A Laplace operator for Poisson manifolds

Abstract

For a Poisson manifold (M,π), equipped with a Riemannian metric g, we define a contravariant analogue of the Laplace operator acting on differential forms. On functions, this operator does not coincide with the “connection Laplacian” defined by means of the contravariant Levi-Civita connection associated to (π,g) unless d(π⌟μg)=0, where μg is the Riemannian volume element. Under this condition and the assumption that M is compact, we establish an analogue of the E. Hopf Lemma, allowing us to use the Bochner technique. We also show that this operator is self-adjoint and has a Weitzenböck type formula. As applications, we get analogues of classical results by Lichnerowicz, Bochner, Meyer and Gallot.