Geodesic
Affine connections on three-dimensional pseudo-Riemannian homogeneous spaces.~I
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2013), pp. 51-68.

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

The goal of this paper is to describe all invariant affine connections on pseudo-Riemannian homogeneous spaces of dimensions 2 and 3. We present a complete local classification of Riemannian homogeneous spaces which is equivalent to the description of effective pairs of Lie algebras supplied with an invariant nondegenerate symmetric bilinear form on the isotropy module. The classification of pseudo-Riemannian homogeneous spaces is given in a separate paper (Part 2). We describe all invariant affine connections together with their curvature and torsion tensors and indicate affine connections on Riemannian homogeneous spaces and Riemannian connections.
Mots-clés : invariant affine connection
Keywords: pseudo-Riemannian homogeneous spaces. 
@article{IVM_2013_12_a4,
     author = {N. P. Mozhei},
     title = {Affine connections on three-dimensional {pseudo-Riemannian} homogeneous {spaces.~I}},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {51--68},
     publisher = {mathdoc},
     number = {12},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2013_12_a4/}
}
TY  - JOUR
AU  - N. P. Mozhei
TI  - Affine connections on three-dimensional pseudo-Riemannian homogeneous spaces.~I
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2013
SP  - 51
EP  - 68
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2013_12_a4/
LA  - ru
ID  - IVM_2013_12_a4
ER  - 
%0 Journal Article
%A N. P. Mozhei
%T Affine connections on three-dimensional pseudo-Riemannian homogeneous spaces.~I
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2013
%P 51-68
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2013_12_a4/
%G ru
%F IVM_2013_12_a4
N. P. Mozhei. Affine connections on three-dimensional pseudo-Riemannian homogeneous spaces.~I. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2013), pp. 51-68. http://geodesic.mathdoc.fr/item/IVM_2013_12_a4/

[1] Levi-Civita T., “Nozione di parallelismo in una varietà qualunque”, Rend. Circ. Mat. Palermo, 42 (1917), 173–205 | DOI

[2] Weyl H., Raum, Zeit, Materie, 6e Aufl., Springer, 1923 | MR

[3] Cartan É., “Les espaces à connexion conforme”, Ann. Soc. Polon. Math., 2 (1923), 171–211

[4] Lumiste Yu. G., “Teoriya svyaznostei v rassloennykh prostranstvakh”, Itogi nauki i tekhn. Algebra. Topologiya. Geometriya, 8, VINITI AN SSSR, M., 1971, 123–168 | MR | Zbl

[5] Kruchkovich G. I., “Klassifikatsiya trekhmernykh rimanovykh prostranstv po gruppam dvizhenii”, UMN, 9:1 (1954), 3–40 | MR | Zbl

[6] Thurston W., “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”, Bull. Amer. Math. Soc., 6 (1982), 357–381 | DOI | MR

[7] Scott P., “The geometries of 3-manifolds”, Bull. London Math. Soc., 15:5 (1983), 401–487 | DOI | MR | Zbl

[8] Shirokov P. A., Izbrannye trudy po geometrii, Izd-vo Kazansk. un-ta, Kazan, 1966

[9] Petrov A. Z., “O geodezicheskom otobrazhenii rimanovykh prostranstv neopredelennoi metriki”, Uchen. zap. Kazansk. un-ta, 109, no. 4, 1949, 7–36

[10] Onischik A. L., Topologiya tranzitivnykh grupp Li preobrazovanii, Izd-vo Fizmatlit, M., 1995

[11] Kobayashi S., Nomizu K., Foundations of differential geometry, v. 1, N.-Y.–London, 1963

[12] Kobayashi S., Nomizu K., Foundations of differential geometry, v. II, N.-Y.–London, 1969 | MR

[13] Mostow G. D., “The extensibility of local Lie groups of transformations and groups on surfaces”, Ann. Math., 52:3 (1950), 606–636 | DOI | MR | Zbl

[14] Lie S., Theorie der Transformationsgruppen, Abschn. III, Teubner, Leipzig, 1893 | Zbl

[15] Komrakov B., Churyumov A., Doubrov B., Two-dimensional homogeneous spaces, Preprint No 17, Univ. Oslo, 1993

[16] Nomizu K., “Invariant affine connections on homogeneous spaces”, Amer. J. Math., 76:1 (1954), 33–65 | DOI | MR | Zbl

[17] Kowalski O., “Riemannian manifolds with homogeneous geodesics”, Boll. Unione Math. Ital. (VII) Ser. B, 5:1 (1991), 189–246 | MR | Zbl

[18] Komrakov B., Tchourioumov A., Mozhey N. et al., Three-dimensional isotropically-faithful homogeneous spaces, Preprints No 35–37, Univ. Oslo, 1993