On Singular Poisson Sternberg Spaces

TL;DR

This work extends cotangent bundle reduction to proper group actions with a single orbit type by developing a singular Sternberg framework built from singular connections. It shows that reduced spaces form stratified fiber bundles over the reduced base $T^{\ast}(M/G)$ and provides a Poisson stratification of the Sternberg space, with symplectic leaves described via reduced curvature and homogeneous Lie–Poisson structures on fiber strata. The theory introduces a singular Ambrose–Singer theorem, a singular Atiyah sequence, and a curvature calculus that yields holonomy contained in $N(H)/H$ and whose curvature governs the holonomy Lie algebra. Collectively, the results yield a concrete, gauge-realized description of singular reductions, enabling explicit Poisson brackets and symplectic leaves on each stratum, and set the stage for addressing the multi-orbit-type general case. This framework has potential impact for explicit reduction procedures in stratified symplectic geometry and geometric mechanics with symmetry.

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.

Theorems & Definitions (38)

• Proposition 1
• Proposition 2
• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Remark 1
• Proposition 3
• proof
• Definition 1