Notes on projective structures with torsion

TL;DR

This paper extends the theory of projective structures to include torsion, describing them via affine connections and Thomas-Whitehead connections, and explores examples with non-trivial torsion but trivial curvature.

Contribution

It generalizes the description of projective structures with torsion using affine and Thomas-Whitehead connections, including the introduction of normal projective connections.

Findings

Describes projective structures with torsion using affine connections.

Introduces normal projective connections and fundamental theorem for Thomas-Whitehead connections.

Provides examples with non-trivial torsion and trivial curvature.

Abstract

We show that projective structures with torsion are described in terms of affine connections in a parallel way as in the torsion-free case which is done by Kobayashi and Nagano. For this, we make use of a bundle of formal frames, which is a generalization of a bundle of frames. We will also describe projective structures in terms of Thomas--Whitehead connections by following Roberts. In particular, we introduce normal projective connections and show the fundamental theorem for Thomas--Whitehead connections regardless the triviality of the torsion. We will study some examples of projective structures of which the torsion is non-trivial while the curvature is trivial. In this article, projective structures are considered to be the same if they have the same geodesics ignoring parameters and the same torsions.