Manifolds of semi-negative curvature

CRISTIAN CONDE - GABRIEL LAROTONDA

PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY

Lugar: Londres; Año: 2009

0024-6115

This paper studies the metric structure of manifolds of semi-negative curvature. Explicitestimates on the geodesic distance and sectional curvature are obtained in thesetting of homogeneous spaces G/K of Banach-Lie groups, and a characterization ofconvex homogeneous submanifolds is given in terms of the Banach-Lie algebras. Asplitting theorem via convex expansive submanifolds is proven, inducing the correspondingsplitting of the Banach-Lie group G. The notion of nonpositive curvaturein Alexandrov´s sense is extended to include p-uniformly convex Banach spaces, andmanifolds of semi-negative curvature with a p-uniformly convex tangent norm fall inthis class of nonpositively curved spaces. Several well-known results, such as existenceand uniqueness of best approximations from convex closed sets, or the Bruhat-Titsxed 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 groupsof bounded linear operators acting on a Hilbert space, and the splittings induced byconditional expectations in such setting.