The subject of this paper is upper bounds on the length of the shortest closed geodesic on simply connected manifolds with non-trivial second homology group. We will give three estimates. The first estimate will explicitly depend on volume and the upper bound for the sectional curvature; the second estimate will depend on diameter, a positive lower bound for the volume, and on the (possibly negative) lower bound on sectional curvature; the third estimate will depend on diameter, on a (possibly negative) lower bound for the sectional curvature and on a lower bound for the simply-connectedness radius. The technique that we develop in order to obtain the last result will also enable us to estimate the homotopy distance between any two closed curves on compact simply connected manifolds of sectional curvature bounded from below and diameter bounded from above. More precisely, let c be a constant such that any metric ball of radius I c is simply connected. There exists a homotopy connecting any two closed curves such that the length of the trajectory of the points during this homotopy has an upper bound in terms of the lower bound of the curvature, the upper bound of diameter and c. [References: 16]
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