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Unformatted text preview: The geometry of reduced cotangent bundles Matthew Perlmutter Departamento de Matem´ atica Instituto Superior T´ ecnico Lisboa joint work with Miguel Rodr´ ıguezOlmos and M. E. SousaDias Supported by the E.U. through the RTN MASIE “Mechanics And Symmetry In Europe” – Typeset by Foil T E X – Regular Hamiltonian G–spaces • ( P ,ω ) symplectic manifold • G Lie group acting freely and properly on P by symplectomorphims • J : P → g * equivariant momentum map i : J 1 (0) , → P π : J 1 (0) → P := J 1 (0) /G then P is a symplectic manifold (MW reduced space) with symplectic form ω defined by π * ω = i * ω – Typeset by Foil T E X – 1 Singular Hamiltonian G–spaces If the freeness assumption is removed then P is no longer a smooth manifold but a stratified Hausdorff space P = F ( H ) ∈ I P P ( H ) (disjoint union), where • I P denotes isotropy lattice of P . • P ( H ) = ( J 1 (0)) ( H ) /G are smooth manifolds • ( J 1 (0)) ( H ) = { z ∈ J 1 (0)  G z is conjugate to H } (orbit types) are smooth manifolds this is called the symplectic stratification of P , because its pieces P ( H ) are symplectic manifolds called symplectic strata , with symplectic reduced form ω ( H ) – Typeset by Foil T E X – 2 given by π ( H ) * G ω ( H ) = i * ( H ) ω ( J 1 (0)) ( H ) i ( H ) / π ( H ) G P P ( H ) The frontier conditions are: 1. P ( K ) ∩ P ( H ) 6 = ∅ ⇔ H 6 K (subconjugation) 2. P ( K ) ∩ P ( H ) 6 = ∅ ⇒ P ( K ) ⊂ P ( H ) In this case we write P ( K ) 6 P ( H ) and say that P ( K ) is incident to P ( H ) . Example: Isotropy Latt., J 1 (0) Stratification Latt., P – Typeset by Foil T E X – 3 The Canoe: S 1 × S 2 → S 2 • ( P ,ω ) = T * S 2 with its canonical symplectic form • S 1 action on S 2 by rotations around the e 3 axis and its cotangent lift on T * S 2 by symplectomorphisms. • Equivariant momentum map: if x · x = 1 and x · p = 0 , then ( x , p ) ∈ T * S 2 , and J ( x , p ) = ( x × p ) · e 3 – Typeset by Foil T E X – 4 Using Invariant Theory the symplectic quotient P is realized as the semialgebraic variety of R 3 called “The Canoe” – Typeset by Foil T E X – 5 Objective: Find finer structure related to the symplectic stratification when P = T * M and G acts by cotangent lifts • Since in the cotangent bundle case P has more structure we want to study how this is reflected in the symplectic quotient P and in the symplectic strata P ( H ) • In particular how close are these sets to cotangent bundles of certain manifolds • To what extent is the symplectic data of the...
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This note was uploaded on 01/04/2012 for the course CDS 280 taught by Professor Marsden,j during the Fall '08 term at Caltech.
 Fall '08
 Marsden,J

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