Geodesic
Parallel and totally geodesic hypersurfaces of solvable Lie groups
Archivum mathematicum, Tome 52 (2016) no. 4, pp. 221-231 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper we consider special examples of homogeneous spaces of arbitrary odd dimension which are given in [5] and [16]. We obtain the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces in both Riemannian and Lorentzian cases.
In this paper we consider special examples of homogeneous spaces of arbitrary odd dimension which are given in [5] and [16]. We obtain the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces in both Riemannian and Lorentzian cases.
DOI : 10.5817/AM2016-4-221
Classification : 53C30, 53C42
Keywords: totally geodesic; parallel; hypersurface; solvable Lie group 
@article{10_5817_AM2016_4_221,
     author = {Nasehi, Mehri},
     title = {Parallel and totally geodesic hypersurfaces of solvable {Lie} groups},
     journal = {Archivum mathematicum},
     pages = {221--231},
     year = {2016},
     volume = {52},
     number = {4},
     doi = {10.5817/AM2016-4-221},
     mrnumber = {3610651},
     zbl = {06674901},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2016-4-221/}
}
TY  - JOUR
AU  - Nasehi, Mehri
TI  - Parallel and totally geodesic hypersurfaces of solvable Lie groups
JO  - Archivum mathematicum
PY  - 2016
SP  - 221
EP  - 231
VL  - 52
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2016-4-221/
DO  - 10.5817/AM2016-4-221
LA  - en
ID  - 10_5817_AM2016_4_221
ER  - 
%0 Journal Article
%A Nasehi, Mehri
%T Parallel and totally geodesic hypersurfaces of solvable Lie groups
%J Archivum mathematicum
%D 2016
%P 221-231
%V 52
%N 4
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2016-4-221/
%R 10.5817/AM2016-4-221
%G en
%F 10_5817_AM2016_4_221
Nasehi, Mehri. Parallel and totally geodesic hypersurfaces of solvable Lie groups. Archivum mathematicum, Tome 52 (2016) no. 4, pp. 221-231. doi: 10.5817/AM2016-4-221

[1] Aghasi, M., Nasehi, M.: Some geometrical properties of a five-dimensional solvable Lie group. Differ. Geom. Dyn. Syst. 15 (2013), 1–12. | MR | Zbl

[2] Aghasi, M., Nasehi, M.: On homogeneous Randers spaces with Douglas or naturally reductive metrics. Differ. Geom. Dyn. Syst. 17 (2015), 1–12. | MR | Zbl

[3] Aghasi, M., Nasehi, M.: On the geometrical properties of solvable Lie groups. Adv. Geom. 15 (4) (2015), 507–517. | DOI | MR | Zbl

[4] Belkhelfa, M., Dillen, F., Inoguchi, J.: Surfaces with parallel second fundamental form in Bianchi Cartan Vranceanu spaces. PDE's, Submanifolds and affine differential geometry, vol. 57, Banach Centre Publishing, Polish Academy Sciences, Warsaw, 2002, pp. 67–87. | MR | Zbl

[5] Božek, M.: Existence of generalized symmetric Riemannian spaces with solvable isometry group. Časopis pěest. mat. 105 (1980), 368–384. | MR | Zbl

[6] Calvaruso, G., Kowalski, O., Marinosci, R.: Homogeneous geodesics in solvable Lie groups. Acta Math. Hungar. 101 (2003), 313–322. | DOI | MR | Zbl

[7] Calvaruso, G., Van der Veken, J., : Lorentzian symmetric three-spaces and the classification of their parallel surfaces. Internat. J. Math. 20 (2009), 1185–1205. | DOI | MR | Zbl

[8] Calvaruso, G., Van der Veken, J., : Parallel surfaces in three-dimensional Lorentzian Lie groups. Taiwanese J. Math. 14 (2010), 223–250. | DOI | MR | Zbl

[9] Calvaruso, G., Van der Veken, J., : Totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups. Results Math. 64 (2013), 135–153. | DOI | MR | Zbl

[10] Chen, B.-Y.: Complete classification of parallel spatial surfaces in pseudo-Riemannian space forms with arbitrary index and dimension. J. Geom. Phys. 60 (2010), 260–280. | DOI | MR | Zbl

[11] Chen, B.-Y., Van der Veken, J., : Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms. Tohoku Math. J. 61 (2009), 1–40. | DOI | MR | Zbl

[12] De Leo, B., Van der Veken, J., : Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces. Geom. Dedicata 159 (2012), 373–387. | DOI | MR | Zbl

[13] Dillen, F., Van der Veken, J., : Higher order parallel surfaces in the Heisenberg group. Differential Geom. Appl. 26 (1) (2008), 1–8. | DOI | MR | Zbl

[14] Inoguchi, J., Van der Veken, J., : Parallel surfaces in the motion groups E(1, 1) and E(2). Bull. Belg. Math. Soc. Simon Stevin. 14 (2007), 321–332. | MR | Zbl

[15] Inoguchi, J., Van der Veken, J., : A complete classification of parallel surfaces in three-dimensional homogeneous spaces. Geom. Dedicata 131 (2008), 159–172. | DOI | MR | Zbl

[16] Kowalski, O.: Generalized Symmetric Spaces. Lecture Notes in Math., vol. 805, Springer-Verlag, Berlin, Heidelberg, New York, 1980. | MR | Zbl

[17] Lawson, H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. of Math. (2) 89 (1969), 187–197. | DOI | MR | Zbl

[18] Nasehi, M.: Parallel and totally geodesic hypersurfaces of 5-dimensional 2-step homogeneous nilmanifolds. Czechoslovak Math. J. 66 (2) (2016), 547–559. | DOI | MR

[19] Simon, U., Weinstein, A.: Anwendungen der De Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie. Manuscripta Math. 1 (1969), 139–146. | DOI | MR | Zbl

Cité par Sources :