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


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 \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{R}P^2 \).


Rigidity positive sectional curvature totally geodesic submanifolds 




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© 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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