Global and local volume bounds and the shortest geodesic loops
The author studies relations between the volume of complete Riemannian manifolds and the length of their shortest non trivial geodesic loops under no curvature assumption. By definition, a geodesic loop based at a point * is a geodesic arc with endpoints *. In theorem A the author provides lower bounds on the diameter, on the volume of the whole manifold and on the volume of sufficiently small balls. In theorem B, by introducing the length of geodesic double loops, the author refines the global volume estimate for 2-essential manifolds but the result does not admit any local version.
scientific article; zbMATH DE number 2147073