Geodesic spheres and symmetries in naturally reductive spaces.

Convexity and Geodesic Metric Spaces

In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metr...

Connections on Naturally Reductive

Proximinality in Geodesic Spaces

Let X be a complete CAT(0) space with the geodesic extension property and Alexandrov curvature bounded below. It is shown that if C is a closed subset of X , then the set of points of X which have a unique nearest point in C is Gδ and of the second Baire category inX. If, in addition,C is bounded, then the set of points ofX which have a unique farthest point in C is dense in X. A proximity resu...

Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.

Connections on Naturally Reductive Spaces, Their Dirac Operator and Homogeneous Models in String Theory

Given a reductive homogeneous space M = G/H endowed with a naturally reductive metric, we study the one-parameter family of connections ∇t joining the canonical and the LeviCivita connection (t = 0, 1/2). We show that the Dirac operator Dt corresponding to t = 1/3 is the so-called “cubic” Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus gen...