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Riemannian structures on $\mathbb{Z}_2^n$-manifolds (2007.07666v1)

Published 15 Jul 2020 in math-ph, gr-qc, math.AG, math.DG, math.MP, and math.QA

Abstract: Very loosely, $\mathbb{Z}_2n$-manifolds are `manifolds' with $\mathbb{Z}_2n$-graded coordinates and their sign rule is determined by the scalar product of their $\mathbb{Z}_2n$-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 $\mathbb{Z}_2n$-manifold, i.e., a $\mathbb{Z}_2n$-manifold equipped with a Riemannian metric that may carry non-zero $\mathbb{Z}_2n$-degree. We show that the basic notions and tenets of Riemannian geometry directly generalise to the setting of $\mathbb{Z}_2n$-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

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