Generic $\xi^{\perp}$-Riemannian Submersions

Yıl 2022, Cilt: 51 Sayı: 2, 390 - 403, 01.04.2022

Ramazan Sarı

https://doi.org/10.15672/hujms.723602

Öz

As a generalization of semi-invariant $\xi ^{\perp }$-Riemannian submersions, we introduce the generic $\xi ^{\perp }$- Riemannian submersions. We focus on the generic $\xi ^{\perp }$-Riemannian submersions for the Sasakian manifolds with examples and investigate the geometry of foliations. Also, necessary and sufficient conditions for the base manifold to be a local product manifold are obtained and new conditions for totally geodesicity are established. Furthermore, curvature properties of distributions for a generic $\xi ^{\perp }$-Riemannian submersion from Sasakian space forms are obtained and we prove that if the distributions, which define a generic $\xi ^{\perp }$-Riemannian submersion are totally geodesic, then they are Einstein.

Anahtar Kelimeler

Riemannian submersion, Sasakian manifolds, second fundamental form of a map, Einstein manifold

Destekleyen Kurum

Amasya University

Proje Numarası

FMB-BAP18-0335

Teşekkür

Thank you to Amasya University for their support

Kaynakça

• [1] M.A. Akyol, Generic Riemannian submersions from almost product Riemannian manifolds, Gazi Univ. J. Sci. 30 (3), 89-100, 2017.
• [2] M.A. Akyol, Conformal Semi-Invariant Submersions from Almost Product Riemann- ian Manifolds, Acta Math. Vietnam. 42, 491-507, 2017.
• [3] M.A. Akyol, Conformal generic submersions, Turkish J. Math. 45, 201-219, 2021.
• [4] M.A. Akyol and Y. Gündüzalp, Semi inavariant semi-Riemannian submersion, Com- mun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 67 (1), 80-92, 2018.
• [5] M.A. Akyol, R. Sarı and E. Aksoy, Semi-invariant $\xi ^{\perp }$−Riemannian submersions from almost contact metric manifolds, Int. J. Geom. Methods Mod. Phys. 14 (5), 1750074, 2017.
• [6] M.A. Akyol and B. Şahin, Conformal semi-invariant submersions, Commun. Con- temp. Math. 19 (2), 1650011, 2017.
• [7] S. Ali and T. Fatima, Generic Riemannian submersions, Tamkang J. Math. 44 (4), 395-409, 2013.
• [8] S. Ali and T. Fatima, Anti-invariant Riemannian submersions from nearly Kaehler manifolds, Filomat, 27 (7), 1219-1235, 2013.
• [9] P. Baird and J.C. Wood, Harmonic morphisms between Riemannian manifolds, Lon- don Mathematical Society Monographs 29, Oxford University Press, The Clarendon Press. Oxford, 2003.
• [10] D.E. Blair, Contact manifold in Riemannain geometry, Lect. Notes Math. 509, Springer-Verlag, Berlin-New York, 1976.
• [11] J.P. Bourguignon and H.B. Lawson, Stability and isolation phenomena for Yang-mills fields, Commun. Math. Phys. 79, 189-230, 1981.
• [12] J.P. Bourguignon and H.B. Lawson, A mathematician’s visit to Kaluza-Klein theory, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, 143-163, 1989.
• [13] C. Dunn, P. Gilkey and J. H. Park, The spectral geometry of the canonical Riemannian submersion of a compact Lie group, J. Geom. Phys. 57 (10), 2065-2076, 2007.
• [14] M. Falcitelli, S. Ianus and A.M. Pastore, Riemannian submersions and related Topics, World Scientific, River Edge, NJ, 2004.
• [15] T. Fatima and S. Ali, Submersions of generic submanifolds of a Kaehler manifold, Arab J. Math. Sci. 20 (1), 119-131, 2014.
• [16] P. Frejlich, Submersions by Lie algebroids, J. Geom. Phys. 137, 237-246, 2019.
• [17] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16, 715-737, 1967.
• [18] Y. Gündüzalp, Anti-invariant semi-Riemannian submersions from almost para- Hermitian manifolds, J. Funct. Spaces, 720623, 2013.
• [19] S. Ianus and M. Visinescu, Kaluza-Klein theory with scalar fields and generalized Hopf manifolds, Class. Quantum Gravity, 4, 1317-1325, 1987.
• [20] S. Ianus and M. Visinescu, Space-time compaction and Riemannian submersions, The Mathematical Heritage of C. F. Gauss, 358-371, River Edge, World Scientific, 1991.
• [21] J.W. Lee, Anti-invariant $\xi ^{\perp }$−Riemannian submersions from almost contact mani- folds, Hacettepe J. Math. Stat. 42 (3), 231-241, 2013.
• [22] M.T. Mustafa, Applications of harmonic morphisms to gravity, J. Math. Phys. 41, 6918-6929, 2000.
• [23] B. O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13, 458-469, 1966.
• [24] F. Özdemir, C. Sayar and H.M. Taştan, Semi-invariant submersions whose total man- ifolds are locally product Riemannian, Quaest. Math. 40 (7), 909-926, 2017.
• [25] K.S. Park, H-semi-invariant submersions, Taiwan. J. Math. 16 (5), 1865-1878, 2012.
• [26] R. Prasad and S. Kumar, Conformal semi-invariant submersion from almost contact manifolds onto Riemannian manifolds, Khayyam J. Math. 5 (2), 77-95, 2019.
• [27] S. Sasaki and Y. Hatakeyama, On differentiable manifolds with contact metric struc- ture, J. Math. Soc. Jpn. 14, 249-271, 1961.
• [28] C. Sayar, H.M. Tastan, F. Özdemir and M.M. Tripathi, Generic submersion from Kaehler manifold, Bull. Malays. Math. Sci. Soc. 43, 809-831, 2020.
• [29] B. Şahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds, Cent. Eur. J. Math. 3, 437-447, 2010.
• [30] B. Şahin, Semi-invariant Riemannian submersions from almost Hermitian manifolds, Canad. Math. Bull. 56, 173-183, 2011.
• [31] B. Şahin, Generic Riemannian Maps, Miskolc Math. Notes, 18 (1), 453-467, 2017.
• [32] B. Watson, G, G’-Riemannian submersions and nonlinear gauge field equations of general relativity, Global Analysis - Analysis on manifolds, Teubner-Texte Math. Teubner, Leipzig, 57, 324-349, 1983.