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

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 $\mathrm{\mid }{M}_1{M}_2\mathrm{\mid }\ge \frac{\pi }{2}$, then $M_i$ is isometric to $\Bbb S^{n_i}/\Bbb Z_h$,${\Bbb C\Bbb P}^{\frac {n_i}2}$or${\Bbb C\Bbb P}^{\frac {n_i}2}/\Bbb Z_2$withthecanonicalmetricwhen$n_i>0$,andthus$M$isisometricto$\Bbb S^n/\Bbb Z_h$,${\Bbb C\Bbb P}^{\frac n2}$or${\Bbb C\Bbb P}^{\frac n2}/\Bbb Z_2$exceptpossiblywhen$n=3$and$M_1$(or$M_2$)$\stackrel{\rm iso}{\cong}\Bbb S^{1}/\Bbb Z_h$with$h\geq 2$or$n=4$and$M_1$(or$M_2$)$\stackrel{\rm iso}{\cong}\Bbb{RP}^2$.