On the geometry of positively curved manifolds with large radius

Abstract

Let $M$ be an $n$-dimensional complete connected Riemannian manifold with sectional curvature $K_M\geq 1$ and radius $\operatorname{rad}(M)>\pi /2$. For any $x\in M$, denote by $\operatorname{rad} (x)$ and $\rho (x)$ the radius and conjugate radius of $M$ at $x$, respectively. In this paper we show that if $\operatorname{rad} (x)\leq \rho (x)$ for all $x\in M$, then $M$ is isometric to a Euclidean $n$-sphere. We also show that the radius of any connected nontrivial (i.e., not reduced to a point) closed totally geodesic submanifold of $M$ is greater than or equal to that of $M$.