Riemannian Structures on Z 2 n -Manifolds

Very loosely, Zn2-manifolds are ‘manifolds’ with Zn2-graded coordinates and their sign rule is determined by the scalar product of their Zn2-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Zn2-manifold, i.e., a Zn2-manifold equipped with a Riemannian metric that may carry non-zero Zn2-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Zn2-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry