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

Abstract

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.

Keywords

• two-dimensional manifold
• three-dimensional manifold
• linear frame bundle
• structure equations
• base and fiber coordinates
• affine torsion