Lectures on Closed Geodesics

237 Pages·1978·6.396 MB·English

by  Wilhelm Klingenberg (auth.)

Grundlehren der
mathematischen Wissenschaften 230
A Series of Comprehensive Studies in Mathematics
Editors
S. S. Chem J. L. Doob J. Douglas, jr.
A. Grothendieck E. Heinz F. Hirzebruch E. Hopf
S. Mac Lane W. Magnus M. M. Postnikov
W. Schmidt D. S. Scott
K. Stein J. Tits B.L. van der Waerden
Managing Editors
B. Eckmann J.K. Moser
Wilhelm Klingenberg
Lectures on
Closed Geodesics
Springer-Verlag
Berlin Heidelberg New York 1978
Wilhelm Klingenberg
Mathematisches Institut der Universitiit Bonn, D-5300 Bonn
AMS Subject Classifications (1970):
Primary: 49 C 05, 49 F 15, 53 C 20, 55 D 35, 58 B 20, 58 E 05, 5H E 10, 58 F 05,58 F 20
Secondary: 34 C 25, 49 B 05,53 B 20, 55 C 30, 55 E 05,57 A 20, 58 D 15, 58 F 15
ISBN-13: 978-3-642-61883-3 e-ISBN-13: 978-3-642-61881-9
DOI: 10.1007/978-3-642-61881-9
Library of Congress Cataloging in Publication Data. Klingenberg, Wilhelm, 1924-. Lectures on closed geodesics
(Grundlehren der mathematischen Wissenschaften; 230). Bihliography: p. IncJudes index. 1. Riemannian manifolds.
2. Curves an surfaces. 1. Tîtle.Il. SerÎes: Die Grundlehren der mathematischen Wissenschaften m Einzeldarstellun
gen: 230. QA649.K54. 516'.362.77-13147.
This work is subject to copyright. All rights are reserved, whcthcr thc whole Of part of the material 15 concerned.
~pecifically those of translation, reprinting, re-use of iIIustrations, hroadcasting, reproduction by photocopying
machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are
made for other than private llse. a fee is payable to the publisher, the amount of the fee tu be determÎned by
agreement with the publisher.
<..IJ by Springer-Verlag Berlin Heidelberg 1978
Softcover reprint ofthe hardco\'er J st edition 1978
Typesettingand printing: Osear Brandstetter Druckerei KG, Wiesbaden. Binding: Konrad Triltsch, Wilrzburg.
2141/3140-543210
To the memory of my father
Preface
The question of existence of c10sed geodesics on a Riemannian manifold and the
properties of the corresponding periodic orbits in the geodesic flow has been the
object of intensive investigations since the beginning of global differential geo
metry during the last century.
The simplest case occurs for c10sed surfaces of negative curvature. Here, the
fundamental group is very large and, as shown by Hadamard [Had] in 1898,
every non-null homotopic c10sed curve can be deformed into a c10sed curve
having minimallength in its free homotopy c1ass. This minimal curve is, up to the
parameterization, uniquely determined and represents a c10sed geodesic.
The question of existence of a c10sed geodesic on a simply connected c10sed
surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this
problem has much in common with the problem ofthe existence of periodic orbits
in the restricted three body problem. Poincare [l.c.] outlined a proof that on an
analytic convex surface which does not differ too much from the standard sphere
there always exists at least one c10sed geodesic of elliptic type, i. e., the corres
ponding periodic orbit in the geodesic flow is infinitesimally stable.
During the following three decades, Birkhoff and Morse, on the one hand, and
Lusternik and Schnirelmann, on the other, developed powerful new methods to
prove the existence of one or possibly several c10sed geodesics on arbitrary
Riemannian manifolds. The principal objects of these studies have always been
Riemannian manifolds, for which the underlying differentiable manifold is a
sphere. Take as the earliest example lacobi's description [la] of the geodesics on
an ellipsoid, in his Lectures on Dynamics in Konigsberg, 1842/43. A later example
is Morse's monograph [Mor 2] from 1934 where, among other things, he put
much effort into an attempt to compute the Z2-Betti numbers of the space of
unparameterized c10sed curves on the n-sphere.
There were, however, few results on the existence of more than one c10sed
geodesic on such manifolds. Probably the most remarkable one is the theorem
due to Lusternik and Schnirelmann [LS 1] in 1929. They showed that on a simply
connected compact surface there exist at least three c10sed geodesics without
self-intersections. Ellipsoids which are not too dissimilar from a round sphere
provide examples of surfaces with exactly three such c10sed geodesics. Note that
there are always infinitely many other prime c10sed geodesics on an ellipsoid; but
they will generally have se1f-intersections.
In 1951, Lyusternik and Fet [LF] proved that at least one c10sed geodesic
exists on every compact Riemannian manifold. Since the 1960's the theory of
c10sed geodesics has been advanced considerably by the work of Alber, Svarc,
VIII Preface
Fet, Gromoll and Meyer, Eliasson and the author. In hindsight it tumed out to be
of particular importance when the author indicated in 1965 [KI 3] how to
reformulate Morse's theory of the space of closed curves on a Riemannian
manifold within the framework of Morse theory on Hilbert manifolds, which had
been established in 1964 by Palais and Smale [PS].
A striking application of this new approach was made in 1969 by Gromoll
and Meyer [GM 2]. They proved that, for every Riemannian metric on a compact
simply connected differentiable manifold M, there exist infinitely many closed
geodesics, provided that the sequence of the rational Betti numbers of the space
AM of parameterized closed curves is unbounded.
As was shown recently by Vigue and Sullivan [VSJ, this hypothesis on the
rational Betti numbers is satisfied if and only if the rational cohomology ring of
the manifold M is not a truncated polynomial ring. Thus, the Gromoll-Meyer
theorem does not apply to the case of Riemannian metrics on the sphere, i.e. it
does not apply to the case which has been the principal object of study in the
theory of closed geodesics during most of its earlier history.
In order to obtain resuIts on the existence of infinitely many closed geo
desics without any hypothesis conceming the topology of the underlying
differentiable manifold, the author has employed the theory of Hamiltonian
systems, and here, in particular, Poincare's and Birkhoffs investigations of the
Hamiltonian f10w near a periodic orbit. (Note that periodic orbits of the geo
desics f10w are in 1 : 1 correspondence with the closed geodesics.) The outcome
was the proof that, generically, there exist infinitely many prime closed geo
desics on every compact Riemannian manifold with finite fundamental group,
cf. [KI 9], [KI 16].
In the course of our work on the present manuscript we succeeded in solving
one of the outstanding problems of the whole theory by proving that there
always - not only generically - exist infinitely many prime closed geodesics on a
compact Riemannian manifold with finite fundamental group. This re suit is
new, even for convex surfaces - the case which was investigated by Poincare
at the very beginning of the theory of closed geodesics more than seven decades
ago.
The proof of this resuIt uses in an essential way the fuU structure of the Hilbert
manifold of closed curves and the associated Morse complex. Only a minor part
is played by the geodesic f1ow. It is to be expected, however, that future work on
generic density properties of period orbits in particular will employ the Poincare
Birkhoff theory of Hamiltonian systems to a greater extent.
The aim of these lectures is to present a reasonably complete, exhaustive and
seIf-contained exposition ofthe methods and results reIating to closed geodesics.
In Chapter 1 we give a concise description of the Hilbert manifold AM of
closed curves on a Riemannian manifold M. AM and its structure are derived
from M functoriaUy and no information is lost by going from M to AM.
Chapter 2 develops the Morse-Lusternik-Schnirelmann theory on M. The
canonical operations of S and of 71 are explained. We conclude with a
2
description of the Morse complex of AM which will play a crucial role later ono
In Chapter 3 we study the geodesic f10w on the tangent bundle TM of a
Riemannian manifold M. Besides offering a rather complete presentation of the
Preface IX
index theorem for closed geodesics we are mainly interested in generic properties
of the geodesic flow.
Chapter 4 is devoted to the principal results of the whole theory. After a
proof of Fet's theorem, which involves the structure of the neighborhood of an
unparameterized closed geodesic, we give a modified proof of a slightly extended
version of the Gromoll-Meyer theorem. This theorem and Sullivan's theory of
the minimal model of AM are then combined to show that there always exist
infinitely many prime closed geodesics on M, if 7t1 M is finite. The crucial step in
the proof is a lemma relating multiplicities of certain pairs of c10sed geodesics.
We conclude the chapter with some generic existence theorems.
Chapter 5 begins with a detailed proof of the theorem of three closed geodesics
- a long sought-after generalization of the classical result of Lusternik and
Schnirelmann on closed surfaces of the type of S2 to arbitrary simply connected
manifolds. This is followed by miscellaneous results on manifolds of elliptic type,
i. e. manifolds with finite fundamental group. The last paragraph is devoted to
manifolds of hyperbolic type, such as those with negative sectional curvature,
these manifolds having a very large fundamental group.
In the Appendix we prove the Lusternik-Schnirelmann theorem on the
o.
existence of three closed geodesics on a surface of genus The proof uses the
Lusternik-Schnirelmann approach to the closed geodesics problems, which is
much more direct and elementary than Morse's approach in its refined version
presented in these lectures. But then, the Morse theory of the Riemannian Hilbert
manifold AM with the functional E is more powerful - witness the results
of Chapter 4.
We have made the Appendix completely independent of the rest of these
lectures. By filling in ali the details and avoiding anything but the most basic facts
of 2-dimensional Riemannian geometry and algebraic topology we present a
proof oftwo classical theorems (besides the Lusternik-Schnirelmann Theorem we
also prove the theorem of Lyusternik and Fet) in such a form as to fit into any
course on Riemannian geometry in general.
These lectures grew out of courses which I have given in various places during
the last ten years. I want to mention in particular Bonn, 1966/67 and 1972/73,
Berkeley, 1971 and 1974, Bombay, 1972, and Minneapolis, 1975.
Here I wish to thank all those who enabled me, through generous invitations,
to lecture and do research at various places and who, at the same time, through
their comments, criticism and encouragement, helped to bring these lectures into
their present form.
1 am indebted to Wolfgang Ziller for his careful and critical reading of a major
part of the manuscript and to Gudlaugur Thorbergsson for helping me in proof
reading. For their tireless efforts in typing the various versions of the manuscript
1 want to thank Ruthie in Berkeley, Kathy in Minneapolis and - most of all -
Cornelia in Bonn.
After long years of labor, I can only hope that my efforts will encourage a
younger generation of mathematicians ta take up where I left off and continue the
attack on the numerous still unsolved problems in the theory of closed geodesics.
Bonn, December 1977 Wilhelm Klingenberg
Contents
Chapter 1. The Hilbert Manifold of Closed Curves . 1
1.1 Hilbert Manifolds . . . . . . . . . . . . . 1
1.2 The Manifold of Closed Curves . . . . . . . 7
1.3 Riemannian Metric and Energy Integral of the Manifold of Closed Curves 15
1.4 The Condition (C) of Palais and Smale and its Consequences. . . . . 22
Chapter 2. The Morse-Lusternik-Schnirelmann Theory on the Manifold of
Closed Curves . . . . . . . . . . . . . . 32
2.1 The Lusternik-Schnirelmann Theory on AM . 32
2.2 The Space ofUnparameterized Closed Curves 40
2.3 Closed Geodesics on Spheres. 46
2.4 Morse Theory on AM. . . 55
2.5 The Morse Complex . . . 65
Chapter 3. The Geodesic Flow 77
3.1 Hamiltonian Systems . . . 77
3.2 The Index Theorem for Closed Geodesics 86
3.3 Properties of the Poincare Map. . . . . 100
3.3 Appendix. The Birkhoff-Lewis Fixed Point Theorem. By J. Moser. 115
Chapter 4. an the Existence of Many Closed Geodesics . 122
4.1 Critical Points in nM and the Theorem of Fet . . 122
4.2 The Theorem of Gromoll-Meyer . . . . . . . . . 132
4.3 The Existence of Infinitely Many Closed Geodesics . 142
4.3 Appendix. The Minimal Model for the Rational Homotopy Type of AM.
By J. Sacks . . . . . . . . . . 156
4.4 Some Generic Existence Theorems . . . . 161
Chapter 5. Miscellaneous Results . . . . . . 167
5.1 The Theorem ofthe Three Closed Geodesics 167
5.2 Some Special Manifolds of Elliptic Type. . 177
5.3 Geodesics on Manifolds of Hyperbolic and Parabolic Type . 188
Appendix. The Theorem of Lusternik and Schnirelmann . . 203
A.1 The Space PM and the Theorem of Lyusternik and Fet . 204
A.2 Closed Curves without Self-intersections on the 2-sphere. 208
A.3 The Theorem of Lusternik and Schnirelmann 210
Bibliography 219
Index . . . . . . . . . . . . . . . . . . . 225
Chapter 1. The Hilbert Manifold of Closed Curves
In this chapter we shall define the Hilbert manifold AM of H1-maps of the
parameterized circle S into a compact Riemannian manifold M.
We will show that AM carries an intrinsically defined Riemannian metric
and a differentiable function E - the energy integral-such that grad E satisfies
the condition (C) of Pa lais and Smale.
This condition allows an extension ofthe classical Morse-Lyusternik-Schnirel
mann theory of a differentiable function on a Euclidean (i. e. finite dimensional)
manifold to such a function on a Hilbert manifold.
1.1 Hilbert Manifolds
A Hilbert manifold M is a topological space with a countable base endowed with
a differentiable atlas where the charts have their image in a fixed separa bie Hilbert
space INI. We also say briefly that M is modeled after INI.
Just as for a Euclidean manifold M there is associated to M the tangent
bundle of M:
an atlas (</>" U')'EA of M gives rise to the atlas (T</>., TU')'EA of TM just as in
the Euclidean case.
We also consider, for a given manifold M, vector bundles
over M with the fibre lE being a Banach space.
That is to say, we have an atlas for M such that, for each chart (</>, U) of that
atlas, a local representation (cP, </>, U) is given, i.e. a commutative diagram
_ cP
7t 1 (U) --> </>(U) x lE
U
2 Chapter 1. The Hilbert Manifold of Closed Curves
of morphisms with the foIIowing properties:
(i) the restriction ifJ of ifJ to the fibre E =n-1(p) over PEU is a topological
p p
linear isomorphism; and
(ii) if (ifJ, ljJ, U) and (ifJ', ljJ', U') are local representations, then the map
ifJ' o ifJ-1 : U ('\ U' --+ L(lE;lE)
is differentiable.
For details see [La], [El 3] and [FK].
If (x, OE ljJ( U) x lE is a local representation of an element of E then we caII
~ the principal part of this representation.
A local representation (ifJ, ljJ, U) of n : E--+ M determines a local representation
(TifJ, ifJ, n-1(U») of the tangent bundle 'CE: TE--+E
Tn-1(U) ~ ljJ(U) xlExlM x lE
If (x, OEljJ (U)xlE then we denote the elements in pr1:~ (x, O by
(x, ~,y, I1)E{(X, O} xlMxlE.
Of fundamental importance is the concept ofa connection in a bundle n : E--+ M.
This is a map
K: TE--+E
such that for a local representation (ifJ, ljJ, U) of Ethere exists a differentiable map
r", : ljJ (U)--+L(lM,lE;lE)
such that the local representative K", : = ifJ o K o TcfJ -1 of K
K",: ljJ(U)xlE xlMx lE--+ljJ(U) xlE
is given by
(x, ~,y, 11)/-+(X, '1 + r ",(x)(y, O).
The r", are called Christoffel symbols.
Note that K is a vector bundle morphism from 'CE into n over n, i.e. the fol
lowing diagram is commutative: