Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry

Abstract

In the study of the collapsed manifolds with bounded sectional curvature, the following two results provide basic tools: a (singular) fibration theorem by K. Fukaya [J. Differential Geom., 1987, 25(1): 139–156] and J. Cheeger, K. Fukaya, and M. Gromov [J. Amer. Math. Soc., 1992, 5(2): 327–372], and the stability for isometric compact Lie group actions on manifolds by R. S. Palais [Bull. Amer. Math. Soc., 1961, 67(4): 362–364] and K. Grove and H. Karcher [Math. Z., 1973, 132: 11–20]. The main results in this paper (partially) generalize the two results to manifolds with local bounded Ricci covering geometry.

Key words： Fibrations; stability for group actions; nilpotent structures; Ricci curvature; bounded Ricci covering geometry

References

1 Anderson M T. Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem. Duke Math J, 1992, 68(1): 67–82

2 Cheeger J. Degeneration of Riemannian Metrics under Ricci Curvature Bounds. Lezioni Fermiane (Fermi Lectures) Scuola Normale Superiore, Pisa, 2001

3 Cheeger J, Colding T H. Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann of Math, 1996, 144(1): 189–237

4 Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. I. J Differential Geom, 1997, 46(3): 406–480

5 Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. III. J Differential Geom, 2000, 54: 37–74

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

7 Cheeger J, Jiang W, Naber A. Rectifiability of singular sets in noncollapsed spaces with Ricci curvature bounded below. arXiv: 1805.07988

8 Chen L, Rong X, Xu S. Quantitative volume space form rigidity under lower Ricci curvature bound II. Trans Amer Math Soc, 2018, 370: 4509–4523

9 Chen L, Rong X, Xu S. Quantitative volume space form rigidity under lower Ricci curvature bound. J Differential Geom, 2019, 113(2): 227–272

10 Colding T H. Shape of manifolds with positive Ricci curvature. Invent Math, 1996, 124(1-3): 175–191

11 Colding T H. Ricci curvature and volume convergence. Ann of Math (2), 1997, 145(3): 477–501

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

13 Fukaya K. Collapsing of Riemannian manifolds to ones of lower dimensions. J Differential Geom, 1987, 25(1): 139–156

14 Fukaya K. A boundary of the set of Riemannian manifolds with bounded curvature and diameter. J Differential Geom, 1988, 28(1): 1–21

15 Fukaya K, Yamaguchi T. The fundamental groups of almost nonnegatively curved manifolds. Ann of Math (2), 1992, 136(2): 253–333

16 Gromov M. Almost flat manifolds. J Differential Geom, 1978, 13: 231–241

17 Grove K, Karcher H. How to conjugate C¹-close group actions. Math Z, 1973, 132: 11–20

18 Huang H, Kong L, Rong X, Xu S. Collapsed manifolds with Ricci bounded covering geometry. arXiv: 1808.03774

19 Huang H, Rong X. Nilpotent structures on collapsed manifolds with Ricci bounded below and local rewinding non-collapsed. Preprint

20 Masur M, Rong X, Wang Y. Margulis lemma for compact Lie groups. Math Z, 2008, 258: 395–406

21 Naber A, Zhang R. Topology and ε-regularity theorems on collapsed manifolds with Ricci curvature bounds. Geom Topol, 2016, 20(5): 2575–2664

22 Palais R S. Equivalence of nearby differentiable actions of a compact group. Bull Amer Math Soc, 1961, 67(4): 362–364

23 Pan J. Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups. arXiv: 1809.10220

24 Pan J. Nonnegative Ricci curvature, stability at infinity and finite generation of fundamental groups. Geom Topol, 2019, 23: 3203–3231

25 Pan J, Rong X. Ricci curvature and isometric actions with scaling nonvanishing property. arXiv: 1808.02329

26 Petersen P, Wei G, Ye R. Controlled geometry via smoothing. Comment Math Helv, 1999, 74: 345–363

27 Rong X. Convergence and collapsing theorems in Riemannian geometry. In: Handbook of Geometric Analysis Vol II. Adv Lect Math (ALM), Vol 13. Beijing/ Somerville: Higher Education Press/International Press, 2010, 193–299

28 Rong X. Manifolds of Ricci curvature and local rewinding volume bounded below. Sci Sin Math, 2018, 48: 791–806 (in Chinese)

29 Rong X. A new proof of the Gromov’s theorem on almost flat manifolds. arXiv: 1906.03377

30 Rong X. Maximally collapsed manifolds with Ricci curvature and local rewinding volume bounded below. Preprint

31 Ruh E. Almost flat manifolds. J Differential Geom, 1982, 17: 1–14