An Isometrical ${\Bbb C\Bbb P}^{n}$-Theorem

Let $M^n\ (n\geq4)$ be a complete Riemannian manifold with $\sec_M\geq 1$, and let $M_i^{n_i}$ ($i=1,2$) be two comlplete totally geodesic submanifolds in $M$. In the paper, we prove that if $n_1+n_2=n-2$ and if the distance $|p_1p_2|\geq\frac{\pi}{2}$ for any $p_i\in M_i$, then $M_i$ is isometric to $\Bbb S^{n_i}$, $\Bbb S^{n_i}/\Bbb Z_h$, ${\Bbb C\Bbb P}^{\frac {n_i}2}$ or ${\Bbb C\Bbb P}^{\frac {n_i}2}/\Bbb Z_2$ with the canonical metric, and thus $M$ is isometric to $\Bbb S^n$, $\Bbb S^n/\Bbb Z_h$, ${\Bbb C\Bbb P}^{\frac n2}$ or ${\Bbb C\Bbb P}^{\frac n2}/\Bbb Z_2$ with the canonical metric except possibly when $n=4$ and $M_1$ or $M_2\stackrel{\rm iso}{\cong}\Bbb{RP}^2$.