Lectures on Closed Geodesics (Grundlehren der mathematischen by W. Klingenberg

By W. Klingenberg

The query of life of c10sed geodesics on a Riemannian manifold and the homes of the corresponding periodic orbits within the geodesic stream has been the item of in depth investigations because the starting of world differential geo­ metry over the past century. the easiest case happens for c10sed surfaces of damaging curvature. the following, the basic team is massive and, as proven through Hadamard [Had] in 1898, each non-null homotopic c10sed curve may 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 tougher. As mentioned by means of Poincare [po 1] in 1905, this challenge has a lot in universal with the matter ofthe lifestyles of periodic orbits within the limited 3 physique challenge. Poincare [l.c.] defined an evidence that on an analytic convex floor which doesn't range an excessive amount of from the traditional sphere there continuously exists at the least one c10sed geodesic of elliptic kind, i. e., the corres­ ponding periodic orbit within the geodesic circulate is infinitesimally stable.

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