Abstract

AbstractWe show that the space P/G of orbits of a proper action of a Lie group G on a locally compact differential space P is a locally compact differential space with quotient topology. Applying this result to reduction of symmetries of Hamiltonian systems, we prove the reduction by stages theorem

Highlights

  • A symplectic manifold is a pair pM, ωq, where M is a manifold and ω is a closed non-degenerate 2-form on M

  • For a proper action of a connected Lie group G on a manifold M, the family N is locally finite, consists of locally closed manifolds, and it defines a stratification of the orbit space R “ M {G

  • Theorems 5 and 6 show that the stratification structure of the orbit space R of a proper action of a connected Lie group G on a manifold M is encoded in the differential structure C8pRq

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Summary

Introduction

A symplectic manifold is a pair pM, ωq, where M is a manifold and ω is a closed non-degenerate 2-form on M. For a proper action of a connected Lie group G on a manifold M , the family N is locally finite, consists of locally closed manifolds, and it defines a stratification of the orbit space R “ M {G. The space R “ M {G of G-orbits of a proper action of a connected Lie group on a manifold M, endowed with the differential structure. Strata of the orbit type stratification of the orbits space R “ M {G of a proper action of G on M are orbits of the family of all vector fields on R. Theorems 5 and 6 show that the stratification structure of the orbit space R of a proper action of a connected Lie group G on a manifold M is encoded in the differential structure C8pRq. the stratification structure is invariant under diffeomorphisms of differential spaces. If G has a smooth proper action on a manifold M , the induced action of G{H on M {H is proper

The Slice Theorem of Palais
Symplectic reduction by stages
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