On Singular Poisson Sternberg Spaces

TL;DR

This paper develops a theory of stratified Poisson Sternberg spaces, extending cotangent bundle reduction to singular cases with single orbit type actions, and introduces singular connections and holonomy theory.

Contribution

It introduces a new framework for stratified Sternberg spaces and singular connections, extending classical reduction and holonomy theories to singular, orbit-type manifolds.

Findings

Stratified symplectic reduced spaces form topological fiber bundles over cotangent bundles.

Poisson stratification of Sternberg spaces is established.

Develops a theory of singular connections and holonomy extending Ambrose-Singer theorem.

Abstract

We obtain a theory of stratified Sternberg spaces thereby extending the theory of cotangent bundle reduction for free actions to the singular case where the action on the base manifold consists of only one orbit type. We find that the symplectic reduced spaces are stratified topological fiber bundles over the cotangent bundle of the orbit space. We also obtain a Poisson stratification of the Sternberg space. To construct the singular Poisson Sternberg space we develop an appropriate theory of singular connections for proper group actions on a single orbit type manifold including a theory of holonomy extending the usual Ambrose-Singer theorem for principal bundles.