An isometrical CPn-theorem

doi:10.1007/s11464-018-0684-1

Front. Math. China

2018, Vol. 13

Issue (2) : 367-398 https://doi.org/10.1007/s11464-018-0684-1

RESEARCH ARTICLE

An isometrical CPn-theorem

Xiaole SU¹, Hongwei SUN², Yusheng WANG¹(

)

¹. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China
². School of Mathematical Sciences, Capital Normal University, Beijing 100037, China

Download: PDF(357 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

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 $| M 1 M 2 | ≥ π / 2$ , then M_i is isometric to $S n i / ? h, ? P n i / 2 / ? 2$ , or $? P n i / 2 / ? 2$ with the canonical metric when n_i>0, and thus, M is isometric to $S n / ? h, ? P n / 2$ , or $? P n / 2 / ? 2$ except possibly when n = 3 and M₁ (or M₂) $≅ i s o S 1 / ? h$ with $h ≥ 2$ or n = 4 and M₁ (or M₂) $≅ i s o ? P 2$ .

Keywords Rigidity positive sectional curvature totally geodesic submanifolds

Corresponding Author(s): Yusheng WANG

Issue Date: 28 March 2018

Cite this article:

Xiaole SU,Hongwei SUN,Yusheng WANG. An isometrical CPn-theorem[J]. Front. Math. China, 2018, 13(2): 367-398.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-018-0684-1
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I2/367

1	Besse A L. Manifolds all of whose Geodesics are Closed. Ergeb Math Grenzgeb, Vol 93. Berlin: Springer, 1978 https://doi.org/10.1007/978-3-642-61876-5
2	Burago Y, Gromov M, Perel' man G. A. D. Alexandrov spaces with curvature bounded below. Uspekhi Mat Nauk, 1992, 47(2): 3–51
3	Cheeger J, Ebin D G. Comparison Theorems in Riemannian Geometry. North-Holland Math Library, Vol 9. Amsterdam: North-Holland Publishing Company, 1975
4	Frankel T. Manifolds of positive curvature. Pacific J Math, 1961, 11: 165–174 https://doi.org/10.2140/pjm.1961.11.165
5	Gromoll D, Grove K. A generalization of Berger’s rigidity theorem for positively curved manifolds. Ann Sci Éc Norm Supér, 1987, 20(2): 227–239 https://doi.org/10.24033/asens.1530
6	Gromoll D, Grove K. The low-dimensional metric foliations of Euclidean spheres. J Differential Geom, 1988, 28: 143–156 https://doi.org/10.4310/jdg/1214442164
7	Grove K, Markvorsen S. New extremal problems for the Riemannian recognition program via Alexandrov geometry. J Amer Math Soc, 1995, 8(1): 1–28 https://doi.org/10.1090/S0894-0347-1995-1276824-4
8	Grove K, Shiohama K. A generalized sphere theorem. Ann of Math, 1977, 106: 201–211 https://doi.org/10.2307/1971164
9	Peterson P. Riemannian Geometry. Grad Texts in Math, Vol 171. Berlin: Springer-Verlag, 1998 https://doi.org/10.1007/978-1-4757-6434-5
10	Rong X C, Wang Y S. Finite quotient of join in Alexandrov geometry. ArXiv: 1609.07747v1
11	Sady R H.Free involutions on complex projective spaces. Michigan Math J, 1977, 24: 51–64 https://doi.org/10.1307/mmj/1029001820
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)
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): 417–423 https://doi.org/10.1007/s11464-013-0341-7
14	Wilhelm F. The radius rigidity theorem for manifolds of positive curvature. J Differential Geom, 1996, 44: 634–665 https://doi.org/10.4310/jdg/1214459225
15	Wilking B. Index parity of closed geodesics and rigidity of Hopf fibrations. Invent Math, 2001, 144: 281–295 https://doi.org/10.1007/PL00005801
16	Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191: 259–297 https://doi.org/10.1007/BF02392966
17	Yamaguchi T. Collapsing 4-manifolds under a lower curvature bound. arXiv: 1205.0323

[1]	Anand DESSAI. Torus actions, fixed-point formulas, elliptic genera and positive curvature[J]. Front. Math. China, 2016, 11(5): 1151-1187.
[2]	Manuel AMANN. Positive curvature, symmetry, and topology[J]. Front. Math. China, 2016, 11(5): 1099-1122.
[3]	Qifeng BAI,Fang LI. Classification of Bott towers by matrix[J]. Front. Math. China, 2016, 11(2): 255-268.
[4]	LIU Zhangjie. Scalar curvature of a class of minimal submanifolds in rank 1 symmetric spaces[J]. Front. Math. China, 2007, 2(3): 417-438.

Viewed

Full text

Abstract

Cited

Shared

Discussed