Curvature-torsion quasitensor of Laptev fundamental-group connection

• Published in: Differential Geometry of Manifolds of Figures
• Main Author: Yu. I. Shevchenko
• Format: Article in Journal/Newspaper
• Language: English; Russian
• Published: Immanuel Kant Baltic Federal University 2020
• Subjects: fundamental group connection; cartan connection; quasitensor of torsion-curvature; torsion-curvature tensor; Mathematics; QA1-939; laptev
• Online Access: https://doi.org/10.5922/0321-4796-2020-51-17; https://doaj.org/article/e3bce185fcf349b6af77487bde3284f6

We consider a space with Laptev's fundamental group connection generalizing spaces with Cartan connections. Laptev structural equations are reduced to a simpler form. The continuation of the given structural equations made it possible to find differential comparisons for the coeffi­cients in these equations. It is proved that one part of these coefficients forms a tensor, and the other part forms is quasitensor, which justifies the name quasitensor of torsion-curvature for the entire set. From differential congruences for the components of this quasitensor, congruences are ob­tained for the components of the Laptev curvature-torsion tensor, which contains 9 subtensors included in the unreduced structural equations. In two special cases, a space with a fundamental connection is a spa­ce with a Cartan connection, having a quasitensor of torsion-curvature, which contains a quasitensor of torsion. In the reductive case, the space of the Cartan connection is turned into such a principal bundle with connec­tion that has not only a curvature tensor, but also a torsion tensor.