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Differential Structure of Orbit Spaces

Published online by Cambridge University Press:  20 November 2018

Richard Cushman  and
Jędrzej Śniatycki
Richard Cushman
Affiliation:
Mathematics Institute, University of Utrecht, Budapestlaan 6, 3508TA Utrecht, The Netherlands email: cushman@math.uu.nl
Jędrzej Śniatycki
Affiliation:
Department of Mathematics and Statistics, University of Calgary, 2500 University Dr. N.W., Calgary, Alberta, T2N 1N4 email: sniat@math.ucalgary.ca
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Abstract

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We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applied to analyze the differential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds.

Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with singular reduced spaces. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed.

Keywords

37J1558A4058D1970H33accessible setsdifferential spacePoisson algebraproper actionsingular reductionsymplectic manifolds
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 4 , 01 August 2001 , pp. 715 - 755
Copyright
Copyright © Canadian Mathematical Society 2001

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