We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering.
A surface which does not admit a length nonincreasing deformation is called metric minimizing. We show that metric minimizing surfaces in CAT(0) spaces are locally CAT(0) with respect to their intrinsic metric.
We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.
In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yaus estimate for weak solutions of the heat equation and prove a sharp Yaus gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(K,N).$
We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $le 3.$