Lectures on Closed Geodesics

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Springer Science & Business Media, Jan 1, 1978 - Mathematics - 227 pages
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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.
  

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Contents

The Hilbert Manifold of Closed Curves
1
12 The Manifold of Closed Curves
7
13 Riemannian Metric and Energy Integral of the Manifold of Closed Curves
15
14 The Condition C of Palais and Smale and its Consequences
22
The MorseLusternikSchnirelmann Theory on the Manifold of Closed Curves
32
22 The Space of Unparameterized Closed Curves
40
23 Closed Geodesics on Spheres
46
24 Morse Theory on AM
55
42 The Theorem of GromollMeyer
132
43 The Existence of Infinitely Many Closed Geodesics
142
The Minimal Model for the Rational Homotopy Type of AM
156
44 Some Generic Existence Theorems
161
Miscellaneous Results
167
52 Some Special Manifolds of Elliptic Type
177
53 Geodesics on Manifolds of Hyperbolic and Parabolic Type
188
The Theorem of Lusternik and Schnirelmann
203

25 The Morse Complex
65
The Geodesic Flow
77
32 The Index Theorem for Closed Geodesics
86
33 Properties of the Poincare Map
100
The BirkhoffLewis Fixed Point Theorem By Jurgen Moser
115
On the Existence of Many Closed Geodesies
122
A1 The Space PM and the Theorem of Lyusternik and Fet
204
A2 Closed Curves without Selfintersections on the 2sphere
208
A3 The Theorem of Lusternik and Schnirelmann
210
Bibliography
219
Index
225
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