An isometrical CPn-theorem

Abstract

Let M^n(n\ge \mathrm{3}) be a complete Riemannian manifold with {{\displaystyle \sec }}_M\ge \mathrm{1}, and let {{\displaystyle M}}_i^{{{\displaystyle n}}_i}(i=\mathrm{1},\mathrm{2}) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n − 2 and if the distance \left|{{\displaystyle M}}_{\mathrm{1}}{{\displaystyle M}}_{\mathrm{2}}\right|\ge \pi /\mathrm{2}, then M_i is isometric to {\mathbb{S}}^{{{\displaystyle n}}_i}/{{\displaystyle \mathrm{\mathbb{Z}}}}_h,{\mathrm{ℂ}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{\mathbb{Z}}}}_{\mathrm{2}}, or {\mathrm{ℂ}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{\mathbb{Z}}}}_{\mathrm{2}} with the canonical metric when n_i>0, and thus, M is isometric to {\mathbb{S}}^n/{{\displaystyle \mathrm{\mathbb{Z}}}}_h,{\mathrm{ℂ}\mathrm{P}}^{n/\mathrm{2}}, or {\mathrm{ℂ}\mathrm{P}}^{n/\mathrm{2}}/{{\displaystyle \mathrm{\mathbb{Z}}}}_{\mathrm{2}} except possibly when n = 3 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}{\mathbb{S}}^{\mathrm{1}}/{{\displaystyle \mathrm{\mathbb{Z}}}}_h with h\ge \mathrm{2} or n = 4 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}\mathrm{ℝ}{\mathrm{P}}^{\mathrm{2}}.

Keywords

Rigidity / positive sectional curvature / totally geodesic submanifolds

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