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 N, then M admits a one-parameter family of metrics g∈ with curvature bounded in absolute value while injectivity radii and the diameters of N -orbits away from the singular set of N uniformly converge to zero as ∈ → 0. Moreover, g∈ is N -invariant away from the singular set. This result extends collapsing results in [CG1], [Fu3] and [G].
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Collapsing construction
- Invariant metrics
- Nilpotent structure