Riemannian Structures on Z 2 n -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.

super twisted products

“…Definition 2.4. (Definition 9 in [3]) An affine connection on a Z 2 -manifold is a Z 2 -degree preserving map…”

“…Definition 2.6. (Definition 11 in [3]) An affine connection on a Riemannian Z 2 -manifold (M, g) is said to be metric compatible if and only if…”

“…On the other hand, in [3], the definition of super warped product spaces was given. Einstein warped products were studied in [7].…”

Linear Z₂ⁿ-Manifolds and Linear Actions

“…Loosely speaking, Z n 2 -manifolds (Z n 2 = Z ×n 2 ) are "manifolds" for which the structure sheaf has a Z n 2 -grading and the commutation rules for the local coordinates comes from the standard scalar product (see [11,13,14,15,18,19,20,21,37] for details). This is not just a trivial or straightforward generalization of the notion of a standard supermanifold, as one has to deal with formal coordinates that anticommute with other formal coordinates, but are themselves not nilpotent.…”

Symplectic $ {\mathbb Z}_2^n $-manifolds

“…As compared with supergeometry, there is a lot more freedom with the degree of a symplectic structure beyond simply even or odd, here understood as the total degree. We remark that Riemannian Z n 2 -manifolds were the subject of our previous paper [12], and in complete parallel with the classical setting, can be viewed as the symmetric cousins of symplectic Z n 2 -manifolds.…”