The query of life of c10sed geodesics on a Riemannian manifold and the houses of the corresponding periodic orbits within the geodesic circulation has been the article of extensive investigations because the starting of world differential geo metry over the past century. the best case happens for c10sed surfaces of detrimental curvature. right here, the basic staff 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 unfastened homotopy c1ass. This minimum curve is, as much as the parameterization, uniquely decided and represents a c10sed geodesic. The query of lifestyles 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 constantly exists at the very least one c10sed geodesic of elliptic kind, i. e., the corres ponding periodic orbit within the geodesic circulate is infinitesimally sturdy.
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