Index aspect in collapsing geometry (Q1881511)

Put the Gromov-Hausdorff topology on the space \(S\) of all isometry classes of complete metric spaces. Consider a sequence of complete pointed Riemannian manifolds of fixed dimension \(n\geq2\) with a lower curvature bound. Gromov showed there is a subsequence which converges to a complete length space. If the Hausdorff dimension of the limit space is less than \(n\), one says the sequence collapses to the limit space. The authors examine these phenomena and define a {collapsing indicator} to describe the locally collapsing phenomena. Results of Fukaya are used to prove a fibration theorem.