×

Totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups. (English) Zbl 1279.53056

The oscillator algebra is the real Lie algebra with four generators \(X,Y,P,Q\) such that the only non-vanishing relations are the following: \[ [X,Y]=P,\quad [Q,X]=\lambda Y\quad \text{and}\quad [Q,Y]=-\lambda X. \] It appears naturally as an algebra of differential operators associated to the harmonic oscillator.
The oscillator group is the unique simply connected Lie group associated to the oscillator algebra. The oscillator group admits a natural left-invariant Lorentz metric.
In this nice paper the authors investigate totally geodesic and parallel hypersurfaces in oscillator groups. The main result is a complete classification of these hypersurfaces, which is stated in Theorem 2 in an explicit coordinate description.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C35 Differential geometry of symmetric spaces
22E99 Lie groups
53C30 Differential geometry of homogeneous manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boucetta, M., Medina, A.: Solutions of the Yang-Baxter equations on orthogonal groups: the case of oscillator groups. Arxiv[Math DG]: 1008.2435v3, 2010.
[2] Bromberg S., Medina A.: Geometry of oscillator groups and locally symmetric manifolds. Geom. Dedicata 106, 97–111 (2004) · Zbl 1088.53045 · doi:10.1023/B:GEOM.0000033845.70512.13
[3] Calvaruso G., Vander Veken J.: Parallel surfaces in three-dimensional Lorentzian Lie groups. Taiwan. J. Math. 14, 223–250 (2010) · Zbl 1194.53019
[4] Calvaruso G., Vander Veken J.: Lorentzian symmetric three-spaces and the classification of their parallel surfaces, Int. J. Math. 20, 1185–1205 (2009) · Zbl 1177.53018
[5] Duran Diaz R., Gadea P.M., Oubiña J.A.: Reductive decompositions and Einstein-Yang-Mills equations associated to the oscillator group. J. Math. Phys. 40, 3490–3498 (1999) · Zbl 0978.53095 · doi:10.1063/1.532902
[6] Gadea P.M., Oubiña J.A.: Homogeneous Lorentzian structures on the oscillator groups. Arch. Math. 73, 311–320 (1999) · Zbl 0954.53029 · doi:10.1007/s000130050403
[7] Levitchev A.V.: Chronogeometry of an electromagnetic wave given by a bi-invariant metric on the oscillator group. Sib. Math. J. 27, 237–245 (1986) · Zbl 0602.53057 · doi:10.1007/BF00969391
[8] Levitchev A.V.: Methods of investigation of the causal structure of homogeneous Lorentz manifolds. Sib. Math. J. 31, 395–408 (1990) · Zbl 0713.53035 · doi:10.1007/BF00970346
[9] Medina A.: Groupes de Lie munis de métriques bi-invariantes. Tôhoku Math. J. 37, 405–421 (1985) · Zbl 0583.53053 · doi:10.2748/tmj/1178228586
[10] Müller D., Ricci F.: On the Laplace-Beltrami operator on the oscillator group. J. Reine Angew. Math. 390, 193–207 (1988) · Zbl 0644.58018
[11] Naitoh H.: Symmetric submanifolds of compact symmetric spaces. Tsukuba J. Math. 10, 215–242 (1986) · Zbl 0619.53033
[12] Streater R.F.: The representations of the oscillator group. Commun. Math. Phys. 4, 217–236 (1967) · Zbl 0155.32503 · doi:10.1007/BF01645431
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.