Conjugate Radius and Sphere Theorem

http://dx.doi.org/10.4153/CMB-1999-026-6
Canad. Math. Bull. 42(1999), 214-220

  Published:1999-06-01
 Printed: Jun 1999

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.