Collapsed Riemannian manifolds with bounded sectional curvature

Abstract

In the last two decades, one of the most important developments in Riemannian geometry is the collapsing theory of Cheeger-Fukaya-Gromov. A Riemannian manifold is called (sufficiently) collapsed if its dimension looks smaller than its actual dimension while its sectional curvature remains bounded (say a very thin flat torus looks like a circle in a bared eyes). We will survey the development of collapsing theory and its applications to Riemannian geometry since 1990. The common starting point for all of these is the existence of a singular fibration structure on collapsed manifolds. However, new techniques have been introduced and tools from related fields have been brought in. As a consequence, light has been shed on some classical problems and conjectures whose statements do not involve collapsing. Specifically, substantial progress has been made on manifolds with nonpositive curvature, on positively pinched manifolds, collapsed manifolds with an a priori diameter bound, and subclasses of manifolds whose members satisfy additional topological conditions e.g. $2$-connectedness.