kobayashi 1261 for those). EXAMPLE 3.18 (Holomorphic maps). Let (M, g) and (N, h) be Ki hlerian manifolds. Denote by m and n the complex dimensions of M and N, respectively. Let (zi) = (z1, ... , zm) and (wi) = (w1, ... , wn) be local complex coordinates systems of M and N, respectively. Then the Kahler metrics g and h in M and N are expressed as

in

g=2

m

g,jdz`dzj, iJ=1

h=2

h,,,Rdzadz"0. a,j=1

If u : M - N denote a C° map from M into N, and if we express u with respect to the local coordinate systems as u(z) = (u1(z', ... , zm), ... , un(z1, ... , zn=)) _ W (z`)),

3. ENERGY OF MAPS AND HARMONIC MAPS

110

we can express the equation of harmonic maps (3.34) for each a as m

gt-

l

192u°i

aziazi

A

ra, A

aua

y

/

+ B,C

19UC r BC(U) suB azi az1

= 0.

Here, the index A runs through 11,... , m,1, ... , rn} and B, C run through { 1, ... , n, 1, ... , n}. 174 and r'BC are, respectively, the con-

nection coefficients of the Levi-Civita connections of V and V' with respect to the local complex coordinate systems.

Since M is a Kahlerian manifold, for any A, r = 0 holds. Sim ilarly, 'rBc = 0 holds for all T BC's other than I ,, since N is also a Ki hlerian manifold. Consequently, with respect the complex coordinate systems, the equation of harmonic maps is given by (3.35) in g..

=,i=1

82ua az=azi + ,y=1

a

OUP oy

u

az Ozj

= 0,

1:5 a!5 n.

Let J denote the almost complex structures in M and N. A C°° map u : M --+ N satisfying J o du = du o J is called a holomorphic map, and similarly, an antiholomorphic map if J o du = -du o J. From the definition, u being holomorphic is equivalent to each ua being a holomorphic function of (z'), namely, aua /az' = 0. Similarly, being antiholomorphic is equivalent to each uA being antiholomorphic,

namely, &a/az* = 0. Hence, we see from (3.35) that a holomorphic or antiholomorphic u : M - N from a Kahlerian manifold M into a Kahlerian manifold N is a harmonic map.

3.5. The second variation formula In this section we obtain the second variation formula regarding the variational problem of the energy of maps. As in the first variation formula in §3.4, let (M, g) be a compact m-dimensional Riemannian manifold, (N, h) an n-dimensional Rie-

mannian manifold, and I = (-e, e) an open interval. Assume that u E C°°(M, N) is a harmonic map from M into N. Let F = {ut}tEl be a COQ variation of u. First, we note the following. Given a variation vector field

dl V= dt _ ut(x) E F(u-'TN)

3.5. THE SECOND VARIATION FORMULA

111

of u, we have the covariant differential VV E I'(TM* (9 u-iTN) with respect to the induced connection in u-1 TN. Furthermore, if we consider the covariant derivative of VV with respect to the connection

V in TM` f& u-1 TN, we obtain VVV E r(TM* e TM' ®u-'TN) as the second order covariant derivative of V. If we express VVV, in local coordinate systems (x`) and (y)) in M and N, respectively, as

vvv = E E vivjv° dxi®dx'®iou, i,j=1 a,3=1

we obtain as the trace of V V V (3.36)

m

m

trace VVV = E E 9ijViVjVo "=1

0uE

r(U-1TN).

i,j=1

On the other hand, given a variation vector field V and the curvature tensor RN of N, if we set

(RN(V,du)du)(x)(v,w) = RN(V(r),du,.(v))du1(w), v,w E TIM, RN(V, du)du gives rise to a C°° section of the vector bundle TM* 0 TM* 0 u-1TN. From this, we define, as the trace of RN (V, du)du, trace RN (V, du)du (3.37)

M

q =jRN

V, du

( &Xi

du

OXJ

E r(u-'TN).

Under the above preparations, we get the following theorem. The idea of the proof for the second variation formula regarding the energy of maps is a sentially the same as in the case of the second variation formula for the energy of curves in Theorem 2.16.

THEOREM 3.19 (The second variation formula). Let u E C°°(M, N) be a harmonic map and let F = {ti }=e, be a C0° variation of U. Then r d2 _ = fM (V, trace (VVV + RN (V, du)du))dicg d 2 E(ua) t=o holds. Here, V and RN are, respectively, the variation vector of u and

the curvature tensor of N, and (, ) denotes the natural fiber metric in the induced bundle u-1T N. Also trace (V V V + RN (V, du)du) is the C°O section of u-1TN defined in (3.36) and (3.37).

3. ENERGY OF MAPS AND HARMONIC MAPS

112

PROOF. As in Theorem 3.8 (the first variation formula), we consider the vector bundle T (M x I) - & F-' T N and the connection V compatible with the natural fiber metric (, in it. First of all, from Theorem 3.8 and Lemma 3.9, we note that

dt

E(ut) _ -

,trace Qdut dµ9

J (

holds for each t E I. By differentiating this equation in t, we get

E(ut)

_-

(3.38)

trace Vdik dµ9

dt

Jrht

/plc out , trace Vdu: }dµ9 /`

(Vttrace Vduc )d

9.

!JJ

V6 represents the covariant differentiation with respect to V in the direction of (0, d/dt) E T(x,c) (M x I). Noting the definition of the curva-

ture tensor R' of N and ((0, d/dt), (8/8x', 0)] = 0, we see, regarding the covariant differentiation with respect to the induced connection

in F-1TN, that

V (O,d/d)V

((i.

(8f 0x1 ,o)du t

= Qi/& ,o)V (o,d/dt)d

+R" holds.

(dttt

a

((i,

0

((0)) ,du((o))) du`(o))

On the other hand, noting Lemma 3.7 and ((0, d/dt),

(8/8x`, 0)] = 0, we get 0(O,d/dt)d

((azi

((o))

=

((os

d ))

_

3.6. THE SECOND VARIATION FORMULA.

113

Hence, we get

V(o,d/dal V(8/e=d,o)dtct ((

_

7 r o/

,o)dtt ((0,d/dt))

+ RN dut

((o))

,

dut

((,0))) dut ((o)). 8x= 8xl

From this, we see readily that (3.39)

Vt trace Vdut = trace

(vv

dutl

+ RN

J

.

If we set t = 0 in (3.38), since u = u0 is a harmonic map, we get r(u) = trace Vdu = 0;

implying that the first term on the right hand side equals 0. On the

other hand, since 'v = att ! _ by definition, using (3.39), the second c-o

term of the right hand side can be rewritten as ,trace (VVV + RN(V, du)du))dµ9; hence, we obtain the desired result.

0

Theorem 3.19 implies that the second variation d2/dt2 E(ut)lt0 of the energy of the map is determined by the tensor product of the variation vector V along the harmonic map u and the curvature tensor RN of the Riemannian manifold N. As seen in §2.5, we were able to investigate the topological structures of Riemannian manifolds of positive curvature using the existence theorem for closed geodesics in compact Riemannian manifolds and the second variation formula for the energy E(c) of curves. In a similar manner, one can study the structures of Rienlannian manifolds using harmonic maps and the second variation formula for the energy of maps. For example, Micallef and Moore [13) recently proved "the topological sphere theorem" for the Riemannian manifolds of positive curvature under a pointwise pinching condition. Their proof utilizes the existing theorem for harmonic spheres regarding the second homotopy group Ira (M) by Sacks and Uhlenbeck [20] in combination with Morse theory on infinite-dimensional manifolds.

Siu and Yau [24], by applying a similar idea to Kir manifolds, solved the so-called the "Frankel conjecture". In their arguments, the

114

3. ENERGY OF MAPS AND HARMONIC MAPS

second variation formula has provided an important tool to measure stability of harmonic maps. For example, see Urakawa [31] for the relationships between the second variation formula and the stability of harmonic maps.

Summary 3.1 Definition of the energy E(u) of a C°° map it a Riemannian manifold M into another Riemannian manifold N. 3.2 A fiber metric and a compatible connection V are naturally defined on the vector bundle TM* ® u-1TN. Using this connection

V, the second fundamental form Vdu and the tension field T(u) _ trace Vdu of u are defined. 3.3 The first variation formula regarding the energy E(u) of a map u and a characterization of the harmonic maps as the critical points of the energy functional E. 3.4 The equation r(u) = 0 for the harmonic maps and examples of harmonic maps. 3.5 The second variation formula regarding the energy E(u) of

a map u.

Exercise 3.1 Let (M, g) be an m-dimensional Riemannian manifold, and let Co(M) denote the vector space of real valued continuous functions defined in M with compact support. Given a locally finite coordinate neighborhood systems {(Ua, 0a)}aEA, let {pa}aEA be a partition of unity subordinate to the open cover {Ua}aEA For a given f E Co(M), set (paf/det(g)) o 0a 1dZ.... d xx . Il9 = aEA

J

. (U-)

Here, (g a) is the matrix consisting of the components of g with respect to the local coordinate system (xa, ... , x,' n,). The integral on the right

hand side represents the Lebesgue integral of a continuous function with compact support defined in the open subset cbaUa in R" `. Then p r o v e that p (f) is uniquely determined independent of the choice of local coordinate systems {(Ua,la)}aEA and partitions of unity subordinate to {Ua }aEA and that µ9 gives a positive Radon measure in M. µ9 is called the standard measure induced from the Riemannian metric g in M. µ9 is also written by h, f dµ9, and called the integral of f over M.

no

EXERCISE

3.2 Let M and N be C1110 manifolds, and let u : M -- N a C°° map from M into N. Denote by u -1 TN the bundle over M induced by u from the tangent bundle TN of N. Given a C'O0 section Y E r(TN),

a C°° section Y a u E r(u-'TN). Prove that the linear connection v' on TN defines a unique linear connection 'V on u-'TN satisfying the following condition. Given a point v E TM and Y E r(TN), 'VvY 0 U

=

holds. 'V is called the induced connection by u in u'1TN from V. 3.3 Let (M, g) be an in-dimensional Riemaunian manifold, and denote by (xi) a local coordinate system of M in a local coordinate neighborhood U. Verify the following. (1) Given a C°° function f E C°° (M), a CoD vector field grad f =

(4f)# on M defined by

g$(grad f(x),v) = tVx(v), v E T.M is called the gradient of f. In U, grad f is given by

E (ii) M (M

g

Of

a=ii

$

(2) Given a C°° vector field X E I'(TM), a C"° function div X defined in M by

div X (x) = trace {v " V,X}, VETAM is called the divenjence of X. If ViXJ's represent the components of

the covariant differential VX of X = 9 1 Xj (x'), the following holds in U:

vixi -

div X = J=1

1

7

with respect to

8k

M FX det(9iJ ) k=1

(3) Given a C°° function f E C°°(M), a COD function A f defined

in M by

f = div grad f

is called the Laplacian of f. A is called the Laplace operator. If Vf V j f's denote the components of the type (0,2) C°Q tensor field

3. ENERGY OF MAPS AND HARMONIC MAPS

116

V ' determined by f with respect to (x3), m

m

&f = > 9'ipipjf = E m

V

4

ij=l

i.j=1

Ek

set

»i

a2 f g=i

axJ m

af

L.. 1 'i

k

k= 1

det(ggj) E 9 Of

k=1

k=1

holds in U.

3.4 (Green's Theorem) Let (M, g) be a compact Riemannian manifold. For any C°° vector field X E r(TM),

f div X f µ9 = 0 holds.

3.5 Let (M, g) be an m,-dimensional Riemannian manifold. Denote by V and V* the Levi-Civita connection of TM and the dual connection in TM* determined by V, respectively. Verify the following-

(1) Given a type (r, s) C°° tensor field T E r(TT'(M)), define the covariant differential VT E r(T,-+1(M)) of T as a generalization of (3.17) by

VT(X, X1, ... , XS, wl, ... ,

X ' T(X1:...

,

X81 w1 I

... ,

tttr)

$

,VxX,,... ,X87w1,... ,wr) i=1

r

- 1: T(X1i...

,

X5, w1, ...

,

wr).

j=1

Given X E r(TM), the covariant derivative VxT of T by X defined by

VTX(Xl, ... , XS, wl, ... , wr) = VT(X, X1, ... , X8, w1, ... , wr)

satisfies the same computational rules as the covariant derivative VxY of vector fields. It also holds that

Vx(T0U) =VxT0U+T®VxU for T E r(T$ (M)) and U E r(T9 (M)).

EXERCISE

117

(2) With respect to a local coordinate system (xi) in M, express respectively. the components of T and VT by T 11:; and VkT Then VkTJ1...J' _ a Tjl...j,

axk ,l. $

;,...ib

a

m

r

Tjl......

il...j.is +

rj6

L1 kt

il...j..-jr i1...{s

b=11=1

e=1 1=1

holds.

3.6 Let (M, g) be an rn-dimensional Riemannian manifold and let R E I'(T3 (M)) be the curvature tensor of M. Then the covariant differential VR E I'(T4 (M)) of R satisfies VR(X, Y, Z, V, w) + VR(Y, Z, X, V, w) + VR(Z,X,Y, V, W) = 0.

In other words, if we express it in a local coordinate system (xi) as m 1:

VR =

VR;kldxt ®dx'

dxk (& dx!

5X 7;'

i,j,k,j,r-1

the following (the second Bianchi identity) holds:

VjRjkj + V j Rrj +

0.

3.7 Let u : M -+ N be a Riemannian submersion from a Riemannian manifold (M, g) into another Riemannian manifold (N, h). Denote by V and V' the Levi-Civita connections of M and N, respectively. Let X, k E F(TM) be the horizontal lifts of X, Y E T(TN). Verify the following:

(1) g(X(z),Y(z)) = h.( )(X(lL(x)),Y(u(x))), z E M. (2) du(]X,Y]) = [X,Y]

(3) VY = (V'xYY +

2[X,Y]

Here, (V'XY) is the horizontal lift of V'XY and [X, Y]1 is the vertical component of [X, k] . 3.8 Regard R2 = C by identifying (x, y) with z = x + uE C. Express the unit three-dimensional sphere S3 C E4 by S3

=

I(z1,z2) E C2 I IziI2 + 122

12

= 11.

Define the Hopf map 46: S3 - S2 by z2) = (2ziz2, Iz1I2 - Iz2I2) E C x R, (zl, z2) E S3. Show that 0 is a harmonic map with respect to the Riemannian metrics on S3 and S2 induced from the standard Euclidean metric.

118

3. ENERGY OF MAPS AND HARMONIC MAPS

3.9 Given an immersion W : M -+ N from a Riemannian manifold (M, g) into another Riemannian manifold (N, h), cp is said to be conformal if cp*h = e2Ag for some C°° function p E COO(M). Assume

that M is of dimension 2. If yo : M - M is a conformal diffeomorphism, then E(u o cp) = E(V) for any C' map u E C°°(M, N), and furthermore, cp is a harmonic map.

3.10 Let (M, g) be a compact two-dimensional Riemannian manifold, and let (N, h) be an n-dimensional Riemannian manifold. Given a COO imbedding u : M -- N, the area of u is defined to be A(u) = IM dAu-h.

Prove that the inequality A(u) < E(u) holds, and that the equality holds if and only if u is conformal.

CHAPTER 4

Existence of Harmonic Maps In this chapter, we consider the existence problem of harmonic maps between compact Riemannian manifolds. Regarded as a generalization of the existence problem of closed geodesics discussed in Chapter 2, whether or not a given map can be deformed to a harmonic map may be ranked as one of the most fundamental questions among geometric variational problems. There is an effective technique called the heat flow method for deforming a given map to a harmonic map. In this chapter, we first explain the approach of the heat flow method. Then, using this method, we prove that any continuous map from a compact Riemannian manifold into a compact Riemannian manifold with nonpositive curvature can be free-homotopically deformed to a harmonic map. This theorem was first proved by FAls and Sampson in 1964. We give, in this chapter, a simpler proof than the original using the inverse function theorem in Banach spaces.

4.1. The heat flow method Let (M, g) and (N, h) be compact Riemannian manifolds of dimension in and n, respectively. Let f E C'30 (M, N) denote a C°° map from M into N. As one might guess, from the examples of harmonic maps seen in §3.4, whether or not f can be continuously deformed to a harmonic map u : M -* N from M into N is a fundamental problem in the study of harmonic maps. For example, in the case where (M, g) is a one-dimensional unit sphere S' C R2, a harmonic map u : S1 -+ N is a closed geodesic in N; hence, the problem is nothing but whether or not a given smooth loop can be deformed continuously

to a closed geodesic in N. In this case, as seen in Theorem 2.23, it is well known that any f E C'00 (S', N) can be deformed to either a constant map or a closed geodesic u : S' -+ N free-homotopic to f. 119

120

4. EXISTENCE OF HARMONIC MAPS

The objective in this chapter is to study the existence of harmonic maps between general compact Riemannian manifolds, and then to prove the following theorem due to Fells and Sampson.

THEOREM 4.1 (Fells-Sampson). Let (M, g) and (N, h) be compact Riemannian manifolds. Assume that (N, h) is of nonpositive

curvature. Then for any f E CO°(M, N), there is a harmonic map : M -- N free-homotopic to f.

u

Here that f a n d 4

are free-homotopic means that there exists

a continuous map u: M x [0,1] - N satisfying u(x, 0) = f (x),

u(x, l) = u,,,,(x),

x E M.

Also N being of nonpositive curvature implies that at each point y E N, the sectional curvature K(o) < 0 for each two-dimensional subspace o C T.N. In what follows, we simply denote it by KN < 0. Unlike the existence theorem (Theorem 2.23) of closed geodesics, the condition on the sectional curvature in Theorem 4.1 is necessary. In fact, an arbitrary f E C°° (M, N) is not always free-homotopic to a harmonic map, unless KN < 0 holds everywhere in N. For example,

Eells and Woods [2] show that any map f : T2 - S2 of mapping degree ±1 from the two-dimensional torus into the two-dimensional sphere is not free-homotopic to a harmonic map regardless of the Riemannian metrics g, h in T2 and S2, respectively. Namely, it is known that there is no harmonic map it : T2 - S2 of mapping degree ±1. In the case of the existence theorem of closed geodesics, as was seen in the proof of Theorem 2.23, we were able to apply the direct method in the variational techniques. Namely, we introduced a functional L which measures the length L(c) of piecewise smooth loops and directly constructed a closed geodesic regarded as a critical point of L from a minimal sequence. However, when M is a more general compact Riemannian manifold other than S, an application of the direct method to the energy functional C°° - R inevitably encounters certain difficulties. Indeed, the reason for the difficulties is that the equation for harmonic maps is essentially a system of nonlinear partial differential equations, as opposed to that the defining equation for geodesics is a system of linear ordinary equations in the tangent bundle TN of N. However, Bells and Sampson were successful in proving Theorem 4.1 using a technique called the heat flow method modeled after Morse theory on infinite-dimensional manifolds.

4.1. THE HEAT FLOW METHOD

121

The key points of the central idea may be described as follows. First, we review the first variation formula of the energy functional E as given in §3.3. Given a C°° map u E C°° (M, N), denote a C° variation of u by F = {ut}tEJ, I = (-e, e), and let

= ut E r(u-'TN) t=o

be the variation vector field. The first variation of E(ut) is given by (4.2)

d E(ut)

(V, r(u))dta9. t=o

M

Here, r(u) is the tension field of u, (, ) is the natural fiber metric in the induced vector bundle u-'TN, and pg is the standard measure in M determined by g. If we regard M = C°° (M, N) as a manifold by ignoring details, the energy functional E : C°° (M, N) - R can also be regarded as a function defined in M. Since the variation F = {ut}tEl of u can then be regarded as determining a curve in M, the variation vector field V E r(u-'TN) defined (4.1) represents nothing but the tangent

vector of this curve F at t = 0. As was seen in §3.3, for a given V E r(u-'TN), a C°° variation F of u was defined; consequently, r(u-'TN) may be regarded as representing the tangent space TuM of M at u E M. Hence, for given W1, W2 E r(u`'TN), ((W1, W2))

=

JM

(WI, W2)dp9

would define an inner product ((, )) in the tangent space Tu,M.

On the other hand, since the first variation d E(ut)t-o of E(ut) is considered to define the derivative dE.(V) of the function E on M in the direction of the tangent vector V, the first variation formula (4.2) can be expressed as

dEu(V) = -((r(u), V)). As is readily seen from the definition of the gradient (see Exercise 3.3 in Chapter 3), this implies that the tension field r(u) of u, indeed, is

nothing but the gradient vector of the functional -E at u; namely, we see that

r(u) = -(grad E)(u).

122

4. EXISTENCE OF HARMONIC MAPS

Consequently, a harmonic map u, which is a critical point of the energy functional, is precisely a singular point (a zero point) of the gradient vector field grad E of E.

rR

FIGURE 4.1. T(u) = -(grad E)(u) Analogous to Morse theory on the finite-dimensional manifolds (see Foundations of Morse theory, Iwanami Shoten, Gendai Sugaku no Kiso [27] ), we may say that the function E in M decreases most efficiently in the direction of -grad E, namely, the tension field r(u). Consequently, one may attempt to deform a given map uo = f E C°° (Al, N) along the flow determined by the tension field r(u) in M as a method to obtain a harmonic map free-homotopic to uo. If, indeed, a deformation as above is possible, its flow ut will be given as a solution to the equation

(4.3)t

= r(ut)

The equation (4.3) is a system of nonlinear parabolic partial dif-

ferential equations, and is nothing but the equation to obtain the integral curves of the tension field r(u) regarded as a vector field in M. Analogous to the so-called classical heat equation (see Appendix §A.2(c)), the method of deforming uo along the solution of (4.3) is, in general, called the heat flow method. Consequently, the existence problem of harmonic maps is reduced to whether or not the deformation of uo along ut reaches a critical point r(um) = 0 of the energy

4.1. THE HEAT FLOW METHOD

123

functional E. Keeping the above in mind, we consider, for a map u : M x [0, T) --+ N, the following initial value problem of a system of nonlinear parabolic partial differential equations (4.4)

f8 (x, t) = r(u(x, t)),

(x, t) E M x (0, T),

tu(xO) = f (x). Here, T > 0 and f E COO (M, N) is a map given as the initial condition. Also we assume that it is continuous in M x 10, T) and is C°° in M x (0, T); namely, u E Ce(M x 10, T), N) n C"- (M x (0, T), N).

A map u satisfying (4.4) is called a solution to the initial value problem (4.4). The system of nonlinear parabolic partial differential equations in (4.4) is called the parabolic equation for harmonic maps. In order to prove Theorem 4.1, given the initial value problem (4.4) of the parabolic equation of harmonic maps, we must show the following:

(1) For any initial value f, (4.4) possesses a solution u : M x Mx10,oo). (2) Set ut (x) = u(x, t) and t -r oo. Then ut converges to a harmonic map u,,, : M - N, and f = uQ and u,,,, are free-homotopic to each other.

In this section, we first assume that (1) holds, and discuss the statement in (2). Let (x¢) and (ye) denote local coordinate systems in M and N, respectively. For a given solution u to (4.4), set ut (x) u(x, t) and define m

E

n

o

e(ut) = 2IdutI2 = 2 4,j=1 a _1

E(ut) =

r

rc(ut) = 2

K(h) = J

e(uu)dµ9,

out

2

=

1

a4

axi 8

,'(ut)dp9.

Here, gtj and h,,,o are, respectively, the components of the Rim metrics g and h in the coordinate systems (xi) and (y'), and gij represents the components of the inverse matrix of (g{?) . We also

4. EXISTENCE OF HARMONIC MAPS

124

express the map u in the local coordinate systems by / /(x', u(2, t) = (u1(x1, ... , m, t), ...'U . , un n (xI, ... , x" , t)) (u

=

t)).

As is clearly seen from the definitions, E(ut) is the energy of each uc E C'°° (M, N), and K(ut) is the kinetic energy of the deformation determined by ut. For the energy density e(ut) and the kinetic energy density rs(ut) of each ut, we have the following formula called the Weitzenbock formula. This Weitzenbock formula is a fundamental equation satisfied by the solutions of the initial value problem (4.4), and it plays an important role in the arguments that follow.

PROPOSMON 4.2. Let u E CO(Mx[O,T),N)nC'°(Mx(O,T),N) be a solution to the parabolic equation for harmonic maps (4.4), and let ut (x) = u(x, t). We have, in M x (0, T), (Weitzenbock formula for e(ut)) (1)

ee

)

=®e(ut)

- IVVut)2

in

m

0. The solution as above is called a local time-dependent solution of (4.13).

For the purpose of discussing the existence of the local timedependent solutions, we rewrite the parabolic equation (4.13) for harmonic maps in a form that is analytically more desirable. To this end, we use the Nash imbedding theorem [151 which shows that an arbitrary compact Riemannian manifold can be isometrically imbedded in Euclidean space of sufficiently high dimension. In other words, we may assume, without loss of generality, that the Riemannian manifold (N, h) is realized as a submanifold of the q-dimensional Euclidean space Rq for a sufficiently large natural number q, and that the Riemannian metric h is nothing but the induced metric from Rq. Let denote such an isometric imbedding, and let N be a tubular neighborhood of the submanifolds t(N) C Rq in Rq. Namely, for a sufficiently small e > 0, N is an open subset (see Exercise 4.3 at the end of this chapter) of Rq defined by

N= {(x, v) I x E t(N), v E Tt(N)l, lvi < F}.

4. EXISTENCE OF HARMONIC MAPS

132

In the tubular neighborhood N, let

7r.N-,t(N) denote the projection; namely, 7r is the map that assigns to each z E N

the closest point in t(N) from z.

Let u : M x [0, T) --i N be a map from M x (0, T) into N c. R9. Regarding u as a R9 valued function, we consider the following initial value problem for the system of parabolic partial differential equations: (4.14)

(&-a) u(x, t) = fI(u)(du, du)(x, t),

(x, t) E M x (0, T ),

u(x, 0) = 10 f (x).

Here, A is the Laplace operator of M and f is the map given as the initial condition of (4.13). fI(u)(du, du) is a vector in R9, and is defined as follows. Let (zA) be the standard coordinate system of Rq and let (x`) be a local coordinate system in M. With respect to them, we express ir(u) and u(x, t), respectively, as 7r(z) = (7r1(zi, ... , z9), .

. . ,

7r9(z1, . . .

,

z4)) = (7rA(zB)),

u(x, t) = (ul (xi, ... , xm, 01. . , u9(xl, ... , xm, t)) = (uA(xi, t)) .

Then the components of H(u) (du, du) are given by m

9

sC j

7rA

auB 7uC

g 8zB8zc (u) ax¢ iJ=1 B,C=1

T.?,

1 < A < q.

As is readily seen from Lemma 3.4, if we denote the second fundamental form of the map it by Vdu, we get (4.15)

II(u) (du, du) = trace Vdir(du, du).

Among the solutions to the initial value problem (4.14), we consider those u : M x [0, T) -i N which satisfy

u E C°(M x [0,T), N) n C2,1(M x (0,T),1V); namely, those which are continuous maps from M x [0, T) into N, and

are, furthermore, of C2 in M and of C' in (0, T). The initial value problems (4.14) and (4.13) are related to each other in the following way.

4.2. EXISTENCE OF LOCAL TIME-DEPENDENT SOLUTIONS

133

PROPOSrri0N 4.6. Let u E Co (M x [0, T), N) nC2" 1(M x (0, T), N).

If u is a solution to the initial value problem (4.14), u(M x [0, T)) C t(N) holds, and u is a solution to the initial value problem (4.13). The converse also holds true.

PRooF. Suppose that u E C°(Mx[0,T),N)nC2,I(Mx(O,T), is a solution of the initial value problem (4.14). First, we verify that u(M x [0, T)) c t(N) holds. To this end, we define a map p : N -+ R4 by

p(z)=z-ar(z), zEN, and a function p:Mx[0,T)-'Rby V(x, t) = l p(u(x, t))12,

(x, t) E M x [0, T).

From the definition of 7r, p(z) = 0 is equivalent to z E t(N). Hence, it suffices to see p(z, t) - 0. Since u(x, 0) = t o f (x) E t(N), we see cp(x, 0) = 0. Also since u is a solution to (4.14), we get at _ at (p(u), p(u)) = 2 dp

\ at

,

p(u))

= 2(dp(Liu -11(u)(du, du)), p(u)), Acp = L (p(u), p(u)) = 2(Op(u), p(u)) + 21Vp(u)I2,

where (, ) is the inner product in R. On the other hand, from a formula for the second fundamental form of composite maps (see Exercise 4.4 at the end of this chapter), we have that

Ap(u) = dp(Lu) + trace Vdp(du, du). Since r(z) + p(z) = z holds from the definition, we have that dir + dp is the identity map and Vdar + Vdp = 0. Noting these together with that the images of dir and p are orthogonal to each other, we get

tap = 2(dp(du) - trace Vdir(du, du), p(u)) + 2IVp(u)12 = 2(dp(Lu - II(du, du)), p(u)) + 21Vp(u) 12. Consequently, we have 00

at

= o - 21Vp(u)I2

'

4. EXISTENCE OF HARMONIC MAPS

134

Green's theorem yields, for each t E (0, T), jM co(, -t)dp9

=

f(, t)dp9

-2

JM

I VP(u) I2dµg < 0.

Hence, we have JM (, t)dug

fM p(., t)dp9 =

implying that cp(x, t) - 0.

Next, we verify the the second half of the assertion. To this

end, let u M x [0, T) --> N be a map from M x [0, T) into N, and set it = t o u. We must show that u is a solution to the initial is a solution to the value problem (4.13), if it : M x [0, T) --+ t initial value problem (4.14). From the definitions, it = t o u and t = 7r o t. Hence, from the formula for the second fundamental form of composition maps, we get Au = trace ©dt(du, du) + dt(r(u) ), trace Vdt = trace Vd7r(dt, dt) + dir (trace Vdt). Since t : N -t IlS9 is an isometric imbedding, noting that trace Vdt is orthogonal to t(N) at each point, we get do (trace Wt.) = 0. These equations yield

dt(r(u)) = L

- traceVd7r(du, du). it

On the other hand, since dt (d at dt

r(u) -

=

au

, we have

) = (f_)u_H(u)(dudu).

As a result, if it is a solution to the initial value problem (4.14), u becomes a solution to the initial value problem (4.13). It can be easily verified that the converse also holds. From Proposition 4.6, we see that we can get a time-dependent local solution to the initial value problem (4.13) by constructing a timedependent local solution to the initial value problem (4.14). Since the equation in the initial value problem is a system of parabolic differential equations with regard to the vector valued function, it is relatively

easy to set up a function space in which existence of solutions is to be discussed. In what follows, we construct a time-dependent local solution to the initial value problem (4.14).

4.2. EXISTENCE OF LOCAL TIME-DEPENDENT SOLUTIONS

135

Following Ladyzenskava. Solonnikov and Ural'eeva [9, p. 71. we set lip a function space in which the existence of solution is treated as follows. Given T > 0, set Q = Al x [0. Tj. Let 0 < a < 1. Given a vector valued function u, Q - R`t. set

IuIQ = sup

(x.t)EQ

(u)("")

=

I41(x. t)I.

I u(.r, t) - u(.r'. t))

sup

')°

(

x#x' (u) (a)

=

lu(a . t)

sup

-

t')I

It - PI",

(x.t).(r.t')EQ

tot'

and define the norms

IUI(

.a,2) =IuIQ IZ61(2+a.1+a/2) =IuIQ

(

4 1 8) .

by

IuIQ

Q

+ (u)(°) + (,)(n/2) + Iat-uIQ + IDxuIQ + IDruIQ (8t'tl)ta/2)

+

+ (Dxa)(1/2+°/2) + (Dru)t"/2)

+ (Otu);.() +

Here, d(x, x') is the distance between .r and in A!. and Otu represents c u/O3t. Also, Dxu and Dxu represent the first order derivative of u in Al and its covariant derivative, respectively. In terms of a local coordinate system (:ri) in Al and the coordinate functions (;y°) in Rq, Du and Dxu are, respectively, defined by

Dxu=du=

q

au"

i=1 a=1

of

d.r i ®

i3

Vy

Dxu=Odu=

oya

i.j=1 cr=1

and IDrulq and ID,2.uI2 are, respectively. given as m 2

D.xuIQ

sup

q

9j

a

u ct

i

(x.t)EAI :.j=1

Ox

q

I D=uI2 =(x.t)EA1 sup

gikyltt7i0jtt"p plu i.j.k.l=1 a=1

4. EXISTENCE OF HARMONIC MAPS

136

With respect to these norms, we define the function spaces Ca'«/2(Q, Rq) and C2+a,1+a/2 (Q,), respectively, by Ca'a/2(Q,Rq) = {u E C°(M x [0, 71) 1 ,C2+a,I+a/2(Q,R9) = {u E C2,1(M x [0,T1) I

IuIQ'

2)

< oa},

ful'+a,1+al2)

< oo},

and set

C2+a,i+a/2(Q, N) = {u E

Ca'a/2(Q, Rq) and

C2+a,1+a/2(Q,

Rq) I u(Q) C N}.

C2+a,1+a/2(Q.

Rq) are Banach spaces with norms respectively. It is easy to verify that Q 4 C2+«,1+«/2(Q, Iq) C,2+a,1+«/2(Q, N) is a closed subset of (see Exercise 4.6 at the end of this chapter). Ca'a/2(Q,1 ) and C'+a}1+a/2(.Q, Rq) are called a Holder space on Q = M x [0, T]. We now prove the following. IuI(«,a/2)'

IuI(2+a,l+a/2),

THEOREM 4.7. Let (M, g) and (N, h) be compact Riemannian manifolds. For any C2+a map f E C2+4(M, N), there exist a positive C2+a,1+a/2(M x [0, el,1V) such number e = e(M, N, f, a) > 0 and u E that u is a solution in M x [0, e) to the initial value problem (4.14). Here, c = e(M, N, f, a) is a constant dependent upon M. N, f and a.

We prove this theorem using the inverse function theorem (see Appendix §A.2(a)) in Bausch spaces. The idea of the inverse function theorem is to reduce solvability of a nonlinear differential equation to solvability of a linearized equation. First, we review the results regarding existence and uniqueness for linear parabolic partial differential equations.

THEOREM 4.8. Let (M, ,g) be a compact Riemannian manifold, and set Q = M x [0, 7']. Given a vector valued function u : Q - R9, let

Lu= A+a be a parabolic partial differential operator, and consider an initial value problem (4.17)

= F(x, t), u(x, 0) = Ax) {Lu(x,t)

(x, t) E M x (0, T),

4.2. EXISTENCE OF LOCAL TIME-DEPENDENT SOLUTIONS

137

Here, the components of Au, a Vu, b u, at u are, respectively, defined

q mg

by

f u',

E aB (x, t) axt

> bB(x, t)uB,

,

B=1

B=1 -=1

If

aB ,

1 < m, 1 < A, B < q,

bB E C 'a/2(Q, JR),

for some 0 < a < 1, then, for any F E Ca'a/2(Q,IR ), f E C2+a(M,Rq), C2+a,l+a/2(Q,R) to (4.17) there exists a unique solution u E F E such that iul(2+a,1+a/2) < Q

holds.

0 such that C2+a',1+a'/2(Q, R) satisfying the following there exists a unique z E IkIQ'a,/2 C'',a'12 with k(x, 0) = 0 and conditions. For any k E < 6, z satisfies P(z)k,

(4.20)

8t(z, 0) = 0.

z(x, 0) = 0,

Here, b = S(M, N, f) is a positive number determined by M, N and f. Now if we set u = v + z and w = P(v), from (4.20), we see that there exists a u E C2+a'.1+« /2(Q, R) satisfying

P(u)(x, t) = (w + k)(x, t), u(x, 0) = f W.

(4.21)

(x, t) E M x (0,1),

Step 3 (Existence of time-dependent local solutions). In order to see the existence of a desired local time-dependent solution, for a given positive number e, consider a C°° function ( : R - R satisfying ((t) = 1 (t < e), ((t) = 0(t > 2e), 0 < C(t) < 1, IK'(t)I 2/e (t E R). We note that w = P(v) E Ca,a/2(Q, R9) C C°',4'/2(Q, $4) and that w(x, 0) = 0 holds from the definition of P(v) and v(x, 0) = f. We can verify through a simple calculation (see Exercise 4.7 at the end of this chapter) that there is a constant C > 0, independent of a and w, such that (4.22)

Ibwl(a',a'/2) < Q

Cc(a-a')/21u,1(a,a/2)

Q

holds.

Set k = -(w. Then K(x, 0) = 0. From (4.22), we have Ikt(',a'/2)

< b for a sufficiently small e. Consequently, there exists a uE x [0, e], R4) such that the following special case of (4.21) holds: C2+a',1+a'/2(M

J P(u)(x, t) = 0, lu(x, 0) = AX).

(x, t) E M x (0, e),

4. EXISTENCE OF HARMONIC MAPS

140

Namely, we have obtained a solution u E to the initial value problem

(A - O)u(z, t) = II(u)(du, du)(z, t), lu(x, o) = AX).

C2+a'.1+a'/2(M

x (0, E], lRq)

(z, t) E M x (0, E),

Since we have

f E C21, (M' V),

n(uu)(du, du.) E Ca,c,12(M x [0, e], IQ),

we see by Theorem 4.8 that uE

C2+a,l+a/2(M

x [01 E],ff4).

Since u(M x [0, e']) C J for a sufficiently small positive number E' (0 < e < E), u is a solution to the initial value problem (4.14) in M x (0, tj. Applying Proposition 4.6, we see that u is a solution to (4.14) in M x [0,,E] . It is also clear from the above proof that E > 0 is a positive number depending on M, N, f and a alone. 0 From Theorem 4.7 and Proposition 4.6, the following clearly follows.

COROLLARY 4.9. Let (M, g) and (N, h) be compact Riemannian manifolds. For a given C2+a maP f E C2+a(M, N),

them exist a positive number T = T (M, N, f, a)

U E C2+a,1+a/2 (M X [0, T], N) such that

ON' (z, t) = r(u(x, 0),

> 0 and

(x, t) E M x (0, T),

I ae(x, 0) = AX) holds. Here, T = T(M, N, f, a) is a constant dependent on M, N, f, a atone.

Noting the result regarding differentiability on the solutions to a linear parabolic partial differential equation, we obtain the Mowing result on existence of time-dependent local solutions to an initial value problem for the parabolic equation for harmonic maps. THEOREM 4.10 (Existence of time-dependent local solutions). Let

(M, g) and (N, h) be compact Riemannian manifolds. For a given

C2+a map f E C2+-(M, N), them exist a positive number

'13. EXISTENCE OF GLOBAL TIME-DEPENDENT SOLUTIONS

T = T(M, N, f,a) > 0 and u E

C2+a,1+a/2(M

141

x f0,T1,N) fl

C° (M x (0, T), N) such that 5 N(x, t) = r (u(x, t)),

(x, t) E M x (O,T),

lu(x,0) = f(x) holds. Here, T = T (M, N, f, a) is a constant dependent on M, N, f, a alone.

PROOF. Let u E C2+a,1+a/2(M x [0,T],N) be the solution in Corollary 4.9. We only have to verify the differentiability of u about each point (x, t) E M x (0, T). As in the proof of Theorem 4.5, denote

by (x') and (y) the local coordinate systems about x and u(x, t), respectively. With respect to these local coordinate systems, the parabolic equation for harmonic maps is expressed for each u° = ya o u as

)'U'

m

n

Y

=,j=1 C1+a,a/2 Noting that the right hand side is from the assumption on u, we see that the theorem (see Appendix §A.2(d)) regarding differentiability on solutions to linear parabolic partial differential equaC3+a,1+°/2. tions implies that u is of This yields that the right hand C4+p,1+a12 side is of C2+-.1+a/2. Then we, in turn, see that u is of Iterating this argument gives us that u is of C°° about each point. 0

4.3. Existence of global time-dependent solutions As was seen in §4.1, in order to prove Theorem 4.1 using the heat flow method, it was necessary to show that the initial value problem of the parabolic equation for harmonic maps (4.23)

J(x,t) = r (u(x, t) ),

(x, t) E M x (0, T ),

lu(x,0) = f(x)

had a solution u : M x [0, oo) - N when T = oo. We call such a solution in M x [0, oo) as above a global time-dependent solution to (4.23). As seen in Theorem 4.10, a local time-dependent solution to (4.23) always exists. However, the parabolic equation for harmonic maps is a system of nonlinear partial differential equations; hence, existence of a global solution is not always guaranteed. In fact, to show the existence of a global solution, it becomes crucial to estimate

4. EXISTENCE OF HARMONIC MAPS

142

the growth rate of the solution u(x, t) in time t. In order to control the effect of the nonlinear terms of the equation, the curvature of the Riemannian manifold N plays an important role. In this section, we investigate the relationships between the existence of global timedependent solutions and the curvatures of M and N.

In what follows, let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. First of all, we discuss the maximal principle for the beat equation as a tool to estimate the growth rate of the solutions to (4.23). Let 0 be the Laplace operator

of M, and Let L = A following holds.

5i

be the heat operator. We verify that the

LEMMA 4.11. Let U E C°(M x [0,T))nC23 (M x (0,T)) be a deal valued function in M x 10, T), which is C2 in M and C' in (0, T). If

u satisfies Lu > 0 in M x (0, T), then

max it = max u

Mx(O,TJ

MX (0)

holds; namely, the maximum value of it in M x [0, T) is attained at a point in M x {0}. PROOF. Let e1, c2 > 0 be positive numbers and set

u(x,t)=u(x,t)-Eit,

Q = M x [0,T-f21.

Regarding the maximum value of ta, we have (4.24)

a u = Mmax} u.

In fact, since u is a continuous function in Q, it attains the maximum

value at a point (x°, t°) in Q. we must show t° = 0. We suppose t° > 0, and induce a contradiction. Since Lu > 0 in M x (0, t) from

-

the assumption, u satisfies at (x°, t°)

0

for ut(x) = u(x, t). Hence, we get the desired result from Lemma

0

4.11.

Propositions 4.12 and 4.13 imply that the growth rate of a solution u to the initial value problem (4.23) is uniformly bounded with respect to time, if N is of nonpositive curvature KN < 0. Namely, we get the following. be

PROPOSITION 4.14. Let u E C2"(Mx [0, T),N)nC°°(Mx(0, T),N) a solution to (4.32). If N is of nonpositive curvature

KN < 0, then, for any 0 < a < 1, there exists a positive number

C=C(M,N,f,a)>0such that

1 u(., t)I c2+a(M,jv) + 1-,j9ui ICt(M,N)
0 at a point z E M, then u is a constant map. (3) If N is of negative curvature KN < 0, u is a constant map or the image of u coincides with the image of a closed geodesic of N. PROOF. (1) From Green's theorem, we have

IM

L e(u)dpg = 0.

Hence, the integral on the right hand side of (4.30) is also 0. On the other hand, each term on the right hand side is nonnegative from the

assumptions .ii"' _ 0 and KN 0 holds, the derivative du. of u at x must equal 0. This implies that e(u)(x) = 0. Since e(u) is a constant, we get e(u) . 0. Hence, u is a constant map. (3) Since Q(du) = 0, we have, with respect to an orthonormal base {e{} for the tangent space T,,M at x E M, (RN (du(e;),du(ep))du(e,),du(e;)) = 0,

1 < i, j < m.

On the other hand, since we have KN < 0 from the assumption, the sectional curvature K(o) of any two-dimensional subspace

4.5. APPLICATIONS TO RIEMANNIAN GEOMETRY

157

o C TTM of T,,M is negative. Hence, du(e;) and du(ej) are never linearly independent. This implies at each x E M that d(x) = dim du,, (TTM) < 1.

M, e(u) 0; hence, u is a constant map. Otherwise, d(ay) = 1. Noting that u is totally geodesic, we readily we that the image of u coincides with the image of a closed geodesic in

Now if d(z) = 0 at an x

0

N.

In Chapter 2, we saw that the fundamental group irl (M) of a Riemannian manifold M of positive curvature is a finite group. In other words, if M is of positive curvature Km > 0, its fundamental group is a small group. On the other hand, the fundamental group of a compact Riemannian manifold of negative curvature is known to be a large group. Here, as an application of harmonic maps, we verify that a nontrivial commutative subgroup of the fundamental group 71 (M) of M with negative curvature is an infinite cyclic group. Namely, the following holds. THIEOREM 4.25 (Preissmann). Let (M, g) be a compact and connected Riemannian manifold, and assume that the sectional curvature KM of M always satisfies Ks,t < 0. Then any nontrivial commutative subgroup of the fundamental group ir1(M) of M is an infinite cyclic group.

PROOF. Let Tl (M, xo) be the fundamental group of M with base xo E M. Furthermore, assume that the two elements a, b of 1r1 (M)

are commutative; namely, ab = ba holds. By the definition, a, b are the homotopy classes of loops with xa as the base point. We express loops representing a, b by the same symbols a, b. Since they are commutative, there is a homotopy f : [0,1] x [0,11 -+, M between loops a - b : [0,1) -+ M and b a : [0, 1] -+ M. Here, this homotopy keeps the base point fixed throughout the deformation from a - b = f 0) to b a = f 1). In other words, noting ,f (0, s) = f (1, s),

s E [0,11,

we readily see from Figure 4.3 that f defines a continuous map T2 - M from a two-dimensional torus T2 into M. Applying Corollary 4.18 to j, we see that f can be deformed free-homotopically to a harmonic map u : T2 -- M. In this case, the loops a - b and b - a are also free-homotopically deformed. The base point is not necessarily fixed throughout this deformation, but we

4. EXISTENCE OF HARMONIC MAPS

158 1

h

a

a

A6

0

a

a

b

FIGURE 4.3

point out that the loops corresponding to a b and b a have the same base point in each stage of the deformation. Since M is of negative curvature KN < 0 from the assumption, Proposition 4.24 implies that either u is a constant map or the image u(T2) of u coincides with the image of a closed geodesic c passing through a point x1 E M in M. Consequently, if u is not a constant map, the loops corresponding to a - b and b a both cover c multiple times in the fundamental group iri (M, x1) of M based at x1. Thus, a b and b a are contained in the infinite cyclic subgroup generated by c. This implies that a b and b - a are both contained in an infinite cyclic subgroup of iri (M, xa). From the above arguments, we see readily that any nontrivial commutative

subgroup of the fundamental group rri(M,XO) of M is an infinite cyclic group.

0

In a like manner using the uniqueness of the harmonic maps, one can show that the set of all isometric transformations of a compact Riemannian manifold of negative curvature forms a small group. In general, it is well known that the set of all isometric transformations of a compact Riemannian manifold M of negative curvature is a Lie group. In particular, if KAJ < 0, this group is a finite group. Namely, the following holds. THEOREM 4.26. Let (All, g) be a connected and compact Riemann-

ian manifold. Furthermore, assume that the sectional curvature KM of M always satisfies K11 < 0. Then the group G of the isometric transformations of M is a finite group.

PROOF. First, we note that an isometric transformation of M homotopic to the identity map is the identity map if K11.t < 0. In fact, let f be an isometric transformation homotopic to the identity

4.5. APPLICATIONS TO RIEMANNIAN GEOMETRY

159

map. Since f is a harmonic map, Theorem 4.22 due to Hartman implies that it is the identity map. From this readily follows that G is discrete. Since G is compact, it is finite.

0

If we note that a holomorphic map between Kahlerian manifolds is harmonic, we get the following. TrEoItzM 4.27. A complex submanifold of a KaJelerian manifold is a minimal submanifold. PROOF. Since M is a complex submanifold of a Kahlereian manifold N, there is an analytic imbedding u : M , N. M is a Kihlerian manifold with the induced metric. Then u becomes a harmonic map from M into N. On the other hand, as seen in Example 3.15, an isometric embedding being harmonic is equivalent to M being a minimal submanifold of N. This gives the desired conclusion. 0

The existence problem of analytic maps between complex manifolds is an important research topic in complex analysis. Especially since analytic maps between Kiihlerian manifolds are harmonic, we

can study the existence problem from the viewpoint of harmonic maps. Indeed, much has been done in this respect. Fbr example, as an application of the theorem of Eells and Sampson, Siu [23] proved the following in 1980. THEOREM 4.28 (Siu). Let N be a Keihleri,an manifold obtained as a quotient manifold of an irreducible bounded symmetric domain. Let N be compact and at least two complez dimensional. Furthermore,

assume that M and N are of the same homptopy type. Then M is biholomophic or anti-biholomorphic to N.

This theorem asserts that the complex structure on a compact Kiihlerian manifold N obtained as a quotient manifold of an irreducible bounded symmetric domain is determined by the homotopy type of the manifold, except for the complex one-dimensional case. It is called the strong rigidity theorem of such Kahleriau manifolds as above. The essential parts of the proof consist of an improvement of the Witzenbock formula for Kahlerian manifolds and the existence

theorem of harmonic maps due to Eells and Sampson. If M and N have the same homotopy type, there is a homptopy equivalence map from M into N. The desired biholomorphic or antibiholomorphic map is obtained by deforming this homotopy equivalence map to a harmonic map. This strong rigidity theorem of Siu is one of the most successful applications of the theory of harmonic maps.

4. EXISTENCE OF HARMONIC MAPS

160

Summary 4.1 The heat flow method and its idea to obtain the critical points of the energy functional e. 4.2 Existence of time-dependent local solutions to the initial value problems of the parabolic equation for harmonic maps. The relationships between the growth rate of solutions and the curvature, and the role of the Weitzenock formula for the estimation. 4.3 Existence of time-dependent global solutions to initial value problems of the parabolic equation for harmonic maps and their convergence to harmonic maps. 4.4 Due to the Eells-Sampson theorem, any continuous map from a compact Riemannian manifold into a compact Riemannianl manifold of nonpositive curvature are free-homotopically deformed to harmonic maps. 4.5 A theorem of Hartman regarding the uniqueness of harmonic maps. A theorem of Preissmann regarding the fundamental group of Riemannian manifolds of negative curvature.

Exercises 4.1

Let (M, g) and (N, h) be Riemannian manifolds of dimen-

sion m and n, respectively. Denote by u E CO° (M, N) a COD map from

M into N. Consider the tensor product TM' 0 u-'TN of the cotangent bundle TM* of M and the induced bundle u 1TN by u from the tangent bundle of N. Denote by V the connection in TM' ® u`1TN compatible with its natural fiber metric (, ). Prove the following (1) Given T E r(TM'(&u-1TN), its second covariant differential

VVT E r(TM' 0 TM' 0 TM' 0 u-TN) is a tensor field of type (0,3) which takes its value in the induced bundle u-1 N. Furthermore,

VVT(x, Y, z) = (V x(V yT))(Z)

- (Vv,ryT)(Z)

holds for x, Y, z e r(TM). (2) Denote by V the connection in TM* 0 u-'TN and by 'V the induced connection in u-'TN. Set RV(X, Y) = VxVy - DY©x - V[X,YJ

R'v(X,Y)

QXVY -I V, VX -, ©(X.YJ.

Also denote by RM, R'' the curvature tensor of M, N, respectively.

EXERCISES

161

For T E F(TAI* (& u-1TN) and X, Y, Z E r(TAI ),

(RV (X,Y)T)(Z) = R'V (X,Y)(T(Z)) - T(R I (X.Y)Z) holds.

(3) With respect to local coordinate systems (x'), (y°), express T, VVT by m

n

?,'dx' (9)

T

i=1 a=1

M

o u,

n

vtvjTk dx' ®drj 0 dxk ®aa o u.

VVT = i,j,k=1 a=1

Then

vtv1Tk - vjVi k =

-

aa0au7 T

RA1ijkT1` + 1=1

RNa.-,.6 x= AX-0 v,i',a=1

holds. Here, R11 jk, R' ,.,,6 are the components of RAr, RN with respect to (x'), (y°). This identity is called the Ricci identity. 4.2 Let (M, g) and (N, h) be Riemannian manifolds of dimension m and n, respectively. Let u : M x [0, T) - N be a solution to the parabolic equation for harmonic maps

,jT(x,t)=,r(u(x,t)),

(x, t) E AI x (0, T).

Set ut(S) = u(x, t). Prove the Weitzenbock formula alC ut)

ar

=

at

"'

2

IV2!Lt I

+

/RN (du(e2)t)

,du(et)}

I2 Here, RN is the curvature tensor of N, and {et} represents an orthonormal base for the tangent space TM at each f o r c(ut) = 11

at

x E Al.

4.3 Let Al denote a compact submanifold of a Riemannian manifold N. Denote by U an open subset of the normal bundle TAI1 of AI consisting of all the normal vectors whose magnitudes are less than e. Show that, for sufficiently small f > 0, the map

exp:U -N obtained by restricting the exponential map to U gives rise to a differential diffeomorphism from U onto the submanifold U = exp(U) of N. This U is called a tubular neighborhood of AI in N.

4. EXISTENCE OF HARMONIC MAPS

162

Let M1, M2, M3 be Riemannian manifolds, and let f, : M1 - M2 and f2: M2 M3 be C°° maps. Regarding the second fundamental form Vd(f2 o fl) and the tension field r(f2 o fl) of the composition f2 o fl,

4.4

Vd(f2 o fi) _ Vdf2(df1, df1) + df2(Vdf1), r(f2 o f1) = trace Vdf2(dfi,df1) + df2(r(f1))

Regarding a COO map u : M - N between R.iemannian manifolds M and N, show that the following are equivalent: (i) The second fundamental form of u satisfies Vdu = 0. (ii) u maps a geodesic c in M onto a geodesic u o c. The aline parameter of c also represents an affine parameter of u o c. Based on these, we call u totally geodesic if Vdu = 0.

hold. 4.5

4.6

Let M be a compact Riemannian manifold and let 0
- o and IIx11= o = x = o;

(i) (ii)

IlAxll = INIIxII,

()

A E R;

Ilx + vll -< Ilxll + IIvII

is given to V, the pair (V, 11 11) of V and II IIr When a norm 11 or simply V, is called a normed space. A normed space V becomes a metric space under the distance p(x, y) = I}x - yll. Hence, we can naturally introduce the notion of convergence in V by defining xn -- x if Hxn - xll = Q. A sequence {x,, } of elements in V is called a Cauchy 11

sequence if Ilxn - Xm 11

0 as rn, n -' ©a. If any Cauchy sequence

APPENDIX A

174

{x,a} converges, i.e., there is an element x E V such that x,, - x, we call V complete. A complete nonmed space V is called a Banach space. A Banach space V is called a Hilbert space, if, to each pair

x, y E V and a real A E R. there corresponds a real number (x, y) satisfying the following conditions: (i)

(x, y) = (y, x);

(ii)

(A1x1 + A2x2, y) = al (X 1, Y) + 1\2(x2. V);

(iii)

IIXI12 = (x, x).

In a Hilbert space, the Schwarz inequality I (x, y) < IIx1111 yll holds.

Let V and W be Banach spaces. A linear operator is a linear map T from a subspace D(T) of V, called the domain of T, into W. A linear operator T is called a continuous operator if Tx. -* Tx as x,, --+ x. T is called a bounded operator if IITxll is bounded for 11x!I < 1. When D(T) = V, being continuous is equivalent to being bounded. In this case, T is called a bounded linear operator. Given a bounded linear operator T, the (operator) norm II T II of T is a real number IITII defined by 11T11 = sup{IITxII Ix E V, IIx1l R9 defined in an open subset of R' is called a linear partial differential operator if it can be expressed as

as(x)D'.

P(x, D) =

(A.3)

101 oo as t - 0, we get the desired conclusion: -m/2 4c1(M)-1t

f

M

(H(x, y, t)

- I/V)2dµg(y) 0, set Q = SZ x (0, T). For a function set a =u )x

{

sup

Iu(x, t) - u(x', t) I I t - t' I

(x,t),(i ,t)EQ x,6x'

sup (u} = (x,t),(x,t')EQ

(u (z, t) - u(x, t`)I

It - t/Ia

t#t' +a'1+a/2) are defined as (4.16) in ChapThe norms IUIQ'a/2) and IUI( Q ter 4. If we define function spaces Ca,a/2(Q), C2+a,1+a/2(Q) C2+a,l+a/2(Q) and Ca'a/2(Q),

as given in §A.2(b) with respect to these norms, the following holds.

THEOREM A.6. (1) Given 0 < a < 1, assume as', b', d E Ca(t ) C2+a,1+a/2 holds, if u E C2"l satisfies and f E Ca'e/2(Q). Then u E the following linear parabolic partial differential equations

(

-

) u(x, t) = .f (x, t).

(2) Let p, q be nonnegativve integers. Given 0, n with I!01 < p,

q, assume Dxaii, Djbt, Dgd E Ca(fl) and DpDt f E Ca,a/2(Q). Then the solutions u to (1) satisfy DgDt u E Ca'0/2 for any 0, n with 101 + 2K < p + 2, K < q + 1. In particular, ail, b, d E C°°(0) and f E C°°(Q) imply that u E C°°(Q). IQI+2K C K, K

Regarding the results mentioned above, see Murata-Kurata [29] and Gilbarg Trudinger [3].

(e)

Schauder estimate. We give here a quick review on the

Schauder estimate of solutions to linear elliptic and parabolic partial differential equations. The estimate was used in §4.3.

Given r>0,setB(0,r)={xERtI IxI