Riemannian geometry of noncommutative super surfaces

TL;DR

This paper develops a Riemannian geometric framework for noncommutative super surfaces, extending classical concepts to the super and noncommutative setting, including metrics, connections, and curvature with identities.

Contribution

It introduces the notions of metric and connections on noncommutative super surfaces, generalizing classical Riemannian geometry to the super noncommutative context.

Findings

Connections are metric-compatible with zero torsion for symmetric super metrics

The super Riemann curvature satisfies noncommutative super Bianchi identities

Examples illustrating the theory are provided and analyzed in detail

Abstract

In this paper, a Riemannian geometry of noncommutative super surfaces is developed which generalizes [4] to the super case. The notions of metric and connections on such noncommutative super surfaces are introduced and it is shown that the connections are metric-compatible and have zero torsion when the super metric is symmetric, giving rise to the corresponding super Riemann curvature. The latter also satisfies the noncommutative super analogue of the first and second Bianchi identities. We also give some examples and study them in details.