Geodesic
On the structure of an affine connection object and the torsion tensor in the bundle of linear frames
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 1, Tome 220 (2023), pp. 99-112.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: bundle of linear frames, structural equations, basic and layer coordinates, Pfaffian derivatives, affine connection, absolute parallelism, flat affine connection, symmetric flat connection.
Mots-clés : torsion of affine connection 
@article{INTO_2023_220_a10,
     author = {K. V. Polyakova},
     title = {On the structure of an affine connection object and the torsion tensor in the bundle of linear frames},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {99--112},
     publisher = {mathdoc},
     volume = {220},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2023_220_a10/}
}
TY  - JOUR
AU  - K. V. Polyakova
TI  - On the structure of an affine connection object and the torsion tensor in the bundle of linear frames
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2023
SP  - 99
EP  - 112
VL  - 220
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2023_220_a10/
LA  - ru
ID  - INTO_2023_220_a10
ER  - 
%0 Journal Article
%A K. V. Polyakova
%T On the structure of an affine connection object and the torsion tensor in the bundle of linear frames
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2023
%P 99-112
%V 220
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2023_220_a10/
%G ru
%F INTO_2023_220_a10
K. V. Polyakova. On the structure of an affine connection object and the torsion tensor in the bundle of linear frames. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 1, Tome 220 (2023), pp. 99-112. http://geodesic.mathdoc.fr/item/INTO_2023_220_a10/

[1] Akivis M. A., Mnogomernaya differentsialnaya geometriya, Kalinin, 1977

[2] Belova O. O., “Svyaznosti trekh tipov v rassloenii nad oblastyu proektivnogo prostranstva”, Differ. geom. mnogoobr. figur., 2003, no. 34, 21–26

[3] Belova O. O., “Tenzor krucheniya podsvyaznosti v rassloenii nad grassmanopodobnym mnogoobraziem tsentrirovannykh ploskostei”, Differ. geom. mnogoobr. figur., 2011, no. 42, 7–11

[4] Belova O. O., “Indutsirovanie analoga svyaznosti Neifelda na grassmanopodobnom mnogoobrazii tsentrirovannykh ploskostei”, Differ. geom. mnogoobr. figur., 2014, no. 45, 23–29

[5] Belova O. O., “Grassmanopodobnoe mnogoobrazie tsentrirovannykh ploskostei”, Mat. zametki., 104:6 (2018), 812–822

[6] Evtushik L. E., Lumiste Yu. G., Ostianu N. M., Shirokov A. P., “Differentsialno-geometricheskie struktury na mnogoobraziyakh”, Itogi nauki i tekhn. Probl. geom., 9 (1979), 5–246

[7] Kagan F. I., “Affinnye svyaznosti na kasatelnom rassloenii”, Izv. vuzov. Mat., 1975, no. 2, 31—42

[8] Kirichenko V. F., Differentsialno-geometricheskie struktury na mnogoobraziyakh, MPGU, M., 2003

[9] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii. T. 1, Nauka, M., 1981

[10] Laptev G. F., “Osnovnye infinitezimalnye struktury vysshikh poryadkov na gladkom mnogoobrazii”, Tr. geom. semin. VINITI., 1 (1966), 139–189

[11] Morozov O. I., Metod podvizhnogo korepera v geometrii differentsialnykh uravnenii, Diss. na soisk. uch. step. d-ra fiz.-mat. nauk, M., 2010

[12] Norden A. P., Prostranstva affinnoi svyaznosti, Nauka, M., 1976

[13] Polyakova K. V., “Spetsialnye affinnye svyaznosti pervogo i vtorogo poryadkov”, Differ. geom. mnogoobr. figur., 2015, no. 46, 114–128

[14] Polyakova K. V., “Tangentsialnoznachnye formy 2-go poryadka”, Mat. zametki., 105:1 (2019), 84–94

[15] Polyakova K. V., “O tenzore krucheniya affinnoi svyaznosti na dvumernom i trekhmernom mnogoobraziyakh”, Differ. geom. mnogoobr. figur., 2021, no. 52, 83–96

[16] Rybnikov A. K., “Ob affinnykh svyaznostyakh vtorogo poryadka”, Mat. zametki., 29:2 (1981), 279–290

[17] Rybnikov A. K., “Ob obobschennykh affinnykh svyaznostyakh vtorogo poryadka”, Izv. vuzov. Mat., 1983, no. 1, 73–80

[18] Stolyarov A. V., “Differentsialnaya geometriya polos”, Itogi nauki i tekhn. Probl. geom., 10 (1978), 25–54

[19] Shevchenko Yu. I., Osnascheniya golonomnykh i negolonomnykh gladkikh mnogoobrazii, Kaliningrad, 1998

[20] Shevchenko Yu. I., “Priemy Lapteva i Lumiste zadaniya svyaznosti v glavnom rassloenii”, Differ. geom. mnogoobr. figur., 2006, no. 37, 185–193

[21] Akivis M. A., Goldberg V. V., Projective Differential Geometry of Submanifolds, North-Holland, 1993

[22] Kolář I., Michor P. W., Slovák J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993

[23] Polyakova K. V., “Generators of flat and symmetric flat affine connections”, Proc. Int. Conf. “Problems of Modern Topology and Its Applications”, Tashkent, 2016, 82–83

[24] Shevchenko Yu., Skrydlova E., “Interpretation of classical affine connection by means Laptev affine connection”, Proc. Int. Conf. “Geometry Days in Novosibirsk–2018”, Sobolev Inst. Math., Novosibirsk, 2018, 28