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5 years ago

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

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

1 MORSE FAMILIES 196 We

1 MORSE FAMILIES 196 We begin by reviewing the theory of generating functions and Morse families for G- invariant Lagrange submanifolds in the neighbourhood of a fixed point of the G action. Further details can be found in [J]. As all our results are local we may identify the manifold (X,x0) with (~n,0) and assume that the action of G on (~n,0) is linear and orthogonal. We shall denote ~n with this action of G by V. We also identify T*V with VSV*, where V* is the dual of V. The orthogonality of the action of G on V implies that V* is isomorphic to V. If (L,0) c (T'V,0) is a G-invariant Lagrange submanifold germ and ~x L : (L,0) ~ (V,0) its associated G-equivariant Lagrange projection, then ker D~XL(0 ) = ToLc~V* is a G-invariant subspace of V* which we denote by W °. Let W* ± denote a G-invariant complement to W* in V* and define W = (W*)* and W ± = (W'Z) *. We can identify V with WSW ±. Let qI ..... qk denote coordinates for W, qk+l ...... qn for W x, Pl ..... Pk for W* and Pk+l ..... Pn for W* x. The existence of G-invariant generating functions for invariant Lagrange submanifolds is given by the following result. Proposition 1.1 There exists a smooth G-invariant function germ S : (W*$W ±, 0) -, ~ such that (L,0) c (V@V*,0) is defined by :- aS qi = i = 1 ........ ,k aPi o~S pj = j = k+l ........ n aqj Conversely, every such function germ generates an invariant Lagrange submanifold germ L such that ker DnL(0 ) c W*. Equality holds if and only if:- rank ~ = O. We refer to k = dim ker D:~L(O) as the comnk of ~L"

197 In the classification of Lagrange projections it is more convenient to replace generating functions by Morse families. This is done by taking Legendre transforms. More precisely, if S(Pl ....... ,Pk,qk+l ....... qn) is a generating function for L, we define a Morse family by:- F(Pl ..... ,Pk,ql ....... qn) = S(PI ....... Pk,qk+l ...... qn) - ~ Piqi • ill The equations for L then become :- ~F ~Pi -- o i= 1 .......... ,k ~P ........... pj j = k+I ....... n. Conversely every such G-invariant function germ F : (W*~BV,0) ~ $1 defines an invariant Lagrange submanifold germ, provided it satisfies the regularity condition:- rank , (0) = k ................. (1.2). 3Pa/~Pb /)PcOqdla,b,c = 1,...,k d d = 1,...,n This ensures that the subset defined by (1.2) is indeed a smooth submanifold. For ker D~L(0 ) = W* we also need:- rank ~F I (0) = 0 ............... (1.3). a pb / ja,b - 1..k We shall restrict the term Morse family to functions germs satisfying (1.2) and (1.3). If V' is a representation of G which has a G-invariant subspace isomorphic to V, then the invariant Morse family F:W*~BV-~ ~ also defines an invariant Lagrange submanifold, L', of T*V'. Let Pl ..... Pn be coordinates on the subspace isomorphic to V and extend these to a system Pl ..... Pn' on V'. Then the equations for L' are obtained by supplementing (1.1) by pj = 0 for j = n+l ..... n'. We will say that the Lagrange submanifold L' is a ~vialextension of L. k

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