Geodesic
Canonical affine connections of the first and second orders
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Tome 203 (2021), pp. 71-83
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We examine affine connections in the bundle of frames of the first and second orders over an $m$-dimensional smooth manifold using the canonical forms of these bundles. We construct the components of the objects of affine connections of the first and second orders determined by zero covariant derivatives of the fiber coordinates of a smooth manifold. These connections are canonical flat affine connections. We examine the objects of deformation from arbitrary affine connections of the first and second orders to the canonical connections of the corresponding orders. Tangent vectors to a smooth manifold are horizontal vectors of the first order for the canonical connections of the first and second orders; these vectors are called first-order canonical vectors. We construct horizontal operators that transform first-order canonical vectors into horizontal vectors of various orders for affine connections of the first and second order. Horizontal vectors are represented as the sums of horizontal vectors of the canonical connection and vertical vectors with the coefficients equal to the components of the deformation tensor from the canonical connection to the given connection.
Keywords: frame bundle, basic coordinates, layer coordinates, affine connection, covariant derivative, horizontal vector. 
@article{INTO_2021_203_a6,
     author = {K. V. Polyakova},
     title = {Canonical affine connections of the first and second orders},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {71--83},
     year = {2021},
     volume = {203},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_203_a6/}
}
TY  - JOUR
AU  - K. V. Polyakova
TI  - Canonical affine connections of the first and second orders
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 71
EP  - 83
VL  - 203
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_203_a6/
LA  - ru
ID  - INTO_2021_203_a6
ER  - 
%0 Journal Article
%A K. V. Polyakova
%T Canonical affine connections of the first and second orders
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 71-83
%V 203
%U http://geodesic.mathdoc.fr/item/INTO_2021_203_a6/
%G ru
%F INTO_2021_203_a6
K. V. Polyakova. Canonical affine connections of the first and second orders. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Tome 203 (2021), pp. 71-83. http://geodesic.mathdoc.fr/item/INTO_2021_203_a6/

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

[2] Belova O. O., “Svyaznosti v rassloeniyakh, assotsiirovannykh s mnogoobraziem Grassmana i prostranstvom tsentrirovannykh ploskostei”, Fundam. prikl. mat., 14:2 (2008), 29–67 | MR

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

[4] Bishop R., Krittenden R., Geometriya mnogoobrazii, Mir, M., 1967

[5] Vasilev A. M., “Differentsialnaya algebra”, Itogi nauki tekhn. Ser. Probl. geom., 10 (1978), 5–23

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

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

[8] Polyakova K. V., “Kovariantnye differentsialy i kovariantnye proizvodnye, assotsiirovannye s poverkhnostyu proektivnogo prostranstva”, Vestn. Balt. feder. un-ta im. I. Kanta., 2013, no. 10, 60–63

[9] Polyakova K. V., “Spetsialnye affinnye svyaznosti 1-go i 2-go poryadkov”, Differ. geom. mnogoobr. figur., 2015, no. 46, 114–128 | Zbl

[10] Polyakova K. V., “O zadanii affinnoi svyaznosti 2-go poryadka vektornoznachnymi formami 1-go, 2-go i 3-go poryadkov”, Differ. geom. mnogoobr. figur., 2016, no. 47, 108–125 | Zbl

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

[12] Polyakova K. V., “Normali na mnogoobrazii i porozhdayuschie ikh otobrazheniya”, Differ. geom. mnogoobr. figur., 2019, no. 50, 110–125 | Zbl

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

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

[15] Shevchenko Yu. I., “Svyaznost v prodolzhenii glavnogo rassloeniya”, Differ. geom. mnogoobr. figur., 1991, no. 22, 117–127 | Zbl

[16] Shevchenko Yu. I., “Priemy Lapteva i Lumiste zadaniya svyaznosti v glavnom rassloenii”, Differ. geom. mnogoobr. figur., 2006, no. 37, 179–187 | Zbl

[17] Catuogno P., “A geometric Itô formula”, Mat. Contemp., 33 (2005), 85–99 | MR

[18] Emery M., “An invitation to second-order stochastic differential geometry”, https://hal.archives-ouvertes.fr/hal-00145073

[19] Janyska J., Kolář I., “On the connections naturally induced on the second order frame bundle”, Arch. Math., 22:1 (1986), 21–28 | MR | Zbl

[20] Polyakova K. V., “Prolongations generated by horizontal vectors”, J. Geom., 110 (2019), 53 | DOI | MR | Zbl

[21] Schwartz L., Géométrie différentielle du 2ème ordre, semi-martingales et équations différentielles stochastiques sur une variété différentielle, Springer, Berlin, 1982 | MR