Bounding geometry of loops in alexandrov spaces

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Abstract

For a path in a compact finite dimensional Alexandrov space X with curv ≥ k, the two basic geometric invariants are the length and the turning angle (which measures the closeness from being a geodesic). We show that the sum of the two invariants of any loop is bounded from below in terms of k, the dimension, diameter, and Hausdorff measure of X. This generalizes a basic estimate of Cheeger on the length of a closed geodesic in a closed Riemannian manifold ([Ch], [GP1, 2]). To see that the above result also generalizes and improves an analog of the Cheeger type estimate in Alexandrov geometry in [BGP], we show that for a class of subsets of X, the n-dimensional Hausdorff measure and rough volume are proportional by a constant depending on n = dim(X).

Original languageEnglish (US)
Pages (from-to)31-54
Number of pages24
JournalJournal of Differential Geometry
Volume92
Issue number1
DOIs
StatePublished - 2012

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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