Convergence of homogeneous manifolds

TL;DR

This paper explores different notions of convergence for homogeneous manifolds, introduces a tractable algebraic approach, and provides a framework for variational and geometric analysis on these spaces.

Contribution

It introduces a new algebraic subset parameterizing homogeneous spaces, linking various convergence notions and facilitating curvature and evolution studies.

Findings

Relationships between infinitesimal, local, and pointed convergence.

A new algebraic subset parameterizing homogeneous manifolds.

Framework for variational problems and geometric flows.

Abstract

We study in this paper three natural notions of convergence of homogeneous manifolds, namely infinitesimal, local and pointed, and their relationship with a fourth one, which only takes into account the underlying algebraic structure of the homogeneous manifold and is indeed much more tractable. Along the way, we introduce a subset of the variety of Lie algebras which parameterizes the space of all n-dimensional simply connected homogeneous spaces with q-dimensional isotropy, providing a framework which is very advantageous to approach variational problems for curvature functionals as well as geometric evolution equations on homogeneous manifolds.