Wide short geodesic loops on closed Riemannian manifolds

Abstract

It is not known whether or not the length of the shortest periodic geodesic on a closed Riemannian manifold [Formula: see text] can be majorized by [Formula: see text], or [Formula: see text], where [Formula: see text] is the dimension of [Formula: see text], [Formula: see text] denotes the volume of [Formula: see text], and [Formula: see text] denotes its diameter. In this paper, we will prove that for each [Formula: see text] one can find such estimates for the length of a geodesic loop with angle between [Formula: see text] and [Formula: see text] with an explicit constant that depends both on [Formula: see text] and [Formula: see text]. That is, let [Formula: see text], and let [Formula: see text]. We will prove that there exists a “wide” (i.e. with an angle that is wider than [Formula: see text]) geodesic loop on [Formula: see text] of length at most [Formula: see text]. We will also show that there exists a “wide” geodesic loop of length at most [Formula: see text]. Here [Formula: see text] is the Filling Radius of [Formula: see text].