Odd symmetric tensors, and an analogue of the Levi-Civita connection for odd symplectic structure

We consider odd Poisson (odd symplectic) structure on supermanifolds induced by an odd symmetric rank $2$ (non-degenerate) contravariant tensor field. We describe the difference between odd Riemannian and odd symplectic structure in terms of the Cartan prolongation of the corresponding Lie algebras, and formulate an analogue of the Levi-Civita theorem for an odd symplectic supermanifold.