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Summary: Negative Ricci Curvature and Isometry Group
Xianzhe Dai
Zhongmin Shen
Guofang Wei §
Abstract
We show that for n-dimensional manifolds with Ricci curvature bounded
between two negative constants the order of their isometry groups is uniformly
bounded by the Ricci curvature bounds, the volume, and the injectivity radius.
We also show that the degree of symmetry (see §2 for definition) is lower semi-
continuous in the Gromov-Hausdorff topology.
1 Introduction
It is now known that negative Ricci curvature does not imply any topological re-
striction on the underlying manifold (Cf. [L]). It does, however, impose geometric
restriction by the classical result of Bochner [Bo]. Namely, the isometry group must
be finite. In this paper we consider quantitative version of Bochner's theorem.
Consider the class of Riemannian n-manifolds satisfying
- Ric - < 0, inj i0, vol V. (1)
(Here the upper bound on the volume is equivalent to an upper bound on the diam-
eter.) By a result of M. Anderson [A] this class is precompact in C1,
topology. We
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