On the structure of an affine connection object and the torsion tensor in the bundle of linear frames

In this paper, we study affine connections in the bundle of linear frame over a smooth manifold based on the structural equations of this bundle. The structure of the components of an affine connection in the bundle of frames over a two-dimensional manifold is obtained by using the layer coordinates whose coefficients are functions of the base coordinates of a point of the manifold. We construct expressions for the components of the torsion tensor for two- and three-dimensional manifolds by using the first-order layer coordinates and functions of the base coordinates. Also, we find expressions for the object of flat connection in terms of the coordinates of absolutely parallel vectors and their Pfaffian derivatives and expressions for the object of symmetric flat connection in terms of the coordinates of absolutely parallel covectors.