Projectively flat and affinely flat parabolic subgroups of special linear groups

TL;DR

This paper investigates the geometric properties of parabolic subgroups within special linear Lie groups, demonstrating their autoparallel nature and conditions for the induced connection to be projectively equivalent to a flat affine connection.

Contribution

It provides a new criterion for when the induced connection on parabolic subgroups is projectively equivalent to a flat affine connection in special linear groups.

Findings

Parabolic subgroups are autoparallel submanifolds.

A criterion for the induced connection to be projectively equivalent to a flat affine connection.

Special linear Lie groups admit a projectively flat affine connection.

Abstract

A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.