Riemannian structures on ${\mathbb{Z}}_2^n$-manifolds

Abstract

Very loosely, ${\mathbb{Z}}_2^n$-manifolds are `manifolds' with ${\mathbb{Z}}_2^n$-graded coordinates and their sign rule is determined by the scalar product of their ${\mathbb{Z}}_2^n$-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 ${\mathbb{Z}}_2^n$-manifold, i.e., a ${\mathbb{Z}}_2^n$-manifold equipped with a Riemannian metric that may carry non-zero ${\mathbb{Z}}_2^n$-degree. We show that the basic notions and tenets of Riemannian geometry directly generalise to the setting of ${\mathbb{Z}}_2^n$-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

17 pages. Comments welcomed

References in corpus (4)

• Once more on parastatistics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie Symmetries of the Lévy-Leblond Equations
• Super-de Sitter and alternative super-Poincaré symmetries
• Fedosov and Riemannian supermanifolds