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On the symmetry of Riemannian manifolds
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Shaoqiang Deng
Published/Copyright:
April 3, 2012
Abstract
Let (M, Q) be an n-dimensional connected Riemannian manifold and 1 ≦ k ≦ n. Then (M, Q) is called k-fold symmetric if given any k tangent vectors ξ1, ξ2, … , ξk at a point x ∈ M, there exists an isometry σ such that σ(x) = x and dσ(ξi) = −ξi, i = 1, 2, … , k. This kind of manifolds with k = 1, usually called weakly symmetric Riemannian manifolds, was introduced by A. Selberg as a weakening of the notion of n-fold symmetric ones, i.e., globally symmetric Riemannian manifolds. It is well known that there are many more weakly symmetric spaces than globally symmetric ones. In this paper, we prove that a connected simply connected 2-fold symmetric Riemannian manifold must be globally symmetric.
Received: 2010-07-20
Revised: 2011-10-04
Published Online: 2012-04-03
Published in Print: 2013-06
©[2013] by Walter de Gruyter Berlin Boston
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