Open AccessAbout the torsion tensor of an affine connection on two-dimensional and three-dimensional manifolds
Author(s) -
Katerina Valentinovna Polyakova
Publication year - 2021
Publication title -
differencialʹnaâ geometriâ mnogoobrazij figur
Language(s) - English
Resource type - Journals
eISSN - 2782-3229
pISSN - 0321-4796
DOI - 10.5922/0321-4796-2021-52-9
Subject(s) - affine connection , affine coordinate system , mathematics , torsion (gastropod) , affine transformation , pure mathematics , connection (principal bundle) , affine group , mathematical analysis , parallel transport , curvilinear coordinates , manifold (fluid mechanics) , affine geometry of curves , affine space , geometry , medicine , mechanical engineering , surgery , engineering
The basis for this study of affine connections in linear frame bundle over a smooth manifold is the structure equations of the bundle. An affine connection is given in this bundle by the Laptev — Lumiste method. The differential equations are written for components of the deformation tensor from an affine connection to the symmetrical canonical one. The expressions for the components of the torsion tensor for two-dimensional and three-dimensional manifolds were found.For a two-dimensional manifold, the affine torsion is a fraction, in the numerator there is a linear combination of two fiber coordinates which coefficients are two functions depending on the base coordinates (the coordinates on the base), and in the denominator there is the determinant composed of the fiber coordinates (the coordinates in a fiber). For a three-dimensional manifold, the arbitrariness of the numerator is determined by nine functions depending on the base coordinates.