Metric-Affine Manifold

TL;DR

This paper explores the properties of metric-affine manifolds, characterized by torsion and nonmetricity, analyzing their geometric implications and proposing a framework for particle dynamics within such manifolds.

Contribution

It introduces the concept of Cartan transport and a compatible connection, extending the understanding of geometric structures in metric-affine manifolds.

Findings

Nonmetricity causes differences between auto parallel and extremal lines.

Torsion modifies the Killing equation and the connection equations.

Cartan transport is introduced as a new concept for particle dynamics.

Abstract

We call a manifold with torsion and nonmetricity the metric-affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport and moving basis. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection. The analysis of the Frenet transport leads to the concept of the Cartan transport and an introduction of the connection compatible with the metric tensor. The dynamics of a particle follows to the Cartan transport. We need additional physical constraints to make a nonmetricity observable.