Symplectic Reduction

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.