5 years ago

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

144 G and Z(H) = C(H) n

144 G and Z(H) = C(H) n H denote the center of H. Note that C(H) is always a subgroup of N(H). Let H ° denote the identity component of H. If H is a dosed subgroup of G then H,H°,N(H) and C(H) axe all dosed Lie subgroups of G. The tangent bundle TM of M has the natural structure of a G-vector bundle over M with G-action defined by gv = Tg(v), v E TM, g E G. If x E M, then T,M is invariant by Gx and so has the natural structure of a G~-representation. If M and N are G manifolds, we let C~(M, N) denote the space of C °o G-equivariant maps from M to N, Diffc~°a(M ) denote the space of Coo G-equivariant diffeomorphisms of M and C~(TM) denote the space of C °o G-eqnivariant vector fields on M. If M is compact (respectively, non-compact) we take the C °o topology (respectively, Whitney Coo topology) on function spaces. We note that all our results hold if we work instead with C r maps, 2 < r < oo, with the proviso that we may sometimes lose one order of differentiabllty In the sequel, we always assume maps, bundles, submanifolds etc. are smooth. We also generally write equivariant rather than G-equivariant when the underlying group G is implicit from the context. Thus, reference to an equivariant vector field on M will always be to an element of C~(TM). Averaging a riemannian metric for M over G, we may and shall assume that M has an equivariant riemannian metric ~ and corresponding structure of a riemannian G-manifold. Denote the exponential map of ~ by exp. Thus, exp : TM 4-4 M will be smooth and G-equivariant. Let Q be an invariant (that is, G-invariant) closed submanifold of M. The normal bundle 7r : N(Q) ~ Q of Q may be identified with the orthogonal complement of TQ in TQM = TMIQ. Using an equivariant compression of N(Q) and restricting exp to N(Q), we may construct an invariant tubular neighbourhood U of Q in M. Specifically, we may construct an equivariant diffeomorphism q : N(Q) ---, U C M such that q = exp on a neighbourhood of the zero section of N(Q). In particular, q restricted to the zero section of N(Q) will be the identity map onto Q. Let z E M and let a denote the G orbit through x. Then, c~ is a smooth compact invariant submanifold of M. Set H = G~ and V = Nx(cr). We note that V is an H-representation and N(ct) is isomorphic to the fiber product G×HV. Let lr : G×HV ~ a denote the associated projection map. From what we have said above, it follows that there exists an equivariant embedding q of G ×H V onto an open invariant neighbourhood U of Or. For y E a, set S u = q(~-l(y)). We call S~ a slice (for the action of G) at y. We recall the following characteristic properties of slices: (s2) (s3) (s4) (ss) S o f3 gSy # 0 if and only if g E Gy. OSy = Sy for all 0 E Gy. oSv = So , for all 0 E G. G.S =U. If a : A C G/Gu ---, G~is a smooth local section of G over an open neighbourhood A of [Gu] in G/G u then the map p~ : Sy × A ~ U defined by p~'(z, a) = a(a)z is an embedding onto the open neighbourhood a(A) • Sy of Su in U.

145 We refer to ,g = {Su[y E a} as a family of slices for the G orbit a. The following well-known result (see Field[6] or Krupa[14]) is basic to many construc- tions involving equlvaxiant maps. Lemma2.1 Let E be a G-vector bundle over M. Let a C M be a G-orbit ands = {S=]z E a} be a family of slices for a. Let z E a and suppose that X is a smooth G=-equivariant section of E]Sx over S=. Then X extends uniquely to a smooth G-equivari- ant section f( of E over G. Sx. Proof: Define fC(gy) = gX(y),g E G,y fi Sx. By the G=-equivariance of X, X is well- defined as a G-equivarlant section of E over G. S~. To verify that )( is smooth it suffices, by G-equivariance, to cheek that X is smooth on a neighbourhood of S=~ in G- S=~. For this, choose a smooth local section a : A C G/G=~ --* G as in ($5). Set r/= (rh, r~) = (f).-1. Thus r/: a(a). S= --* S= x A. For z E a(A) . S=~, ff(z) = ~(z)X(ra(z)). Hence, X is smooth on a( A ). S=. | We also note (see [6], [14]) Lemma 2.2 Let E be a G-vector bundle over M. Let z E M and v E .E°= =. Then there exists s E C~(E) such that s(z) = v. Conversely, if s E C~'(E), then s(z) E E2=, all zEM. 3 Homogeneous Spaces and Local Sections Most of this section is based on Field[7, 8]. An alternative presentation may be given along the lines of Krupa[14], Let G be a compact Lie group. Let Gr denote the group of right translations of G. Thus, Gr ~ G and the action of Gr on G is given by: g(k) = kg -1, k E G, g E Gr. Let Gc denote the group of transformations of G defined by conjugation. That is, we regard Gc ~ G with action defined on G by: g(k) = gkg -1, k E G, g E Gr. We recall that a semi-direct product structure can be defined on G x G by: (a,b)(c,d) = (ac, bada-1), a,b,c,d E G. Let T(G) denote G x G with this composition. Observe that T(G) acts on G by: (a,b)g = aga-lb -1, (a,b) E T(G), g 6 G. Obviously, Gc and G, may be identified with the subgroups G × {e}, {e} x G of T(G). With these identifications, Gr

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