A parametrized compactness theorem under bounded Ricci curvature

Abstract

We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter, and lower bounded injectivity radius.

Key words： Gromov-Hausdorff distance; almost Riemannian submersion; parametrized compactness; fiber bundle

References

1 Anderson M T. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent Math, 1990, 102(1): 429–445

2 Berestovskii V N, Guijarro L. A metric characterization of Riemannian submersions. Ann Global Anal Geom, 2000, 18(6):577–588

3 Cheeger J. Comparison and Finiteness Theorems for Riemannian Manifolds. Ph D Thesis. Princeton: Princeton Univ, 1967

4 Cheeger J. Finiteness theorems for Riemannian manifolds. Amer J Math, 1970, 92(1): 61–74

5 Cheeger J, Fukaya K, Gromov M. Nilpotent structures and invariant metrics on collapsed manifolds. J Amer Math Soc, 1992, 5(2): 327–372

6 Dai X Z, Wei G F. A comparison-estimate of Topogonov type for Ricci curvature. Math Ann, 1995, 303(2): 297–306

7 Dai X Z, Wei G F, Ye R G. Smoothing Riemannian metrics with Ricci curvature bounds. Manuscripta Math, 1996, 90(1): 49–61

8 Fukaya K. Collapsing Riemannian manifolds to ones with lower dimensions. J Differential Geom, 1987, 25: 139–156

9 Fukaya K. A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters. J Differential Geom, 1988, 28: 1–21

10 Green R E, Wu H. Lipschitz convergence of Riemannian manifolds. Pacific J Math, 1988, 131: 119–141

11 Gromov M. Structures métriques pour les variétés riemanniennes. Textes Math, 1. Paris: CEDIC/Fernand Nathan, 1981

12 Hamilton R S. Three-manifolds with positive Ricci curvature. J Differential Geom, 1982, 17(2): 255–306

13 Jiang Z H, Li X, Xu S C. Stability of nilpotent structures of collapsed manifolds on the same scale. Preprint, 2017

14 Kapovitch V. Perelman’s stability theorem. In: Metric and Comparison Geometry. Surveys in Differential Geometry, Vol XI. Boston: International Press, 2007, 103–136

15 Kasue A. A convergence theorem for Riemannian manifolds and some applications. Nagoya Math J, 1989, 114: 21–51

16 O’Neill B. The fundamental equations of a submersion. Michigan Math J, 1966, 13(4): 459–469

17 Perelman G. Alexandrov spaces with curvatures bounded from below II. Preprint, 1991

18 Peters S. Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds. J Reine Angew Math, 1984, 349: 77–82

19 Peters S. Convergence of Riemannian manifolds. Compos Math, 1987, 62(1): 3–16

20 Petersen P. Riemannian Geometry.3rd ed. Grad Texts in Math, Vol 171. Berlin: Springer, 2016

21 Postnikov M M. Geometry VI: Riemannian Geometry. Encyclopaedia Math Sci, Vol 91. Berlin: Springer-Verlag, 2001

22 Rong X C. Convergence and collapsing theorems in Riemannian geometry. In: Ji L Z, Li P, Schoen R, Simon L, eds. Handbook of Geometric Analysis, Vol II. Adv Lect Math, Vol 13. Beijing/Boston: Higher Education Press/International Press, 2010, 193–298

23 Rong X C, Xu S C. Stability of almost submetries. Front Math China, 2011, 6(1): 137–154

24 Rong X C, Xu S C. Stability of e^ε-lipschitz and co-lipschitz maps in Gromov-Hausdorff topology. Adv Math, 2012, 231(2): 774–797

25 Tapp K. Bounded Riemannian submersions. Indiana Univ Math J, 2000, 49(2): 637–654

26 Tapp K. Finiteness theorems for submersions and souls. Proc Amer Math Soc, 2002, 130(6): 1809–1817

27 Walczak P. A finiteness theorem for Riemannian submersions. Ann Polon Math, 1992, 57: 283–290

28 Walczak P. Erratum to the paper A finiteness theorem for Riemannian submersions. Ann Polon Math, 1993, 58: 319

29 Wu J Y. A parametrized geometric finiteness theorem. Indiana Univ Math J, 1996, 45(2): 511–528

30 Xu S C. Stability theorems on almost submetries. Ph D Thesis. Beijing: Capital Normal Univ, 2010

31 Xu S C. Local estimate on convexity radius and decay of injectivity radius in a Riemannian manifold. Commun Contemp Math, July, 2017