Lagrangian and Clairaut anti-invariant semi-Riemannian submersions in para-Kaehler geometry

Yılmaz Gündüzalp; Murat  Polat

doi:10.5269/bspm.64600

Yılmaz Gündüzalp Dicle University
Murat Polat Dicle University https://orcid.org/0000-0003-1846-0817

Abstract

Purpose of this article is to examine some geometric features of Clairaut anti-invariant semi-Riemannian submersions from para-Kaehler manifold to a Riemannian manifold. We give Lagrangian semi-Riemannian submersion in para-Kaehler space froms. Then, we investigate under what conditions Clairaut submersions can become anti-invariant semi-Riemannian submersions. After, we obtain conditions for totally geodesic on vertical and horizontal distributions. We also supply a non-trivial example of Clairaut submersion

Downloads

Download data is not yet available.

Author Biographies

Yılmaz Gündüzalp, Dicle University

Mathematics Education

Murat Polat, Dicle University

Department of Mathematics

References

Akyol, M. A. and S. ahin, B., Conformal anti-invariant submersions from almost Hermitian manifolds, Turk. J. Math., 40, 43-70, (2016).

Akyol, M. A., Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistics, 46(2), 177-192, (2017).

Allison, D., Lorentzian Clairaut submersions, Geometriae Dedicate, 63, 309-319, (1996).

Baditoiu, G. and Ianus, S., Semi-Riemannian submersions from real and complex pseudo-hyperbolic spaces, Diff. Geom. and appl. 16, 79-84, (2002).

Baditoiu, G. and Ianus, S., Semi-Riemannian submersions with totally umbilic fibres, Rendiconti del CircoloMatematico di Palermo Serie II, Tomo LI (2002), pp. 249-276.

Bishop, R. L., Clairaut submersions, Differential Geometry (in Honer of Kentaro Yano), Kinokuniya, Tokyo, 21-31, (1972).

Bourguignon, J. P. and Lawson, H. B., A mathematician’s visit to Kaluza- Klein theory, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, 143-163, (1989).

Bourguignon, J. P. and Lawson, H. B., Stability and isolation phenomena for Yang-Mills fields, Comm. Math. Phys. 79, 189-230, (1981).

Caldarella, A. V., On para-quaternionic submersions between para-quaternionic Kähler manifolds, Acta Applicandae Mathematicae 112, 1-14, (2010).

Chen, B-Y., Lagrangian H-umbilical submanifolds of para-Kahler manifolds, Taiwanese J. Math. 15 (6), 2483-2502, (2011).

Cruceanu, V., Fortuny, P. and Gadea, P. M., A Survey on Paracomplex Geometry, Rocky Mountain J. Math. 26(1): 83-115, (1996).

Falcitelli, M., Ianus, S. and Pastore, A. M., Riemannian Submersions and Related Topics, World Scientific, (2004).

Gilkey, P., Itoh, M. and and Park, J. H., Anti-invariant Riemannian Submersions: A Lie-theoretical Approach, Taiwanese J. Math., 20(4), 787-800, (2016).

Gündüzalp, Y. and S. ahin, B., Paracontact semi-Riemannian submersions, Turkish J.Math. 37(1), 114-128, (2013).

Gündüzalp, Y. and S. ahin, B., Para-contact para-complex semi-Riemannian submersions, Bull. Malays. Math. Sci. Soc., 37(1), 139-152, (2014).

Gündüzalp, Y., Anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds, Journal of Function Spaces abd Applications, ID 720623, (2013).

Gündüzalp, Y. Anti-invariant pseudo-Riemannian submersions and Clairaut submersions from paracosymplectic manifolds, Mediterranean Journal of Mathematics, 16(4), 1-18, (2019).

Gray, A., Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16, 715-737, (1967).

Ianus, S. and Visinescu, M., Kaluza-Klein theory with scalar fields and generalised Hopf manifolds, Classical Quantum Gravity 4, 1317-1325, (1987).

Ianus, S. and Visinescu, M. Space-time compactification and Riemannian submersions, The mathematical heritage of C.F. Gauss, World Sci. Publ., River Edge, NJ, 358-371, (1991).

Ianus, S., Mazzocco, R. and Vilcu, G. E., Riemannian submersions from quaternionic manifolds, Acta Appl. Math. 104, 83-89, (2008).

Ianus, S., Vilcu, G. E. and Voicu, R. C., Harmonic maps and Riemannian submersions between manifolds endowed with special structures, Banach Center Publications 93, 277-288, (2011).

Erken, I.K. and Murathan, C., Anti-Invariant Riemannian Submersions from Cosymplectic Manifolds onto Riemannian Manifolds, Filomat 97, No. 7, pp. 1429–1444, (2015).

Lee, J., Park, J. H., S. ahin, B. and Song, D. Y., Einstein conditions for the base of anti-invariant Riemannian submersions and Clairaut submersions, Taiwanese J. Math. 19(4), 1145-1126, (2015).

Lee, J. W., Anti-Invariant ?-Riemannian Submersions from Almost Contact Manifolds, Hacettepe Journal of Mathematics and Statistics, 42(2), 231-241, (2013).

O‘Neill, B., The fundamental equations of a submersion, Michigan Math. J. 13, 459-469, (1966).

O‘Neill, B., Semi-Riemannian Geometry with Application to Relativity, Academic Press, New York, 1983.

Park, K.S., H-anti-invariant submersions from almost quaternionic Hermitian manifolds, Czechoslovak Mathematical Journal 67(2), 557-578, (2017).

S. ahin, B., Anti-invariant Riemannian submersions from almost Hermitian manifolds, Central European J.Math, 8(3), 437-447, (2010).

S. ahin, B., Riemannian submersions, Riemannian maps in Hermitian geometry, and their applications, Academic Press, (2017).

Tas. tan, H. M. and Gerdan, S., Clairaut anti-invariont submersions from Sasakian and Kenmotsu manifolds, Mediterr. J. Math. 14(6), (2017).

Watson, B., Almost Hermitian submersions, J. Differential Geom. 11, 147-165, (1976).