Some Notes Concerning Riemannian Submersions and Riemannian Homogenous Spaces

Year 2019, Volume: 12 Issue: 1, 116 - 125, 27.03.2019

Mehmet Gülbahar , Erol Kılıç , Sadık Keleş

https://doi.org/10.36890/iejg.545856

Abstract

Riemannian submersions between Lie groups and Riemannian homogeneous spaces are

investigated. With the help of connections, some characterizations dealing these spaces are

obtained.

Keywords

Riemann submersion, Lie group, Riemannian homogeneous space, connection

References

• [1] Agricola, I., Ferreira, A. C., Tangent Lie groups are Riemannian naturally reductive spaces. Adv. in Appl. Clifford Algebras 27 (2017), 895-911.
• [2] Arvanitogeorgos, A., An introduction to Lie groups and the geometry of homogeneous spaces. American Mathematical Soc. 22, 2003.
• [3] Arvanitoyeorgos, A., Lie transformation groups and geometry. In Proceedings of the Ninth International Conference on Geometry, Integrability and Quantization (pp. 11-35). Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, 2008.
• [4] Berdinsky, D. A., Taimanov, I. A., Surfaces in three-dimensional Lie groups. Siberian Math. J. 46(6) (2005), 1005–1019.
• [5] Besse, A. L., Einstein manifolds. Springer Science, Business Media, 2007.
• [6] Falcitelli, M., Ianus, S., Pastore, A. M., Riemannian submersions and related topics. World Scientific Company, 2004.
• [7] Fegan, H. D., Introduction to compact Lie groups. Vol. 13, World Scientific Publishing Company, 1991.
• [8] Ferus, D., Symmetric submanifolds of Euclidean space. Mathematische Annalen 247(1) (1980), 81-93.
• [9] Guijarro, L., Walschap, G., When is a Riemannian submersion homogeneous?. Geometriae Dedicata 125(1) (2007), 47-52.
• [10] Gülbahar, M., Eken Meriç S., Kılıç, E., Sharp inequalities involving the Ricci curvature for Riemannian submersions. Kragujevac J. Math. 42(2) (2017), 279-293. [11] Hsiang, W.Y., Lawson Jr, H. B., Minimal submanifolds of low cohomogeneity. J. Differential Geom. 5(1-2) (1971), 1-38.
• [12] Kirillov, A. A., Elements of the theory of representations. Springer-Verlag, Berlin, Heidelberg, New York 1976.
• [13] Kobayashi, S., Submersions of CR-submanifolds. Tohoku Math. J. 89 (1987), 95-100.
• [14] Megia, I. S. M., Which spheres admit a topological group structure. Rev. R. Acad. Cienc. Exactas Fıs. Quım. Nat. Zaragoza 62 (2007), 75-79.
• [15] O’Neill, B., Semi-Riemannian geometry with applications to relativity. Academic press, United Kingdom (1983).
• [16] Pro, C., Wilhelm, F., Flats and submersions in non-negative curvature. Geometriae Dedicata 161(1) (2012), 109-118.
• [17] Ranjan, A., Riemannian submersions of compact simple Lie groups with connected totally geodesic fibres. Mathematische Zeitschrift 191(2) (1986), 239-246.
• [18] Sahin, B., Riemannian submersions, Riemannian maps in Hermitian geometry and their applications. Academic Press 2017.
• [19] Sepanski, M. R., Compact lie groups. Springer Science, Business Media, 2007.
• [20] Tapp, K., Flats in Riemannian submersions from Lie groups. Asian J. of Math. 13(4) (2009), 459-464.