Sphere Theorem for Manifolds with Positive Curvature
  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 $S^{n}.$ 
Bazanfare Mahaman