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

TL;DR

This paper extends Riemannian geometry concepts to $Z_2^n$-manifolds, demonstrating that fundamental notions and theorems generalize to this higher-graded setting, with insights into similarities and differences with supergeometry.

Contribution

It introduces and develops the theory of Riemannian $Z_2^n$-manifolds, establishing foundational geometric principles in this new graded context.

Findings

Fundamental theorem of Riemannian geometry extends to $Z_2^n$-manifolds

Basic notions of Riemannian geometry generalize to $Z_2^n$-setting

Identifies similarities and differences with supergeometry

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.