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# Conjugate Radius and Sphere Theorem

Published:1999-06-01
Printed: Jun 1999
• Seong-Hun Paeng
• Jong-Gug Yun
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## Abstract

Bessa [Be] proved that for given $n$ and $i_0$, there exists an $\varepsilon(n,i_0)>0$ depending on $n,i_0$ such that if $M$ admits a metric $g$ satisfying $\Ric_{(M,g)} \ge n-1$, $\inj_{(M,g)} \ge i_0>0$ and $\diam_{(M,g)} \ge \pi-\varepsilon$, then $M$ is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius.
 MSC Classifications: 53C20 - Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53C21 - Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]