An Isometrical \Bbb C\Bbb Pⁿ-Theorem

2015/06/11 by Xiaole Su, Su, Xiaole, Hongwei Sun +3
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1506.03535

Abstract

Let Mⁿ (n≥3) be a complete Riemannian manifold with \sec_M≥ 1, and let M_i^n_i (i=1,2) be two comlplete totally geodesic submanifolds in M. We prove that if n₁+n₂=n-2 and if the distance |M₁M₂|≥\fracπ2, then M_i is isometric to \Bbb S^n_i/\Bbb Z_h, \Bbb C\Bbb P^\frac n_i2 or \Bbb C\Bbb P^\frac n_i2/\Bbb Z₂ with the canonical metric when n_i>0, and thus M is isometric to \Bbb Sⁿ/\Bbb Z_h, \Bbb C\Bbb P^{\frac n2} or \Bbb C\Bbb P^{\frac n2}/\Bbb Z₂ except possibly when n=3 and M₁ (or M₂) \stackrel\rm iso≅\Bbb S¹/\Bbb Z_h with h≥ 2 or n=4 and M₁ (or M₂) \stackrel\rm iso≅\BbbRP².

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