[Paper Review] A cotangent bundle slice theorem

Tanya Schmah|ArXiv.org|Sep 9, 2004

Homotopy and Cohomology in Algebraic Topology22 references4 citations

TL;DR

This paper presents a constructive cotangent bundle slice theorem for proper cotangent-lifted Lie group actions, extending the Hamiltonian slice theorem of Marle and Guillemin-Sternberg to points with fully isotropic momentum values. The key contribution is a new, explicit splitting of the symplectic normal space that refines reconstruction equations and enables local normal forms for symplectic reduced spaces.

ABSTRACT

This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to all proper cotangent-lifted actions, around points with fully-isotropic momentum values. We also present a ``tangent-level'' commuting reduction result and use it to characterise the symplectic normal space of any cotangent-lifted action. In two special cases, we arrive at splittings of the symplectic normal space, which lead to refinements of the reconstruction equations (bundle equations) for a Hamiltonian vector field. We also note local normal forms for symplectic reduced spaces of cotangent bundles.

Motivation & Objective

• To extend the Hamiltonian slice theorem to cotangent-lifted group actions with proper, non-free symmetries.
• To provide a constructive, geometric normal form for the symplectic structure near points with fully isotropic momentum values.
• To characterize the symplectic normal space of cotangent-lifted actions using a novel 'tangent-level' commuting reduction technique.
• To refine the reconstruction equations (bundle equations) in special cases, particularly for relative equilibria in simple mechanical systems.
• To establish cotangent-bundle-specific local normal forms for symplectic reduced spaces, improving understanding of singular reduction.

Proposed method

• Apply Palais' slice theorem in the configuration space Q to construct a Gq-invariant slice A transverse to the G-orbit through q.
• Lift the resulting diffeomorphism to the cotangent bundle, yielding a G-equivariant symplectic diffeomorphism from T*Q to T*(G ×_{Gq} A).
• Unroll the twisted product T*(G ×_{Gq} A) into the untwisted product T*(G × A), introducing two commuting actions: cotangent lift of left multiplication and Gq-twist.
• Perform tangent-level commuting reduction on T*(G × A) to characterize the symplectic normal space Ns as a reduced space of T*(G × B) for a suitable subspace B ⊂ A.
• Construct a symplectic tube via a cotangent lift of a Riemannian exponential map, ensuring the construction is explicit and avoids reliance on Darboux’s theorem or constant rank embeddings.
• Verify uniqueness of the construction via Lemma 5.3, which establishes a canonical property of the symplectic tube.

Experimental results

Research questions

• RQ1 Can a constructive Hamiltonian slice theorem be developed for cotangent-lifted actions beyond the free case?
• RQ2 How can the symplectic normal space be characterized in terms of tangent-level reduction for cotangent-lifted group actions?
• RQ3 What splittings of the symplectic normal space arise in special cases such as when the configuration isotropy group is contained in the momentum isotropy group?
• RQ4 How do the new splittings refine the reconstruction equations (bundle equations) in the context of relative equilibria of simple mechanical systems?
• RQ5 What are the local normal forms for symplectic reduced spaces in the cotangent bundle setting, and how do they relate to singular reduction?

Key findings

• A new cotangent bundle slice theorem (Theorem 5.6) is established for all proper cotangent-lifted actions at points with fully isotropic momentum values (Gμ = G), providing a constructive, symplectic local model.
• The symplectic normal space Ns admits a cotangent-bundle-specific splitting Ns ≅ T*B in two cases: when Gz ⊂ Gμ and when the point is purely in the group direction (i.e., z|A = 0).
• The splitting in the case Gz ⊂ Gμ generalizes the Montgomery-Marsden-Ratiu splitting for free actions, extending its applicability to non-free, proper actions.
• For relative equilibria of simple mechanical systems, the new splitting leads to a refined version of the reconstruction equations (bundle equations), improving their structure and applicability.
• The construction yields explicit local normal forms for symplectic reduced spaces, as shown in Theorem 4.8 and Remark 4.15, which are specific to the cotangent bundle geometry.
• An alternative construction of the symplectic tube is given in Theorems 5.11 and 5.12, showing it arises as a cotangent lift of a simple map between twisted products, and the first explicit computation of such a tube is provided in the example at the end of Section 5.