Geodesic
Left-invariant metrics on a tensor bundle of type $(2,0)$ over a Lie group
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 4, pp. 146-155 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we construct vertical and horizontal lifts of left-invariant vector fields. We establish necessary and sufficient conditions for the horizontal lift of a left-invariant vector field to be a left-invariant field. From a left-invariant metric on $G$, we build a left-invariant metric on $T^2_0G$.
Keywords: left-invariant metric, tensor bundle over a Lie group, vertical and horizontal lifts, left-invariant vertical and horizontal distributions. 
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     author = {N. A. Opokina},
     title = {Left-invariant metrics on a~tensor bundle of type $(2,0)$ over {a~Lie} group},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {146--155},
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     volume = {154},
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     url = {http://geodesic.mathdoc.fr/item/UZKU_2012_154_4_a12/}
}
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N. A. Opokina. Left-invariant metrics on a tensor bundle of type $(2,0)$ over a Lie group. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 4, pp. 146-155. http://geodesic.mathdoc.fr/item/UZKU_2012_154_4_a12/

[1] Shapukov B. N., “Tenzornye rassloeniya”, Pamyati Lobachevskogo posvyaschaetsya, Sb. nauch. st., Izd-vo Kazan. un-ta, Kazan, 1992, 104–125 | MR

[2] Opokina N. A., “Kasatelnye i tenzornye rassloeniya tipa $(2,0)$ nad gruppoi Li”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 147, no. 1, 2005, 138–147 | Zbl

[3] Postnikov M. M., Lektsii po geometrii. Semestr 4. Differentsialnaya geometriya, Nauka, M., 1988, 496 pp. | MR | Zbl

[4] Shapukov B. N., Zadachi po gruppam Li i ikh prilozheniyam, NITs “RKhD”, M., 2002, 256 pp.

[5] Opokina N. A., “Levaya svyaznost na tenzornom rassloenii tipa $(2,0)$ nad gruppoi Li”, Izv. vuzov. Matem., 2006, no. 11, 77–82 | MR

[6] Eizenkhart L. P., Nepreryvnye gruppy preobrazovanii, Editorial URRS, M., 2004, 362 pp.

[7] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, v 2 t., v. 1, Nauka, M., 1987, 344 pp. | MR

[8] Postnikov M. M., Lektsii po geometrii. Semestr 5. Rimanova geometriya, Faktorial, M., 1998, 496 pp.

[9] Gavrilov S. P., “Geodezicheskie levoinvariantnykh metrik na svyaznoi dvumernoi neabelevoi gruppe Li”, Gravitatsiya i teoriya otnositelnosti. Sb. st., 18, Izd-vo Kazan. un-ta, Kazan, 1981, 28–44 | MR