Collapsing Construction with Nilpotent Structures

Abstract

A fundamental result concerning collapsed manifolds with bounded sectional curvature is the existence of compatible local nilpotent symmetry structures whose orbits capture all collapsed directions of the local geometry [CFG]. The underlying topological structure is called an N-structure of positive rank. We show that if a manifold M admits such an N-structure \({\mathcal{N}}\), then M admits a one-parameter family of metrics g∈ with curvature bounded in absolute value while injectivity radii and the diameters of \({\mathcal{N}}\)-orbits away from the singular set of \({\mathcal{N}}\) uniformly converge to zero as \(\epsilon \rightarrow 0\). Moreover, g∈ is \({\mathcal{N}}\)-invariant away from the singular set. This result extends collapsing results in [CG1], [Fu3] and [G].