Inradius collapsed manifolds

Takao Yamaguchi, Zhilang Zhang

Published 2015-12-26Version 1

In this paper, we study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds with boundary. Actually we show that the inradius collapse occurs when the limit space is a topological closed manifold, for instance. In the general case, we determine the limit spaces of inradius collapsed manifolds as Alexandrov spaces with curvature uniformly bounded below. When the limit space has co-dimension one, we completely determined the topology of inradius collapsed manifold in terms of singular $I$-bundles. Genral inradius collapse to almost regular spaces are also characterized. In the case of unbounded diameters, we prove that the number of boundary components of inradius collapsed manifolds is at most two. We also discuss the case of non inradius collapse and obtain information on the limit space and a fiber bundle theorem.