Riemannian Structures on Z 2 n -Manifolds

Andrew James Bruce and Janusz Grabowski
Additional contact information
Andrew James Bruce: Mathematics Research Unit, University of Luxembourg, Esch-sur-Alzette, L-4364 Luxembourg, Luxembourg
Janusz Grabowski: Institute of Mathematics, Polish Academy of Sciences, 00-656 Warsaw, Poland

Mathematics, 2020, vol. 8, issue 9, 1-23

Abstract: Very loosely, Z 2 n -manifolds are ‘manifolds’ with Z 2 n -graded coordinates and their sign rule is determined by the scalar product of their 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 Z 2 n -manifold, i.e., a Z 2 n -manifold equipped with a Riemannian metric that may carry non-zero Z 2 n -degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of 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.

Keywords: Z 2 n -manifolds; Riemannian structures; affine connections (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/9/1469/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/9/1469/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:9:p:1469-:d:407046

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().