Normal frames for non-Riemannian connections

TL;DR

This paper demonstrates that normal frames, similar to normal coordinates, can be constructed for non-Riemannian connections, even when torsion and non-metricity are present, extending classical concepts.

Contribution

It introduces a method to construct normal frames for non-Riemannian connections, broadening the applicability of normal coordinate techniques beyond Riemannian geometry.

Findings

Normal frames can be constructed for non-Riemannian connections.

Normal frames retain key features of normal coordinates.

Construction is possible despite torsion and non-metricity.

Abstract

The principal properties of geodesic normal coordinates are the vanishing of the connection components and first derivatives of the metric components at some point. It is well-known that these hold only at points where the connection has vanishing torsion and non-metricity. However, it is shown that normal frames, possessing the essential features of normal coordinates, can still be constructed when the connection is non-Riemannian.