Sphere Theorem for Manifolds with Positive Curvature

Abstract

In this paper, we prove that, for any integer $n\ge 2,$ there exists an ${\epsilon }_n\ge 0$ so that if $M$ is an n-dimensional complete manifold with sectional curvature $K_M\ge 1$ and if $M$ has conjugate radius bigger than $\frac{\pi }{2}$ and contains a geodesic loop of length $2(\pi -{\epsilon }_n),$ then $M$ is diffeomorphic to the Euclidian unit sphere $S^n.$