Singular cosphere bundle reduction

TL;DR

This work develops a refined framework for singular reduction of cosphere bundles at the zero momentum value. Building on cotangent-lift and contact reduction theory, it introduces the coisotropic-legendrian (C-L) stratification, which preserves bundle structure and is compatible with base orbit-type stratifications, yielding a stratified bundle $k^0: \mathcal{C}_0\to Q/G$. For base actions that are almost semifree, the reduced space $\mathcal{C}_0$ decomposes into a cosphere bundle over the principal stratum plus Legendrian seams over singular orbits, providing a transparent geometric and topological picture. The paper also contains explicit computations for representative examples, illustrating how the stratifications relate and how the Reeb dynamics project to the reduced spaces. Overall, the results offer a finer, bundle-aware description of singular cosphere bundle reduction with practical descriptions of strata, their frontiers, and their contact-geometric structure.

Abstract

This paper studies singular contact reduction for cosphere bundles at the zero value of the momentum map. A stratification of the singular quotient, finer than the contact one and better adapted to the bundle structure of the problem, is obtained. The strata of this new stratification are a collection of cosphere bundles and coisotropic or Legendrian submanifolds of their corresponding contact components.

Figures (6)

• Figure 1: Diagram defining $\eta_{(H)}$
• Figure 2: Diagram defining $\eta_{(H) \succ (L)}$
• Figure 3: The contact reduced space as a parabola fibrating over a half-closed line.
• Figure 4: Isotropy and stratification lattices for the $\mathbb{T} ^2$ action on $\mathbb{R} ^4$.
• Figure 5: The ambient space of $\mathcal{C}_0$

Theorems & Definitions (30)

• Remark 2.1
• Theorem 2.1
• Definition 3.1
• Remark 3.1
• Theorem 3.1
• Lemma 4.1
• proof
• Remark 4.1
• Theorem 4.1
• Lemma 4.2