Projective real calculi and Levi-Civita connections

TL;DR

This paper investigates the conditions under which Levi-Civita connections exist in real calculi over projective modules, focusing on complex vector modules over matrix algebras and providing explicit criteria for their existence.

Contribution

It establishes necessary and sufficient conditions for Levi-Civita connections in real calculi over projective modules, including specific cases involving complex vector modules and matrix algebras.

Findings

Existence depends on the Lie algebra of hermitian derivations.

Necessary and sufficient conditions are provided for specific modules.

Explicit projection coefficients determine Levi-Civita connection existence.

Abstract

Based on its central role in the framework of real calculi, the existence of the Levi-Civita connection for real calculi over projective modules is studied, with a special emphasis placed on the simple module of N-dimensional complex vectors over the algebra of complex N-by-N matrices. It is demonstrated that existence of the Levi-Civita connection in this case depends on the Lie algebra g of hermitian derivations, and necessary and sufficient conditions for the possibility of constructing a real calculus for which there exists a Levi-Civita connection are given in this case. Furthermore, in the general case of real calculi over projective modules, necessary and sufficient conditions for the existence of the Levi-Civita connection are given in terms of explicit projection coefficients.