Riemannian Structures on <inline-formula> <mml:math id="mm1" display="block"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula>-Manifolds

Abstract

Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-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 Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

Keywords

• Z2n-manifolds%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula><mml:math id="mm9" display="block"><mml:semantics><mml:mrow><mml:msubsup><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula></named-content>-manifolds
• Riemannian structures
• affine connections