Planar space with projective connection

Yu. I. Shevchenko ; E. V. Skrydlova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 2, Tome 180 (2020), pp. 113-119

Résumé

A projective space in which a linear group acts ineffectively allows one to construct the corresponding space of Cartan projective connection. We show that the structure equations of the Cartan space allow one to obtain differential equations for the components of the projective curvature-torsion tensor. This tensor contains the torsion tensor, the extended torsion tensor, and the affine curvature-torsion tensor. An analog of the Bianchi identities is found. A generalizable algorithm for constructing structure equations of the space of Cartan projective connection is formulated. Using a generalized algorithm, we construct the structure equations of the planar space of projective connection, whose special cases are the ruled space of Akivis projective connection, point space of Cartan projective connection, and its dual hyperplanar space of projective connection. We also prove that the curvature-torsion tensor of a plane space with projective connection has three subtensors, one of which is an analog of the torsion tensor of the Cartan space.

Détail

Keywords: space of Cartan projective connection, curvature-torsion tensor, analog of Bianchi identities, ruled space of projective connection, planar space of projective connection.