Fedosov and Riemannian supermanifolds

TL;DR

This paper explores the geometric structures of supermanifolds, focusing on symplectic and metric generalizations, and reveals differences between even and odd cases, including curvature properties.

Contribution

It provides a detailed analysis of even and odd supermanifolds with symplectic and Riemannian structures, highlighting their relations and curvature characteristics.

Findings

Even supermanifolds have a rich set of geometric structures.

Odd Riemannian supermanifolds can only have constant curvature.

Odd Riemannian and Fedosov supermanifolds are characterized by a scalar curvature tensor.

Abstract

Generalizations of symplectic and metric structures for supermanifolds are analyzed. Two types of structures are possible according to the even/odd character of the corresponding quadratic tensors. In the even case one has a very rich set of geometric structures: even symplectic supermanifolds (or, equivalently, supermanifolds with non-degenerate Poisson structures), even Fedosov supermanifolds and even Riemannian supermanifolds. The existence of relations among those structures is analyzed in some details. In the odd case, we show that odd Riemannian and Fedosov supermanifolds are characterized by a scalar curvature tensor. However, odd Riemannian supermanifolds can only have constant curvature.