Frontiers of Mathematics in China

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

An isometrical ℂP n -theorem

  • Xiaole Su
  • Hongwei Sun
  • Yusheng WangEmail author
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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  1. 1.
    Besse A L. Manifolds all of whose Geodesics are Closed. Ergeb Math Grenzgeb, Vol 93. Berlin: Springer, 1978Google Scholar
  2. 2.
    Burago Y, Gromov M, Perel0man G. A. D. Alexandrov spaces with curvature bounded below. Uspekhi Mat Nauk, 1992, 47(2): 3–51Google Scholar
  3. 3.
    Cheeger J, Ebin D G. Comparison Theorems in Riemannian Geometry. North-Holland Math Library, Vol 9. Amsterdam: North-Holland Publishing Company, 1975zbMATHGoogle Scholar
  4. 4.
    Frankel T. Manifolds of positive curvature. Pacic J Math, 1961, 11: 165–174MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gromoll D, Grove K. A generalization of Berger’s rigidity theorem for positively curved manifolds. Ann Sci Éc Norm Supér, 1987, 20(2): 227–239MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gromoll D, Grove K. The low-dimensional metric foliations of Euclidean spheres. J Differential Geom, 1988, 28: 143–156MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grove K, Markvorsen S. New extremal problems for the Riemannian recognition program via Alexandrov geometry. J Amer Math Soc, 1995, 8(1): 1–28MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grove K, Shiohama K. A generalized sphere theorem. Ann of Math, 1977, 106: 201–211MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Peterson P. Riemannian Geometry. Grad Texts in Math, Vol 171. Berlin: Springer- Verlag, 1998CrossRefGoogle Scholar
  10. 10.
    Rong X C, Wang Y S. Finite quotient of join in Alexandrov geometry. ArXiv: 1609.07747v1Google Scholar
  11. 11.
    Sady R H. Free involutions on complex projective spaces. Michigan Math J, 1977, 24: 51–64MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Su X L, Sun H W, Wang Y S. Generalized packing radius theorems of Alexandrov spaces with curvature ⩾ 1. Commun Contemp Math, 2017, 19(3): 1650049 (18 pp)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sun Z Y, Wang Y S. On the radius of locally convex subsets in Alexandrov spaces with curvature ⩾ 1 and radius > π=2. Front Math China, 2014, 9(2): 417–423MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wilhelm F. The radius rigidity theorem for manifolds of positive curvature. J Differential Geom, 1996, 44: 634–665MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wilking B. Index parity of closed geodesics and rigidity of Hopf brations. Invent Math, 2001, 144: 281–295MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191: 259–297MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yamaguchi T. Collapsing 4-manifolds under a lower curvature bound. arXiv: 1205.0323Google Scholar

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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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