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

TL;DR

This paper classifies complete Riemannian manifolds with sectional curvature at least 1 that contain two totally geodesic submanifolds satisfying specific dimension and distance conditions, showing they are isometric to certain symmetric spaces.

Contribution

It proves an isometrical classification theorem for manifolds with curvature ≥ 1 containing two special totally geodesic submanifolds, extending classical sphere theorems.

Findings

Manifolds are isometric to spheres or complex projective spaces with canonical metrics.

Submanifolds are isometric to quotients of spheres or complex projective spaces.

The classification holds under the condition that the sum of submanifold dimensions is n-2 and their distance is at least π/2.

Abstract

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 $\Bbb S^{n_i}/\Bbb Z_h$, ${\Bbb C\Bbb P}^{\frac {n_i}2}$ or ${\Bbb C\Bbb P}^{\frac {n_i}2}/\Bbb Z_2$ with the canonical metric when $n_i>0$, and thus $M$ is isometric to $\Bbb S^n/\Bbb Z_h$, ${\Bbb C\Bbb P}^{\frac n2}$ or ${\Bbb C\Bbb P}^{\frac n2}/\Bbb Z_2$ except possibly when $n=3$ 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 $M_2$) $\stackrel{\rm iso}{\cong}\Bbb{RP}^2$.