Connections on Metriplectic Manifolds

In this note we discuss conditions under which a linear connection on a manifold equipped with both a symmetric (Riemannian) and a skew-symmetric (almost-symplectic or Poisson) tensor field will preserve both structures. If (M, g) is a (pseudo-)Riemannian manifold, then classical results due to T. Levi-Civita, H. Weyl and E. Cartan [7] show that for any (1, 2) tensor field T i jk which is skew-symmetric by lower indices, there exists a unique linear connection Γ preserving the metric (∇ Γ g = 0), with T as its torsion tensor: T i kj = 1 2 (Γ i jk − Γ i kj). It has also been shown [4] that given any symmetric (by lower indices) (1, 2) tensor S i jk on a symplectic manifold (M, ω), there exists a unique linear connection preserving ω which has S as its symmetric part, i.e., S i jk = 1 2 (Γ i jk + Γ i kj). Moreover, it is known [9] that if ω is a regular Poisson tensor on M , then there always exists a linear connection on M with respect to which ω is covariantly constant. Such connections are called Poisson connections, and can be chosen to coincide with the Levi-Civita connection of the metric g (if g is Riemannian) in certain cases. Considering these results, one is naturally led to the question: Given a skew-symmetric (0, 2) tensor ω, and a (pseudo-)Riemannian metric g on a manifold M , when do there exist linear connections preserving ω and g simultaneously: ∇ Γ ω + ∇ Γ g = 0 ? (1) Motivated by the terminology of P.J. Morrison [6], we call the a manifold equipped with both a (pseudo-)Riemannian metric g and a skew-symmetric (2, 0) tensor P a metriplectic manifold, and a connection which preserves both tensors will be called a metriplectic connection. In the first section we restrict ourselves to the case in which both ω = P −1 and g are nondegenerate, that is ω is almost-symplectic and g is Riemannian. We combine the results from [7] and [4] to derive a necessary condition for a connection Γ to be a metriplectic connection. We also discuss the form of Γ in the almost-Hermitian and symplectic cases. The main result of this section is the following Proposition Any connection Γ with symmetric part Π and torsion T that preserves both a Riemannian metric …