An Isometrical ${\mathbb{CP}}^n$-Theorem

Abstract

Let $M^n(n\ge 3)$ be a complete Riemannian manifold with ${\sec }_M\ge 1$, and let $M_i^{n_i}$ ($i=1,2$) be two comlplete totally geodesic submanifolds in $M$. We prove that if $n_1+n_2=n-2$ and if the distance $|M_1{M}_2|\ge \frac{\ddot{I}}{€}2$, then $M_i$ is isometric to ${\mathbb{S}}^{n_i}/{\mathbb{Z}}_h$, ${\mathbb{CP}}^{\frac{n_i}{2}}$ or ${\mathbb{CP}}^{\frac{n_i}{2}}/{\mathbb{Z}}_2$ with the canonical metric when $n_i>0$, and thus $M$ is isometric to ${\mathbb{S}}^n/{\mathbb{Z}}_h$, ${\mathbb{CP}}^{\frac{n}{2}}$ or ${\mathbb{CP}}^{\frac{n}{2}}/{\mathbb{Z}}_2$ except possibly when $n=3$ and $M_1$ (or $M_2$) $\stackrel{\mathrm{iso}}{\cong }{\mathbb{S}}^1/{\mathbb{Z}}_h$ with $h\ge 2$ or $n=4$ and $M_1$ (or $M_2$) $\stackrel{\mathrm{iso}}{\cong }{\mathbb{RP}}^2$.

26 pages