In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume. The related questions that will also be studied are the following: given a contractible k-dimensional sphere in M~n, how "fast" can this sphere be contracted to a point, if π_i(M~n) = {0} for 1 ≤ i < k. That is, what is the maximal length of the trajectory described by a point of a sphere under an "optimal" homotopy? Also, what is the "size" of the smallest non-contractible k-dimensional sphere in a (k - 1)-connected manifold M~n providing that M~n is not k-connected?
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