Previous |  Up |  Next

Article

Keywords:
totally geodesic; parallel; hypersurface; solvable Lie group
Summary:
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.
References:
[1] Aghasi, M., Nasehi, M.: Some geometrical properties of a five-dimensional solvable Lie group. Differ. Geom. Dyn. Syst. 15 (2013), 1–12. MR 3073067 | Zbl 1331.53071
[2] Aghasi, M., Nasehi, M.: On homogeneous Randers spaces with Douglas or naturally reductive metrics. Differ. Geom. Dyn. Syst. 17 (2015), 1–12. MR 3367072 | Zbl 1333.53068
[3] Aghasi, M., Nasehi, M.: On the geometrical properties of solvable Lie groups. Adv. Geom. 15 (4) (2015), 507–517. DOI 10.1515/advgeom-2015-0025 | MR 3406478 | Zbl 1328.53062
[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 1972463 | Zbl 1029.53071
[5] Božek, M.: Existence of generalized symmetric Riemannian spaces with solvable isometry group. Časopis pěest. mat. 105 (1980), 368–384. MR 0597914 | Zbl 0475.53045
[6] Calvaruso, G., Kowalski, O., Marinosci, R.: Homogeneous geodesics in solvable Lie groups. Acta Math. Hungar. 101 (2003), 313–322. DOI 10.1023/B:AMHU.0000004942.87374.0e | MR 2017938 | Zbl 1057.53041
[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 10.1142/S0129167X09005728 | MR 2574312 | Zbl 1177.53018
[8] Calvaruso, G., Van der Veken, J., : Parallel surfaces in three-dimensional Lorentzian Lie groups. Taiwanese J. Math. 14 (2010), 223–250. DOI 10.11650/twjm/1500405737 | MR 2603452 | Zbl 1194.53019
[9] Calvaruso, G., Van der Veken, J., : Totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups. Results Math. 64 (2013), 135–153. DOI 10.1007/s00025-012-0304-4 | MR 3095133 | Zbl 1279.53056
[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 10.1016/j.geomphys.2009.09.012 | MR 2587393 | Zbl 1205.53061
[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 10.2748/tmj/1238764545 | MR 2501861 | Zbl 1182.53018
[12] De Leo, B., Van der Veken, J., : Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces. Geom. Dedicata 159 (2012), 373–387. DOI 10.1007/s10711-011-9665-1 | MR 2944538 | Zbl 1247.53076
[13] Dillen, F., Van der Veken, J., : Higher order parallel surfaces in the Heisenberg group. Differential Geom. Appl. 26 (1) (2008), 1–8. DOI 10.1016/j.difgeo.2007.11.001 | MR 2393968 | Zbl 1142.53017
[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 2341567 | Zbl 1125.53040
[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 10.1007/s10711-007-9222-0 | MR 2369197 | Zbl 1136.53016
[16] Kowalski, O.: Generalized Symmetric Spaces. Lecture Notes in Math., vol. 805, Springer-Verlag, Berlin, Heidelberg, New York, 1980. MR 0579184 | Zbl 0431.53042
[17] Lawson, H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. of Math. (2) 89 (1969), 187–197. DOI 10.2307/1970816 | MR 0238229 | Zbl 0174.24901
[18] Nasehi, M.: Parallel and totally geodesic hypersurfaces of 5-dimensional 2-step homogeneous nilmanifolds. Czechoslovak Math. J. 66 (2) (2016), 547–559. DOI 10.1007/s10587-016-0274-x | MR 3519620
[19] Simon, U., Weinstein, A.: Anwendungen der De Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie. Manuscripta Math. 1 (1969), 139–146. DOI 10.1007/BF01173099 | MR 0246234 | Zbl 0172.46701
Partner of
EuDML logo