The differentiable structure of three remarkable diffeomorphism groups

Abstract

The goal of this paper is to find the for three well-known Lie algebras: the globally Hamiltonian vector fields, the infinitesimal quantomorphisms and the homogeneous real-valued functions of degree one on the cotangent bundle minus the zero section. In all our considerations the underlying manifold is assumed to be compact without boundary. Before explaining and motivating our results, a very brief review of the group structure of diffeomorphism groups is in order ([9, 10, 18]). If M, N are compact, boundaryless, finite-dimensional manifolds, rN: TN--,N, the tangent bundle of N, an HS-map, s>(dimM)/2, from M to N has by definition all derivatives of order (dim M)/2. Then the space IP(M, N) of all such maps is a Hilbert manifold whose tangent space at every point is the Hilbert space T I. H~(M, N) = {gelid(M, TN) I ~:N ~ g =f} If E---, M is a vector bundle, the same construction works for all HS-sections HS(E) of E, s > (dim M)/2. If N = IR, HS(M, N) will be denoted by C~(M, IR). Let ~ + I ( M ) denote the diffeomorphisms of M of Sobolev class H s+l, i.e. r t ~ + l ( M ) if and only if t/ is bijective and t/, t ] l : M---,M are of class H ~+1. ~s+l(M) is a topological group, and since it is open in Hs+I(M,M), it is also a Hilbert manifold. Right multiplication R~: @~+ ~(M) ~ + I(M), R,(~) = ~ o t/ is C ~ for each t / ~ Y +~(M) and if t/E@ s+k+l(M), left multiplication L , : ~ + t ( M ) ~ + I ( M ) , L~(~)=t/o~ is C k. The inversion map t/~--~t/-1 in ~ + t(M) is only continuous. The tangent maps of R, and L, at e, the identity of ~ + a ( M ) , are given by Tr TeL,(X)=T~oX, where X~Yys+I(M) =HS+I(TM), the set of all H~+l-vector fields on M. The tangent space at e, Tr coincides with Y'~+~(M), which is a Sobolev space. The bracket [X, Y] of X, y ~ f s + I(M ) is however only of class H ~ (one derivative is lost). The usual bracket of vector fields is the Lie algebra bracket of 5F ~+ X(M), i.e. if J(,