Let M be a simply connected Riemannian manifold in Mk,vD(n), the space of closed Riemannian manifolds of dimension n with sectional curvature bounded below by k, volume bounded below by v, and diameter bounded above by D. Let c be the smallest positive real number such that any closed curve of length at most 2d can be contracted to a point over curves of length at most cd, where d is the diameter of M. In this paper, we show that under these hypotheses there exists a computable rational function, G(n, k, v, D), such that any continuous map of Sl to Ωp,qM, the space of piecewise differentiable curves on M connecting p and q, is homotopic to a map whose image consists of curves of length at most exp(cexp(G(n,k,v,D))). In particular, for any points p,q∈M and any integer m>0, there exist at least m geodesics connecting p and q of length at most mexp(cexp(G(n,k,v,D))). © Mathematica Josephina, Inc. 2025.