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Riemannian Structures on Z 2 n -Manifolds

Author

Listed:
  • 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)

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.

Suggested Citation

  • Andrew James Bruce & Janusz Grabowski, 2020. "Riemannian Structures on Z 2 n -Manifolds," Mathematics, MDPI, vol. 8(9), pages 1-23, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1469-:d:407046
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