LINEAR CONNECTIONS ON EXTENDED SPACE-TIME

TL;DR

This paper proposes a generalized modification of Kaluza-Klein theory incorporating noncommutative internal geometries, analyzing how linear connections influence the structure of these geometries and providing a counter-example.

Contribution

It introduces a broad modification of Kaluza-Klein theory to include arbitrary noncommutative geometries and examines the restrictions imposed by linear connections.

Findings

Linear connections impose restrictions on noncommutative geometries.

A counter-example demonstrates limitations of the proposed extension.

The theory accommodates arbitrary finite noncommutative internal geometries.

Abstract

A modification of Kaluza-Klein theory is proposed which is general enough to admit an arbitrary finite noncommutative internal geometry. It is shown that the existence of a non-trival extension to the total geometry of a linear connection on space-time places severe restrictions on the structure of the noncommutative factor. A counter-example is given.