Manifolds of semi-negative curvature

CRISTIAN CONDE; GABRIEL LAROTONDA

PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY

London Mathematical Society

Lugar: London, UK; Año: 2010 p. 670 - 704

0024-6115

This paper studies the metric structure of manifolds of semi-negative curvature. Explicit estimates on the geodesic distance and sectional curvature are obtained in the setting of homogeneous spaces $G/K$ of Banach-Lie groups, and a characterization of convex homogeneous submanifolds is given in terms of the Banach-Lie algebras. A splitting theorem via convex expansive submanifolds is proven, inducing the corresponding splitting of the Banach-Lie group $G$. The notion of nonpositive curvature in Alexandrov´s sense is extended to include $p$-uniformly convex Banach spaces, and manifolds of semi-negative curvature with a $p$-uniformly convex tangent norm fall in this class of nonpositively curved spaces. Several well-known results, such as existence and uniqueness of best approximations from convex closed sets, or the Bruhat-Tits fixed point theorem, are shown to hold in this setting, without dimension restrictions. Finally, these notions are used to study the structure of the classical Banach-Lie groups of bounded linear operators acting on a Hilbert space, and the splittings induced by conditional expectations in such setting.