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Frontiers of Mathematics in China

, Volume 13, Issue 2, pp 367–398 | Cite as

An isometrical ℂP n -theorem

  • Xiaole Su
  • Hongwei Sun
  • Yusheng Wang
Research Article

Abstract

Let M n (n ⩾ 3) be a complete Riemannian manifold with sec M ⩾ 1, and let \(M_i^{n_i }\) (i = 1, 2) be two complete totally geodesic submanifolds in M. We prove that if n1 + n2 = n − 2 and if the distance |M1M2| ⩾ π/2, then M i is isometric to \(\mathbb{S}^{n_i } /\mathbb{Z}_h\), \(\mathbb{C}P^{n_i /2}\), or \(\mathbb{C}P^{n_i /2} /\mathbb{Z}_2 \) with the canonical metric when n i > 0; and thus, M is isometric to S n /ℤ h , ℂPn/2, or ℂPn/2/ℤ2 except possibly when n = 3 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{S}^1 /\mathbb{Z}_h \) with h ⩾ 2 or n = 4 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{R}P^2 \).

Keywords

Rigidity positive sectional curvature totally geodesic submanifolds 

MSC

53C20 

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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of EducationBeijingChina
  2. 2.School of Mathematical SciencesCapital Normal UniversityBeijingChina

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