Manifolds of semi-negative curvature

TL;DR

This paper extends the concept of nonpositive curvature to infinite-dimensional Banach manifolds with p-uniformly convex tangent norms, establishing key geometric and fixed point results applicable to Banach-Lie groups and operator groups.

Contribution

It introduces a generalized nonpositive curvature framework for Banach manifolds, including explicit curvature estimates and characterizations of convex submanifolds, with applications to Banach-Lie groups.

Findings

Existence and uniqueness of best approximations in this setting

Fixed point theorems for semi-negative curvature spaces

Splitting theorems for Banach-Lie groups

Abstract

The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and 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. Homogeneous spaces G/K of Banach-Lie groups of semi-negative curvature are also studied, explicit estimates on the geodesic distance and sectional curvature are obtained. 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. 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.