Symplectic Reduction

More documents

Recommendations

Info

144 23 THE MARSDEN-WEINSTEIN-MEYER THEOREM Theorem 23.5 (Slice Theorem) Let G be a compact Lie group acting on a manifold M such that G acts freely at p E M. For sufficiently small ,-, tj : G x S, -+ M maps G x S, diffeomorphically onto a G-invariant neighborhood U of the G-orbit through p. The proof of this slice theorem is sketched further below. Corollary 23.6 If the action of G is free at p, U. then the action is free on Corollary 23.7 The set of points where G acts freely is open. Corollary 23.8 The set G x S,, U is G-invariant. Hence, the quotient - - UIG S, is smooth. Conclusion of the proof that MIG is a manifold and -7r : M -+ MIG is a smooth fiber map. For p E M, let q = 7r(p) E MIG. Choose a G-invariant neighborhood U of p as in the slice theorem: U -- G'x S (where S = S, for an appropriate 6). Then 7r(U) UIG = =: V is an open neighborhood of q in MIG. By the slice theorem, S 4 V is a homeomorphism. We will use such neighborhoods V as charts on MIG. To show that the transition functions associated with these charts are smooth, consider two G-invariant open sets U1, U2 in M and corresponding slices S1, S2 of the G-action. Then S12= S, nU2, S21 S2 nUl = are both slices for the G-action on U, nU2. To compute the transition map S12 -+ S21, consider the diagram S12 -24 id x S12 G x S12 U, nU2 . S2, - + id x S2, G x S2, Then the composition is smooth. S12-+ U, n U2 -L G x S21 24 S21 Finally, we need to show that 7r : M -+ MIG is a smooth fiber map. For P E M, q = 7r(p), choose a G-invariant neighborhood U of the G- orbit through p of the form q : G x S -4 U. Then V = UIG -- S is the corresponding neighborhood of q in MIG: MD U 17 GxS ' GxV 7r MIG D V v
23.3 Proof of the Marsden- Weinstein-Meyer Theorem 145 Since the projection on the right is smooth, 7r is smooth. Exercise. Check that the transition functions for the bundle defined by 7r are smooth. Sketch for the proof of the slice theorem. We need to show that, for - sufficiently small, 71 : G x S, -+ U is a diffeomorphism where U C M is a G-invariant neighborhood of the G-orbit through p. Show that: (a) d7l(id,p) is bijective. (b) Let G act on G x S by the product of its left action on G and trivial action on S. Then 71: G x S --+ M is G-equivariant. (c) d7l is bijective at all points of G x Ipl. This follows from (a) and (b). (d) The set G x Jp} is compact, and q : G x S -+ M is injective on G x JpJ with d7I bijective at all these points. By the implicit function theorem, there is a neighborhood Uo of G x Jp} in G x S such t'hat q maps U0 diffeomorphically onto a neighborhood U of the G-orbit through p. (e) The sets G x S.., varying 6, form a neighborhood base for G x Jp} in G x S. So in (d) we may take Uo = G x S,.. 23.3 Proof of the Marsden-Weinstein-Meyer Theorem Since G acts freely on p-'(0) dpp is surJective for all p E y-1(0) 0 is a regular value '(0) is a submanifold of codimension = dim G for the first two Parts of the Marsden-Weinstein-Meyer theorem it is enough to apply the third ingredient from Section 23.2 to the free action of G on P'(0). At p E p-1(0) the tangent space to the orbit TpOp is an isotropic subspace of the symplectic vector space (Tp M, wp), i.e., Tp Op g (Tp Op). (Tp Op)' = ker dltp = Tp p - 1 (0) . The lemma (second ingredient) gives a canonical symplectic structure on the quotient Tpy-'(0)1Tp0p. The point A E Mred y-'(O)IG has tangent = space T[p] Mred 1 _- Tpp- (0) ITp Op. Thus the lemma defines a nondegenerate 2-form. LUred On Mred. This is well-defined because w is G-invariant.
Page 1 and 2: Part IX Symplectic Reduction The ph
Page 3: - p, 23.2 Ingredients 143 Q is nond
Page 7 and 8: 2)-dimensional . . ,such 24 Reducti
Page 9 and 10: - 24.3 Reduction for Product Groups
Page 11 and 12: (zi, . . 24.5 Orbifolds 151 Example
Page 13: HOMEWORK 18 153 4. Locate all point

Symplectic Reduction

Create successful ePaper yourself

Delete template?

Save as template?