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A cotangent bundle slice theorem

Tanya Schmah
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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.

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Tanya Schmah. 2004-09-09. A cotangent bundle slice theorem. https://arxiv.org/abs/math/0409148

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