About the torsion tensor of an affine connection on two-dimensional and three-dimensional manifolds

• Published in: Differential Geometry of Manifolds of Figures
• Main Author: K.V. Polyakova
• Format: Article in Journal/Newspaper
• Language: English; Russian
• Published: Immanuel Kant Baltic Federal University 2021
• Subjects: two-dimensional manifold; three-dimensional manifold; linear frame bundle; structure equations; base and fiber coordinates; affine torsion; Mathematics; QA1-939; laptev
• Online Access: https://doi.org/10.5922/0321-4796-2021-52-9; https://doaj.org/article/c83b81b888cd4e7b960a168a3fb83058

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.