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Differential Geometry of Manifolds

2021 №52

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Polyakova K.V.

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

DOI
10.5922/0321-4796-2021-52-9
Pages
83-96
two-dimensional manifold three-dimensional manifold linear frame bundle structure equations base and fiber coordinates affine torsion

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.

Reference

1.   Akivis, M. A.: Multidimensional differential geometry. Kalinin (1977).

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.  Laptev, G. F.: Fundamental infinitesimal structures of higher or­ders on a smooth manifold. Tr. Geom. Sem., 1, 139—189 (1966).

4.  Polyakova, K. V.: Special affine connection of the 1st and 2nd or­ders. DGMF. Kaliningrad. 46, 114—128 (2015).

5.  Polyakova, K. V.: Second-Order Tangent-Valued Forms. Math. Notes, 105:1, 71—79 (2019).

6.  Rybnikov, A. K.: Affine connections of second order. Math. Notes, 29:2, 143—149 (1981).

7.  Rybnikov, A. K.: Second-order generalized affine connections. Soviet Math. (Izv. Vuzov), 27:1, 84—93 (1983).

8.  Shevchenko, Yu. I.: Laptev and Lumiste methods for the specifca­tion of a connection in a principal bundle. DGMF. Kaliningrad. 37, 179—187 (2006).

9.  Belova, O. O.: Generalized affine connections associated with the space of centered planes. Mathematics, 9 (7), 782 (2021). https://doi. org/10.3390/math9070782.