Let M be a simply connected Riemannian manifold in \({\mathscr {M}}_{k,v}^D(n)\) , 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 \(S^l\) to \(\Omega _{p,q}M\) , 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 \(\exp (c\exp (G(n,k,v,D)))\) . In particular, for any points \(p,q \in M\) and any integer \(m>0\) , there exist at least m geodesics connecting p and q of length at most \(m\exp (c\exp (G(n,k,v,D)))\) .
