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Let $M^n\ (n\geq3)$ 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$M$. We prove that if $n_1+n_2=n-2$ and if the distance$|M_1M_2|\geq\frac{\pi}{2}$, then $M_i$ is isometric to $\Bbb S^{n_i}/\BbbZ_h$, ${\Bbb C\Bbb P}^{\frac {n_i}2}$ or ${\Bbb C\Bbb P}^{\frac {n_i}2}/\BbbZ_2$ with the canonical metric when $n_i>0$, and thus $M$ is isometric to $\BbbS^n/\Bbb Z_h$, ${\Bbb C\Bbb P}^{\frac n2}$ or ${\Bbb C\Bbb P}^{\frac n2}/\BbbZ_2$ except possibly when $n=3$ and $M_1$ (or $M_2$) $\stackrel{\rmiso}{\cong}\Bbb S^{1}/\Bbb Z_h$ with $h\geq 2$ or $n=4$ and $M_1$ (or $M_2$)$\stackrel{\rm iso}{\cong}\Bbb{RP}^2$.

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