A map between metric spaces is called a submetry if it mapsballs of radius R around a point onto balls of the sameradius around the image point. We show that when the domain and targetspaces are complete Riemannian manifolds, submetries correspond toC
1,1 Riemannian submersions. We also study someconsequences of this fact, and introduce the notion of submetries with asoul.

A Metric Characterization of Riemannian Submersions

This article is intended to be an introduction to the theory of metric spaces of curvature bounded below, and a survey of recent results. Time and space constraints have prevented it from being as comprehensive as we originally had planned, but we hope that this article will provide a good beginning point for a student or mathematician interested in this area, and a common reference for future papers on the subject. Through Section 9 we have tried to present good enough sketches of nearly all arguments, so that a dedicated reader can fill in the remaining details without too much difficulty. In Section 9 we provide fewer details, and in the last sections we mostly survey known results.

Metric Spaces of Curvature ⩾ k

We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut δ -homogeneous spaces in the case of Riemannian manifolds and prove that they constitute a new proper subclass of geodesic orbit (g.o.) spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian spaces.

On δ-homogeneous Riemannian manifolds

Uniform universal covers of uniform spaces

We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact, positively curved Alexandrov spaces with maximal symmetry rank.

/pdf/orientation-and-symmetries-of-alexandrov-spaces-with-14cgmyw7xd.pdf

Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature

We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces.

Homogeneous spaces of curvature bounded below

Geometric groups. I

Covering group theory for locally compact groups

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. Each coverable space has a generalized universal cover with universal and lifting properties. Associated with this generalized universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces. Keywords: universal cover, uniform space, geodesic space, fundamental group

Valera Berestovskii

Papers

Uniform universal covers of uniform spaces

Homogeneous spaces of curvature bounded below

Geometric groups. I

Covering group theory for locally compact groups

Generalized Universal Covers of Uniform Spaces