Abstract: Speaker's results on Alexandrov spaces of bounded curvature: 1)Metric foundations of Riemannian geometry; 2)Definitions of Alexandrov spaces of curvatures bounded from above and/or below in terms of isometric embedding of quads of spaces points into three-dimensional spaces of constant curvature; 3)Some special constructions of alexandrov spaces of bounded curvature; 4) Applications of these constructions to a) the solution to K.Borsuk problem on metrization of polytopes with local uniqueness of shortest arcs and b) proof of equivalence of Poincare conjecture and possibility to introduce special locally CAT(1)-metrics on 4-dimensional sphere.

Cite: Berestovskii V.N.
Alexandrov spaces of bounded curvature and some their applications
Воркшоп по геометрическому анализу, посвященный 110-летию со дня рождения А. Д. Александрова 23-25 Nov 2022