Stability of almost submetries

doi:10.1007/s11464-010-0076-7

Front Math Chin

2011, Vol. 6

Issue (1) : 137-154 https://doi.org/10.1007/s11464-010-0076-7

RESEARCH ARTICLE

Stability of almost submetries

Xiaochun RONG^1,2, Shicheng XU²(

)

1. Mathematics Department, Rutgers University, New Brunswick, NJ 08903, USA; 2. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Download: PDF(266 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

In this paper, we consider a triple of Gromov-Hausdorff convergence: Ai→dGHA, Bi→dGHB and maps f_i : A_i → B_i converge to a map f : A → B, where A_i are compact Alexandrov n-spaces and B_i are compact Riemannian m-manifolds such that the curvature, diameter and volume are suitably bounded (non-collapsing). When f is a submetry, we give a necessary and sufficient condition for the sequence to be stable, that is, for i large, there are homeomorphisms, Ψ_i : A_i → A, Φ_i : B_i → B such that f ? Ψ_i = Φ_i ? f_i. When f is an ?-submetry with ?>0, we obtain a sufficient condition for the stability in the case that A_i are Riemannian manifolds. Our results generalize the stability/finiteness results on fiber bundles by Riemannian submersions and by submetries.

Keywords Gromov-Hausdorff convergence Alexandrov space stability fiber bundle (almost) submetry

Corresponding Author(s): XU Shicheng,Email:shichengxu@gmail.com

Issue Date: 01 February 2011

Cite this article:

Xiaochun RONG,Shicheng XU. Stability of almost submetries[J]. Front Math Chin, 2011, 6(1): 137-154.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-010-0076-7
https://academic.hep.com.cn/fmc/EN/Y2011/V6/I1/137

1	Berestovskii V N, Guijarro L. A metric characterization of Riemannian submersions. Ann Global Anal Geom, 2000, 18(6): 577-588 doi: 10.1023/A:1006683922481
2	Burago Y, Gromov M, Perel’man G. A.D. Alexandrov spaces with curvature bounded below. Uspekhi Mat Nauk, 1992, 47: 3-51
3	Cheeger J. Finiteness theorems for Riemannian manifolds. Amer J Math, 1970, 92: 61-75 doi: 10.2307/2373498
4	Cheeger J, Fukaya K, Gromov M. Nilpotent structures and invariant metrics on collapsed manifolds. J Amer Math Soc, 1992, 5: 327-372
5	Fukaya K. Hausdorff convergence of Riemannian manifolds and its applications. In: Recent Topics in Differential and Analytic Geometry. Adv Stud Pure Math, 18-I. Boston: Academic Press, 1990, 143-238
6	Green R E, Wu H. Lipschitz convergence of Riemannian manifolds. Pacific J Math, 1988, 131: 119-141
7	Gromov M, Lafontaine J, Pansu P. Structures metriques pour les varietes riemannienes. Paris: CedicFernand, 1981
8	Grove K, Petersen P. A radius sphere theorem. Invent Math, 1993, 112: 577-583 doi: 10.1007/BF01232447
9	Kapovitch V. Perelman’s stability theorem. In: Surveys in Differential Geometry, XI. Somerville: Int Press, 2007, 103-136
10	Perel’man G. Alexandrov spaces with curvatures bounded from below II. Preprint , 1991
11	Perel’man G. Elements of Morse theory on Aleksandrov spaces. St Petersburg Math J , 1993, 5: 205-213
12	Peters S. Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds. J Reine Angew Math, 1984, 349: 77-82 doi: 10.1515/crll.1984.349.77
13	Petersen P. Riemannian Geometry. Grad Texts in Math, Vol 171. Berlin: Springer- Verlag, 2006
14	Petrunin A. Semiconcave functions in Alexandrov’s geometry. In: Cheeger J, Grove K, eds. Surveys in Differential Geometry, Vol XI. Somerville: Int Press, 2007, 137-202
15	Rong X. Convergence and collapsing theorems in Riemannian geometry. In: Ji L Z, Li P, Schoen R, Simon L, eds. Handbook of Geometric Analysis, Vol II. Beijing-Boston: Higher Education Press and International Press, 2010, 193-298
16	Tapp K. Bounded Riemannian submersions. Indiana Univ Math J, 2000, 49(2): 637-654 doi: 10.1512/iumj.2000.49.1705
17	Tapp K. Finiteness theorems for submersions and souls. Proc Amer Math Soc, 2002, 130(6): 1809-1817 doi: 10.1090/S0002-9939-01-06244-X
18	Walczak P. A finiteness theorem for Riemannian submersions. Ann Polon Math, 1992, 57: 283-290
19	Walczak P. Erratum to the paper A finiteness theorem for Riemannian submersions. Ann Polon Math, 1993, 58: 319
20	Wu J Y. A parametrized geometric finiteness theorem. Indiana Univ Math J, 1996, 45(2): 511-528 doi: 10.1512/iumj.1996.45.1130
21	Yamaguchi T. Collapsing and pinching under a lower curvature bound. Ann of Math, 1991, 133: 317-357 doi: 10.2307/2944340
22	Yamaguchi T. A convergence theorem in the geometry of Alexandrov spaces. Actes de la Table Ronde de Geometrie Differential (Luminy, 1992), Semin Congr, 1. 1996, 601-642

[1]	Ze LI. Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case[J]. Front. Math. China, 2020, 15(5): 923-957.
[2]	Hongzhi HUANG. Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry[J]. Front. Math. China, 2020, 15(1): 69-89.
[3]	Guangqiang LAN, Fang XIA. General decay asymptotic stability of neutral stochastic differential delayed equations with Markov switching[J]. Front. Math. China, 2019, 14(4): 793-818.
[4]	Xueping LI. Lipschitz Gromov-Hausdorff approximations to two-dimensional closed piecewise flat Alexandrov spaces[J]. Front. Math. China, 2019, 14(2): 349-380.
[5]	Liping XU, Jiaowan LUO. Global attractiveness and exponential decay of neutral stochastic functional differential equations driven by fBm with Hurst parameter less than 1/2[J]. Front. Math. China, 2018, 13(6): 1469-1487.
[6]	Xiang LI, Shicheng XU. A parametrized compactness theorem under bounded Ricci curvature[J]. Front. Math. China, 2018, 13(1): 67-85.
[7]	Zhi LI, Litan YAN, Xianghui ZHOU. Global attracting sets and stability of neutral stochastic functional differential equations driven by Rosenblatt process[J]. Front. Math. China, 2018, 13(1): 87-105.
[8]	Fernando GALAZ-GARCÍA. A glance at three-dimensional Alexandrov spaces[J]. Front. Math. China, 2016, 11(5): 1189-1206.
[9]	Zhiqiang ZHOU,Jingtang MA. Lattice Boltzmann methods for solving partial differential equations of exotic option pricing[J]. Front. Math. China, 2016, 11(1): 237-254.
[10]	Gengen ZHANG,Aiguo XIAO. Exact and numerical stability analysis of reaction-diffusion equations with distributed delays[J]. Front. Math. China, 2016, 11(1): 189-205.
[11]	Junjun LIAO,Xiangjun WANG. Stability of stochastic differential equation with linear fractal noise[J]. Front. Math. China, 2014, 9(3): 495-507.
[12]	Yusheng WANG,Zhongyang SUN. Radius of locally convex subsets in Alexandrov spaces with curvature≥1 and radius>π/2[J]. Front. Math. China, 2014, 9(2): 417-423.
[13]	Weichao GUO,Jiecheng CHEN. Stability of nonlinear Schrödinger equations on modulation spaces[J]. Front. Math. China, 2014, 9(2): 275-301.
[14]	Pengzhan HUANG, Yinnian HE, Xinlong FENG. Convergence and stability of two-level penalty mixed finite element method for stationary Navier-Stokes equations[J]. Front Math Chin, 2013, 8(4): 837-854.
[15]	Jianguo CAO, Bo DAI, Jiaqiang MEI. A quadrangle comparison theorem and its application to soul theory for Alexandrov spaces[J]. Front Math Chin, 2011, 6(1): 35-48.

Viewed

Full text

Abstract

Cited

Shared

Discussed