Extended Schouten classification for non-Riemannian geometries

TL;DR

This paper extends the Schouten classification to analyze generalized connections in non-Riemannian geometries, clarifying the roles of torsion, non-metricity, and asymmetric metrics at first-order approximation.

Contribution

It introduces an extended Schouten classification framework for non-Riemannian geometries, including explicit formulas for inverse structure matrices and autoparallel trajectories.

Findings

Derived the inverse structure matrix in the linearized regime

Defined autoparallel trajectories for generalized connections

Clarified the contribution of connection components at first-order approximation

Abstract

A generalized connection, including Christoffel coefficients, torsion, non-metricity tensor and metric-asymmetricity object, is analyzed according to the Schouten classification. The inverse structure matrix is found in the linearized regime, autoparallel trajectories are defined and the contribution of the components of the connection are clarified at first-order approximation.