Affiliation:
1. Immanuel Kant Baltic Federal University
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.
Publisher
Immanuel Kant Baltic Federal University
Subject
Geology,Ocean Engineering,Water Science and Technology
Reference9 articles.
1. 1. Akivis, M. A.: Multidimensional differential geometry. Kalinin (1977).
2. 2. Evtushik, L. E., Lumiste, Yu. G., Ostianu, N. M., Shirokov, A. P.: Differential-geometric structures on manifolds. J. Soviet Math. 14:6, 1573—1719 (1980).
3. 3. Laptev, G. F.: Fundamental infinitesimal structures of higher orders on a smooth manifold. Tr. Geom. Sem., 1, 139—189 (1966).
4. 4. Polyakova, K. V.: Special affine connection of the 1st and 2nd orders. DGMF. Kaliningrad. 46, 114—128 (2015).
5. 5. Polyakova, K. V.: Second-Order Tangent-Valued Forms. Math. Notes, 105:1, 71—79 (2019).