The Journal of Geometric Analysis
, Volume 5, Issue 3, pp 419426
First online:
Spaces on and beyond the boundary of existence
 Peter PetersenAffiliated withDepartment of Mathematics, University of CaliforniaDepartment of Mathematics, SUNY Stony Brook
 , Frederick WilhelmAffiliated withDepartment of Mathematics, University of CaliforniaDepartment of Mathematics, SUNY Stony Brook
 , Shunhui ZhuAffiliated withDepartment of Mathematics, University of CaliforniaDepartment of Mathematics, SUNY Stony Brook
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In this note we discuss various questions on whether or not quotients of Riemannian manifolds by Lie groups can be the GromovHausdorff limits of manifolds with certain curvature bounds. In particular we show that any quotient of a manifold by a Lie group is a limit of manifolds with a lower curvature bound; this answers a question posed by Burago, Gromov, and Perelman. On the other hand, we prove that not all such spaces are limits of manifolds with absolute curvature bounds. We also give examples of spaces with curvature ≥1 that are not limits of manifolds with curvature ≥δ > 1/4.
Math Subject Classification
53C20Key Words and Phrases
GromovHausdorff convergence Alexandrov spaces orbifolds GromovHausdorff limits of Riemannian manifolds Title
 Spaces on and beyond the boundary of existence
 Journal

The Journal of Geometric Analysis
Volume 5, Issue 3 , pp 419426
 Cover Date
 199509
 DOI
 10.1007/BF02921805
 Print ISSN
 10506926
 Online ISSN
 1559002X
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 53C20
 GromovHausdorff convergence
 Alexandrov spaces
 orbifolds
 GromovHausdorff limits of Riemannian manifolds
 Authors

 Peter Petersen ^{(1)} ^{(2)}
 Frederick Wilhelm ^{(1)} ^{(2)}
 Shunhui Zhu ^{(1)} ^{(2)}
 Author Affiliations

 1. Department of Mathematics, University of California, 90024, Los Angeles, CA
 2. Department of Mathematics, SUNY Stony Brook, 11794, Stony Brook, NY