Notes on projective structures with torsion
Taro Asuke
TL;DR
The paper extends the classical Cartan description of projective structures to include torsion by employing a bundle of formal frames and Thomas–Whitehead TW-connections. It proves a normal, Ricci-flat TW-connection exists for a given projective structure and relates this to affine data, providing explicit formulas for the normal connection components and the induced curvature/torsion. A framework for comparing TW-connections via structural equivalence is developed, revealing how torsion can be varied without altering the unparameterized geodesics and the projective structure. The work also furnishes concrete examples with nontrivial torsion and zero curvature, especially on T^2, demonstrating that torsionful but flat projective structures are abundant and classifiable. Overall, the results broaden the toolkit for studying projective geometry and its applications to foliations and transversely projective structures by incorporating torsion in a systematic, canonical way.
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.