An isometrical CPn-theorem

Abstract

Let M^n(n\ge \mathrm{3}) be a complete Riemannian manifold with {{\displaystyle \sec }}_M\ge \mathrm{1}, and let {{\displaystyle M}}_i^{{{\displaystyle n}}_i}(i=\mathrm{1},\mathrm{2}) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n − 2 and if the distance \left|{{\displaystyle M}}_{\mathrm{1}}{{\displaystyle M}}_{\mathrm{2}}\right|\ge \pi /\mathrm{2}, then M_i is isometric to {\mathbb{S}}^{{{\displaystyle n}}_i}/{{\displaystyle \mathrm{\mathbb{Z}}}}_h,{\mathrm{ℂ}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{\mathbb{Z}}}}_{\mathrm{2}}, or {\mathrm{ℂ}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{\mathbb{Z}}}}_{\mathrm{2}} with the canonical metric when n_i>0, and thus, M is isometric to {\mathbb{S}}^n/{{\displaystyle \mathrm{\mathbb{Z}}}}_h,{\mathrm{ℂ}\mathrm{P}}^{n/\mathrm{2}}, or {\mathrm{ℂ}\mathrm{P}}^{n/\mathrm{2}}/{{\displaystyle \mathrm{\mathbb{Z}}}}_{\mathrm{2}} except possibly when n = 3 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}{\mathbb{S}}^{\mathrm{1}}/{{\displaystyle \mathrm{\mathbb{Z}}}}_h with h\ge \mathrm{2} or n = 4 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}\mathrm{ℝ}{\mathrm{P}}^{\mathrm{2}}.

Key words： Rigidity; positive sectional curvature; totally geodesic submanifolds

References

1 Besse A L. Manifolds all of whose Geodesics are Closed. Ergeb Math Grenzgeb, Vol 93. Berlin: Springer, 1978

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

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

6 Gromoll D, Grove K. The low-dimensional metric foliations of Euclidean spheres. J Differential Geom, 1988, 28: 143–156

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

8 Grove K, Shiohama K. A generalized sphere theorem. Ann of Math, 1977, 106: 201–211

9 Peterson P. Riemannian Geometry. Grad Texts in Math, Vol 171. Berlin: Springer-Verlag, 1998

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

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

14 Wilhelm F. The radius rigidity theorem for manifolds of positive curvature. J Differential Geom, 1996, 44: 634–665

15 Wilking B. Index parity of closed geodesics and rigidity of Hopf fibrations. Invent Math, 2001, 144: 281–295

16 Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191: 259–297

17 Yamaguchi T. Collapsing 4-manifolds under a lower curvature bound. arXiv: 1205.0323