Author | Klingenberg, Wilhelm. author |
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Title | Lectures on Closed Geodesics [electronic resource] / by Wilhelm Klingenberg |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1978 |
Connect to | http://dx.doi.org/10.1007/978-3-642-61881-9 |
Descript | XI, 230 p. online resource |
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 Poincarรฉ 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