Lectures on Closed Geodesics by W. Klingenberg

By W. Klingenberg

The query of life of c10sed geodesics on a Riemannian manifold and the houses of the corresponding periodic orbits within the geodesic movement has been the article of in depth investigations because the starting of worldwide differential geo­ metry over the last century. the best case happens for c10sed surfaces of detrimental curvature. the following, the elemental team is huge and, as proven by way of Hadamard [Had] in 1898, each non-null homotopic c10sed curve will be deformed right into a c10sed curve having minimallength in its loose homotopy c1ass. This minimum curve is, as much as the parameterization, uniquely decided and represents a c10sed geodesic. The query of life of a c10sed geodesic on a easily hooked up c10sed floor is far more challenging. As mentioned via Poincare [po 1] in 1905, this challenge has a lot in universal with the matter ofthe life of periodic orbits within the constrained 3 physique challenge. Poincare [l.c.] defined an explanation that on an analytic convex floor which doesn't range an excessive amount of from the traditional sphere there consistently exists not less than one c10sed geodesic of elliptic kind, i. e., the corres­ ponding periodic orbit within the geodesic circulate is infinitesimally stable.

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Corresponding to every conjugacy class of. 1 of the fundamental group 11:1M of M we have a connected component A' of A=AM which does not contain the set AO ofpoint curves. Claim. The function E assumes its infimum K' on A', and E- 1(K')nA' consists of closed geodesics. Proof Recall that the set A' is a ,p-family. Its critical value K' is positive since a curve c with E(c), and hence L(c) (the length of c), sufficiently small will be null homotopic. If cEE- 1(K')nA' were not a critical point of E, then IIgrad E(c)111 >0 and E(,psC)

3) shall we be able to prove the existence of an infinite number of prime closed geodesics by bringing into play some hitherto unobserved features for the Morse theory on AM. Cf. 1) for the existence of 2n-1 "short" closed geodesics. e. the space AS" of parameterized circles on S". In order to define AS" we assume S" to be embedded in the standard way as the hypersurface of constant curvature equal to 1 in IRn +1 . A cirele on S" is either a point curve or the intersection of S" with a 2-plane of IRn+l having distance < 1 from the origin.

Hence K~K'. Now assume that K=K'. Put CrAnE-1(K)=K. For any open neighborhood OIJ of K in A - AO, there exists UEW with lui c OIJuA K - , cf. 2). We can choose a subdivision of the singular simplices a of u, which is so fine that either lai c OIJ or lalcA"-. e. since AM is locally contractible, CrAnE-1(K) carries a non-trivial cycle of dim ,. Indeed, otherwise there would exist a cocyc1e 'lE' with carrier 'lnOlJ=~. But then and therefore K' < K. 11 CoraUary. Let Wi' 1 ~i~r, be pairwise subordinated homology classes of A mod AO, dim wi=ki , O~kl < ...

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