Degenerations of Riemannian manifolds

TL;DR

This survey explains the geometric and topological structures of Riemannian manifolds under collapsing conditions, emphasizing curvature assumptions without covering applications.

Contribution

It provides a comprehensive exposition of collapsing theory in Riemannian geometry, focusing on geometric and topological aspects under various curvature conditions.

Findings

Detailed description of collapsed and non-collapsed regions

Analysis of geometric structures under curvature assumptions

Foundational overview for further research in collapsing theory

Abstract

This is an expositiry article on collapsing theory in Riemannian geometry written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applications of collapsing theory to Riemannian geometry are NOT discussed in this survey.