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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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Volume 2013, Issue 680


On the symmetry of Riemannian manifolds

Shaoqiang Deng
Published Online: 2012-04-03 | DOI: https://doi.org/10.1515/crelle.2012.040


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 (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.

About the article

Received: 2010-07-20

Revised: 2011-10-04

Published Online: 2012-04-03

Published in Print: 2013-06-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2013, Issue 680, Pages 235–256, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle.2012.040.

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©[2013] by Walter de Gruyter Berlin Boston.Get Permission

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