The Journal of Geometric Analysis

, Volume 5, Issue 3, pp 419–426 | Cite as

Spaces on and beyond the boundary of existence

  • Peter Petersen
  • Frederick Wilhelm
  • Shun-hui Zhu


In this note we discuss various questions on whether or not quotients of Riemannian manifolds by Lie groups can be the Gromov-Hausdorff 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


Key Words and Phrases

Gromov-Hausdorff convergence Alexandrov spaces orbifolds Gromov-Hausdorff limits of Riemannian manifolds 


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Copyright information

© Mathematica Josephina, Inc. 1995

Authors and Affiliations

  • Peter Petersen
    • 1
    • 2
  • Frederick Wilhelm
    • 1
    • 2
  • Shun-hui Zhu
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaLos Angeles
  2. 2.Department of MathematicsSUNY Stony BrookStony Brook

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