Lectures on Closed Geodesics

W. Klingenberg (Autor)

Buch | Softcover

XI, 230 Seiten

2011 | 1. Softcover reprint of the original 1st ed. 1978
Springer Berlin (Verlag)
978-3-642-61883-3 (ISBN)

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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.

1. The Hilbert Manifold of Closed Curves.- 1.1 Hilbert Manifolds.- 1.2 The Manifold of Closed Curves.- 1.3 Riemannian Metric and Energy Integral of the Manifold of Closed Curves.- 1.4 The Condition (C) of Palais and Smale and its Consequences.- 2. The Morse-Lusternik-Schnirelmann Theory on the Manifold of Closed Curves.- 2.1 The Lusternik-Schnirelmann Theory on ?M.- 2.2 The Space of Unparameterized Closed Curves.- 2.3 Closed Geodesics on Spheres.- 2.4 Morse Theory on ?M.- 2.5 The Morse Complex.- 3. The Geodesic Flow.- 3.1 Hamiltonian Systems.- 3.2 The Index Theorem for Closed Geodesics.- 3.3 Properties of the Poincaré Map.- 3.3 Appendix. The Birkhoff-Lewis Fixed Point Theorem. By J. Moser.- 4. On the Existence of Many Closed Geodesics.- 4.1 Critical Points in ?M and the Theorem of Fet.- 4.2 The Theorem of Gromoll-Meyer.- 4.3 The Existence of Infinitely Many Closed Geodesics.- 4.3 Appendix. The Minimal Model for the Rational Homotopy Type of ?M. By J. Sacks.- 4.4 Some Generic Existence Theorems.- 5. Miscellaneous Results.- 5.1 The Theorem of the Three Closed Geodesics.- 5.2 Some Special Manifolds of Elliptic Type.- 5.3 Geodesics on Manifolds of Hyperbolic and Parabolic Type.- Appendix. The Theorem of Lusternik and Schnirelmann.- A.2 Closed Curves without Self-intersections on the 2-sphere.- A.3 The Theorem of Lusternik and Schnirelmann.

Erscheint lt. Verlag	13.10.2011
Reihe/Serie	Grundlehren der mathematischen Wissenschaften
Zusatzinfo	XI, 230 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	381 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Curvature • Geschlossene Geodätische • manifold • Riemannian manifold • riemannian manifolds • Riemannsche Mannigfaltigkeit
ISBN-10	3-642-61883-9 / 3642618839
ISBN-13	978-3-642-61883-3 / 9783642618833
Zustand	Neuware