Odd Symmetric Tensors, and an Analogue of the Levi-Civita Connection for an Odd Symplectic Supermanifold

TL;DR

This paper explores odd symplectic structures on supermanifolds, introduces an analogue of the Levi-Civita connection for these structures, and compares them to odd Riemannian structures using Lie algebra prolongations.

Contribution

It formulates an analogue of the Levi-Civita theorem specifically for odd symplectic supermanifolds, expanding geometric understanding in supergeometry.

Findings

Difference between odd Riemannian and odd symplectic structures clarified

An analogue of Levi-Civita connection for odd symplectic supermanifolds established

Lie algebra prolongation used to characterize structures

Abstract

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.