Bounding geometry of loops in Alexandrov spaces

Details Bounding geometry of loops in Alexandrov spaces Bounding geometry of loops in Alexandrov spaces

2010-08-16

arxiv.org/abs/1008.2745v2

Mathematics

Differential Geometry

Scientific paper

For a path in a compact finite dimensional Alexandrov space $X$ with curv $\ge \kappa$, 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 $\kappa$, the dimension, diameter and Hausdorff measure of $X$. This generalizes a basic estimate of Cheeger on the length of a closed geodesic in closed Riemannian manifold ([Ch], [GP1,2]). We also show that the $n$-dimensional Hausdorff measure and rough volume of $X$ are proportional by a constant depending on $n=\dim(X)$. This implies that the above result generalizes and improves an analogous of the Cheeger type estimate in Alexandrov geometry in [BGP].

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