Levi-Civita connections for a class of noncommutative minimal surfaces
Abstract
We study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for which there is a natural concept of torsion, and we prove that there exist metric and torsion-free connections for every minimal surface in this class.
Keywords
Cite
@article{arxiv.2102.04698,
title = {Levi-Civita connections for a class of noncommutative minimal surfaces},
author = {Joakim Arnlind},
journal= {arXiv preprint arXiv:2102.04698},
year = {2021}
}