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Published online by Cambridge University Press: 24 March 2006
In this paper, we prove that, for any integer $n\ge 2,$ and any $\delta > 0$ there exists an $\epsilon(n,\delta) \ge 0$ such that if $M$ is an $n$-dimensional complete manifold with sectional curvature $K_M \ge 1$ and if $M$ has conjugate radius $\rho \ge\frac{\pi}{2}+\delta $ and contains a geodesic loop of length $2(\pi-\epsilon(n,\delta))$ then $M$ is diffeomorphic to the Euclidian unit sphere $\mathbb{S}^{n}.$
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