Lectures on Closed Geodesics (Grundlehren der mathematischen - download pdf or read online

By W. Klingenberg

ISBN-10: 3642618839

ISBN-13: 9783642618833

The query of lifestyles of c10sed geodesics on a Riemannian manifold and the houses of the corresponding periodic orbits within the geodesic move has been the item of in depth investigations because the starting of world differential geo metry over the last century. the easiest case happens for c10sed surfaces of detrimental curvature. the following, the basic team is massive and, as proven by means of 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 made up our minds and represents a c10sed geodesic. The query of lifestyles of a c10sed geodesic on a easily hooked up c10sed floor is way more challenging. As mentioned by way of Poincare [po 1] in 1905, this challenge has a lot in universal with the matter ofthe life of periodic orbits within the limited 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 normal sphere there continuously exists at the least one c10sed geodesic of elliptic style, i. e., the corres ponding periodic orbit within the geodesic circulation is infinitesimally stable.

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