Acta mathematica scientia, Series B >
FUNDAMENTAL GROUPS OF CLOSED POSITIVELY CURVED MANIFOLDS WITH ALMOST DISCRETE ABELIAN GROUP ACTIONS
Received date: 2006-12-28
Revised date: 2008-03-22
Online published: 2010-01-20
Supported by
Supported partially by NSFC (10826052, 10671018).
Let M be a closed n-manifold of positive sectional curvature. Assume that $M$ admits an effective isometrical
T1× Zpk-action with p prime. The main result of the article is that if k=1 for n=3 or k>n+1/4 for n ≥ 5, then there exists a positive constant $p(n)$, depending only on n, such that π1(M) is cyclic if p ≥ p(n).
Key words： fundamental groups; positive sectional curvature; group actions
WANG Yu-Sheng . FUNDAMENTAL GROUPS OF CLOSED POSITIVELY CURVED MANIFOLDS WITH ALMOST DISCRETE ABELIAN GROUP ACTIONS[J]. Acta mathematica scientia, Series B, 2010 , 30(1) : 203 -207 . DOI: 10.1016/S0252-9602(10)60037-9
[1] Wolf J A. The Spaces of Constant Curvature. New York: McGraw-Hill, 1967
[2] Grove K, Searle C. Positively curved manifolds with maximal symmetry-rank. J Pure Appl Alg, 1994, 91: 137--142
[3] Deng S. On two point homogeneous finsler spaces. Acta Math Sci, 2008, 28A(6): 1218--1221
[4] Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191: 259--297
[5] Rong X. Positively curved manifolds with almost maximal symmetry rank. Geom Dedi, 2002, 59: 157--182
[6] Rong X. Fundamental group of positively curved manifolds admitting compatible local torus actions. Asian J Math, 2005, 4: 545--560
[7] Rong X, Wang Y. Fundamental group of manifolds with positive curvature and torus actions. Geom Dedi, 2005, 113: 165--184
[8] Frank P, Rong X, Wang Y. Fundamental groups of positively curved manifolds with symmetry. Preprint
[9] Fang F, Rong X. Positively curved manifolds with maximal discrete symmetry rank. Amer J Math, 2004, 126: 227--245
[10] Bredon G. Introduction to Compact Transformation Groups. Pure and Applied Mathematics, Vol 46. New York, London: Academic Press, 1972
[11] Gromov M. Curvature, diameter and Betti numbers. Comme Math Helv, 1981, 56: 179--195
[12] Brown K S. Cohomology of Groups. New York: Springer-Verlag, 1982
[13] Kobayashi S. Transformation Groups in Differential Geometry. New York: Springer-Verlag, 1972
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