Geometric Analysis of Riemannian Submersions: Curvature Tensors and Total Umbilic Fibers

Abstract

In this paper, we investigate the geometric properties of Riemannian submersions, providing a comprehensive analysis of various curvature tensors, all associated with a new type of semi-symmetric non-metric connection. We also investigate the behavior of these curvatures in cases where Riemannian submersions have total umbilic fibers.

Keywords

• Curvature tensors
• Riemannian submersion
• Semi-symmetric non-metric connection
• Total umbilic fibers

References

1. [1] Ahsan, Z., Siddiqui, S. A.: Concircular curvature tensor and fluid spacetimes. Int. J. Theor. Phys. 48, 3202-3212 (2009).
2. [2] Akyol, M. A., Ayar, G.: New curvature tensors along Riemannian submersions. Miskolc Mathematical Notes. 24(3), 1161–1184 (2023).
3. [3] Akyol, M. A., Beyendi, S.: Riemannian submersions endowed with a semi-symmetric non-metric connection. Konuralp J. Math. 6(1), 188-193 (2018).
4. [4] Berestovskii, V. N., Guijarro, L.: A metric characterization of Riemannian submersions. Ann Global Anal. Geom. 18, 577-588 (2000).
5. [5] Chaubey, S. K., Yildiz, A.: Riemannian manifolds admitting a new type of semi-symmetric non- metric connection. Turk. J. Math. 43(4), 1887-1904 (2019).
6. [6] Demir, H., Sari, R.: Riemannian submersions with quarter-symmetric non- metric connection. J. Eng. Technol. Appl. Sci. 6(1), 1-8 (2021).
7. [7] Demir, H., Sari, R.: Riemannian submersions endowed with a semi-symmetric metric connection. Euro-Tbil. Math. J. 99-108 (2022).
8. [8] Doric, M., Petrovic-Torgasev, M.: Verstraelen, L. Conditions on the conharmonic curvature tensor of Kaehler hypersurfaces in complex space forms. Publ. Inst. Math. (N.S). 44 (58), 97-108 (1988).

9. [9] Eken Meric¸, S¸ ., G¨ulbahar, M., Kılıc¸, E.: Some inequalities for Riemannian submersions. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N. S.). 63, 1-12 (2017).
10. [10] Escobales, Jr., Richard, H .: Riemannian submersions with totally geodesic fibers. J. Diff. Geom. 10, 253-276 (1975).
11. [11] Falcitelli, M., Ianus, S., Pastore, A. M.: Riemannian Submersions and Related Topics,World Scientific. (2004).
12. [12] Friedmann, A.: Schouten J.A.: U¨ ber die Geometrie der halbsymmetrischen U¨ bertagungen. Mathematische Zeitschrift. 21, 211-223 (1924).
13. [13] Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16(7), 715-737 (1967).
14. [14] Gundmundson, S.: An Introduction to Riemannian Geometry. Lecture Notes in Mathematics. University of Lund, Faculty of Science (2014).
15. [15] Hall, G.: Einstein’s geodesic postulate, projective relatedness and Weyl’s projective tensor. In Mathematical physics: Proceedings of the 14th Regional Conference. 27-35 (2017).
16. [16] Ianus, S., Mazzocco, R., Vilcu, G. E.: Riemannian submersions from quaternionic manifolds. Acta Appl. Math. 104, 83-89 (2008).
17. [17] Karatas, E., Zeren, S., Altin, M.: Riemannian submersions endowed with a new type of semi-symmetric non-metric connection, Therm. Sci. 27 (4B), 3393-3403 (2023).
18. [18] Mishra, R. S.: H􀀀Projective curvature tensor in K¨ahler manifold. Indian J. Pure Appl. Math. 1, 336-340 (1970).
19. [19] Narita, F.: Riemannian submersion with isometric reflections with respect to the fibers, Kodai Math. J. 16, 416-427 (1993).
20. [20] Ojha, R. H.: A note on the M􀀀projective curvature tensor. Indian J. Pure Appl. Math. 8 (12), 1531-1534 (1975).
21. [21] O’Neill, B.: The Fundamental Equations of a Submersions. Michigan Math. J. 13, 459-469 (1966).
22. [22] Pak, E.: On the pseudo-Riemannian spaces. J. Korean Math. Soc. 6, 23-31 (1969).
23. [23] Pokhariyal, G. P., Mishra, R. S.: Curvature tensors and their relativistic significance II. Yokohama Math. J. 19 (2), 97-103 (1971).
24. [24] S¸ ahin, B.: Riemannian submersions from almost Hermitian manifolds. Taiwanese J. math. 17, 629-659 (2013).
25. [25] S¸ ahin, B.: Riemannian submersions, Riemannian maps in Hermitian Geometry and their applications. Elsevier, London (2017).
26. [26] Yano, K.: On semi-symmetric metric connections. Rev. Roum. Math. Pures Appl. 15, 1579-1586 (1970).