DIFFERENTIAL GEOMETRY/PDE SEMINAR

Wednesday, March 7 Padelford C-36 3:45 PM

Spaces of curvature bounded below: manifold vs nonmanifolds (mini-survey)

Yuri Burago (Steklov Mathematics Institute, St. Petersburg)

Loosely speaking, Alexandrov Spaces (AS) of curvature less than k (also AS(k) ) are length spaces in which all sufficiently small triangles are “thinner” than triangles with the same side lengths on a simply connected surface of constant curvature k. Such spaces have been intensively studied for the past several decades. The cases of k ≤ 0 and k > 0 are very different.

In the former case, for simply connected spaces, the local curvature conditions imply the same conditions for all (arbitrarily large) triangles (the Alexander – Bishop theorem). As a result almost all classical theorems on Riemannian manifolds of nonpositive (and strictly negative) curvature have been generalized for AS. In addition, AS of negative curvature are Gromov hyperbolic spaces.

For AS of curvature ≤ k, k > 0 there are almost no global results. Even local (structural) theory of such spaces is of great interest.

I am going to present some new results in the AS theory obtained basically by St. Petersburg mathematicians S. Buyalo, S. Ivanov, N. Kosovskiy and myself. These results, in particular, show that properties of the Alexandrov spaces that are manifolds are essentially different from those of general AS.