Conformal anti-invariant submersions from cosymplectic manifolds

Abstract

We introduce conformal anti-invariant submersions from cosymplectic manifolds onto Riemannian manifolds. We give an example of a conformal anti-invariant submersion such that characteristic vector field $\xi$ is vertical. We investigate the geometry of foliations which are arisen from the definition of a conformal submersion and show that the total manifold has certain product structures. Moreover, we examine necessary and sufficient conditions for a conformal anti-invariant submersion to be totally geodesic and check the harmonicity of such submersions.

Keywords

• Cosymplectic manifold,Conformal submersion,Conformal anti-invariant submersion

References

1. Akyol, M. A., Sar R. and Aksoy E. Semi-invariant $\xi^\perp$-Riemannian submersions from almost contact metric manifolds, Int. J. Geom. Methods Mod. Phys, DOI: 10.1142/S0219887817500748, 2017.
2. Akyol, M. A. and “ahin, B. Conformal anti-invariant submersions from almost Hermitian manifolds, Turkish Journal of Mathematics, 40 (1), 43-70, 2016.
3. Akyol, M. A. and “ahin, B. Conformal semi-invariant submersions, Commun. Contemp. Math. 19, 1650011, 22 pp, 2017.
4. Blair, D. E. Contact manifold in Riemannain geometry, Lecture Notes in Math. 509 (Springer-Verlag, Berlin-New York), 1976.
5. Blair, D. E. The theory of quasi-Sasakian structure, J. Differential Geom. 1, no. 3-4, 331-345, 1967.
6. Baird, P. and Wood, J. C. Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, 29 (Oxford University Press, The Clarendon Press. Oxford), 2003.
7. Cengizhan, M. and Erken I. K. Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian submersions, Filomat. 29 (7), 1429-1444, 2015.
8. Erken, I. K. and Cengizhan, M. Anti-invariant Riemannian submersions from Sasakian manifolds, arxiv:1302.4906.

9. Chinea, D. Harmonicity on maps between almost contact metric manifolds. Acta Math. Hungar. 126 (4), 352-365, 2010.
10. Chinea, D. Harmonicity of holomorphic maps between almost Hermitian manifolds. Canad. Math. Bull. 52 (1), 18-27, 2009.
11. Chinea, D. On horizontally conformal $(\varphi,\varphi')$-holomorphic submersions. Houston J. Math. 34 (3), 721-737, 2008.
12. Fuglede, B. Harmonic Morphisms Between Riemannian Manifolds, Ann. Inst. Fourier (Grenoble) 28, 107-144, 1978.
13. Falcitelli, M., Ianus, S. and Pastore, A. M.Riemannian submersions and Related Topics. World Scientic, River Edge, NJ, 2004.
14. Gundmundsson, S. The Geometry of Harmonic Morphisms, Ph.D. Thesis, University of Leeds, 1992.
15. Gundmundsson, S. and Wood, J. C. Harmonic Morphisms between almost Hermitian man- ifolds. Boll. Un. Mat. Ital. B. 11 (2), 185-197, 1997.
16. Gündüzalp, Y. Anti-invariant semi-Riemannian submersions from almost para Hermitian manifolds, Journal of Function Spaces and Applications, 1 (7), http://dx.doi.org/10.1155/2013/720623, 2013.
17. Gray, A. Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16, 715-737, 1967.
18. Ishihara, T. A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. kyoto Univ. 19, 215-229, 1979.
19. Jin, D. H. and Lee, J. W. Conformal anti-invariant submersions from hyperkahler manifolds, JP Journal of Geometry and Topology, 19 (2), (2016), 161-183.
20. Lee, J. W. Anti-invariant $\xi^\perp$-Riemannian submersions from almost contact manifolds, Hacettepe Journal of Mathematics and Statistic, 42 (3), 231-241, 2013.
21. Ludden, G. D. Submanifolds of cosymplectic manifolds, J. Differential Geom. 4, 237-244, 1970.
22. Olzsak, Z. On almost cosymplectic manifolds, Kodai Math J. 4, 239-250, 1981.
23. O'Neill, B. The fundamental equations of a submersion. Mich. Math. J. 13, 458-469, 1966.
24. Ornea, L. and Romani, G. The fundamental equations of a conformal submersions, Beitrague Z. Algebra and Geometrie Contributions Algebra and Geometry, 34 (2), 233- 243, 1993.
25. Ponge, R. and Reckziegel, H. Twisted products in pseudo-Riemannian geometry, Geom. Dedicata. 48 (1), 15-25, 1993.
26. Shahid, A. and Tanveer, F. Anti-invariant Riemannian submersions from nearly Kählerian manifolds, Filomat, 27 (7), 1219-1235, 2013.
27. Şahin, B. Anti-invariant Riemannian submersions from almost Hermitian manifolds, Central European J. Math, No. 3, 437-447, 2010.
28. Ş“ahin, B. Semi-invariant Riemannian submersions from almost Hermitian manifolds, Canad. Math. Bull, 56 (1), 173-182, 2013.
29. Şahin, B. Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie, 54 (102), No. 1, 93-105, 2011.
30. Şahin, B. Riemannian submersions from almost Hermitian manifolds, Taiwanese J. Math. 17, 629-659, 2012.
31. Ş“ahin, B. Riemannian submersions, Riemannian maps in Hermitian Geometry and Their Applications, (London, Elsevier, Academic Press, 2017).
32. Taştan, H. M. On Lagrangian submersions, Hacet. J. Math. Stat. 43 (6), 993-1000, 2014.
33. Watson, B. Almost Hermitian submersions, J. Differential Geometry, 11 (1), 147-165, 1976.