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Closed geodesics on Riemannian manifolds. (English) Zbl 0539.53003

Regional Conference Series in Mathematics 53. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0703-X). iii, 79 p. (1983).
The booklet is published as a lecture note of a regional conference on the theory of closed geodesics held at Univ. of Florida in August 1982. As one of the founder of this field, the author mainly gives an exposition of the (equivariant) Morse theory on the space of closed curves in the first half of the booklet. It consists of three chapters: Chapter I introduces the structure of a Hilbert manifold and the energy function E on the space of \(H^ 1\) curves in a Riemannian manifold M. Chapter II deduces the condition (C) for E and for the submanifold \(\Lambda\) M of closed curves in the Hilbert manifold and Chapter III investigates E up to second order on neighborhoods of critical points. Morse theory thus introduced on the pair (\(\Lambda\) M, E) is similar to the ordinary infinite dimensional Morse theory except for the natural orientation reversing action of \(Z_ 2\) and the starting point shifting action of \(S^ 1\) on M which yields critical submanifolds in general, because E is invariant under the action. The appendix to Chapter III treats these actions. The latter half of the booklet is devoted to apply the framework to the enumeration and existence problems of closed geodesics. In Chapter IV the pinching theorem is treated, (originally introduced by the author himself) and some other topics obtained by A. Lyusternik are presented. In Chapter V, the existence problem of infinitely many closed geodesics is treated somewhat intuitively. One could expect forthcoming elaboration on this topic too.
Reviewer: V.Shikata

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C22 Geodesics in global differential geometry
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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