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 language||English (US)|
|Number of pages||24|
|Journal||Journal of Differential Geometry|
|State||Published - 2012|
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology