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Is a derivation in each factor: {F Á G, H} = F Á {G, H} þ G Á {F, H}, for all F, G, H 2 C1 (P). The notion of Poisson manifolds was rediscovered many times under different names, starting with Lie, Dirac, Pauli, and others. The name Poisson manifold was coined by Lichnerowicz. For any H 2 C1 (P), the Hamiltonian vector field XH is defined by XH ðFÞ ¼ fF; Hg; F 2 C1 ðPÞ ½3 i = 1, . . , n. This finite-dimensional Hamiltonian system is a system of ordinary differential equations for which there are well-known existence and uniqueness theorems, that is, it has locally unique smooth solutions, depending smoothly on the initial conditions.

More precisely, if (M, g) is a compact Riemannian Inequalities in Sobolev Spaces 35 manifold of dimension n, and 1 p < n, then the ? inequality for the embedding H 1, p (M) & Lp (M) reads as: there exists K > 0 such that for any u 2 H 1, p (M), Z p=p? Z Z ? jujp dvg K jrujp dvg þ jujp dvg ½5 M M M where dvg is the Riemannian volume element with respect to g. When (M, g) is no longer compact, the Sobolev embedding theorem might become false. A nontrivial key observation is that a Sobolev inequality like [5] on a complete manifold (M, g) implies the existence of a uniform (with respect to the center) lower bound for the volume of balls of radius 1.

C1 (P), {. }) is a Lie algebra, that is, {. } : C1 (P) Â C1 (P) ! C1 (P) is bilinear, skew-symmetric and satisfies the Jacobi identity {{F, G}, H} þ {{H, F}, G} þ {{G, H}, F} = 0 for all F, G, H 2 C1 (P) and 2. {. } satisfies the Leibniz rule, that is, { . } is a derivation in each factor: {F Á G, H} = F Á {G, H} þ G Á {F, H}, for all F, G, H 2 C1 (P). The notion of Poisson manifolds was rediscovered many times under different names, starting with Lie, Dirac, Pauli, and others. The name Poisson manifold was coined by Lichnerowicz.