Conformal anti-invariant submersions from cosymplectic manifolds

Yıl 2017, Cilt: 46 Sayı: 2, 177 - 192, 01.04.2017

Mehmet Akif Akyol

Öz

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.

Anahtar Kelimeler

Cosymplectic manifold, Conformal submersion, Conformal anti-invariant submersion

Kaynakça

• 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.
• Akyol, M. A. and “ahin, B. Conformal anti-invariant submersions from almost Hermitian manifolds, Turkish Journal of Mathematics, 40 (1), 43-70, 2016.
• Akyol, M. A. and “ahin, B. Conformal semi-invariant submersions, Commun. Contemp. Math. 19, 1650011, 22 pp, 2017.
• Blair, D. E. Contact manifold in Riemannain geometry, Lecture Notes in Math. 509 (Springer-Verlag, Berlin-New York), 1976.
• Blair, D. E. The theory of quasi-Sasakian structure, J. Differential Geom. 1, no. 3-4, 331-345, 1967.
• Baird, P. and Wood, J. C. Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, 29 (Oxford University Press, The Clarendon Press. Oxford), 2003.
• Cengizhan, M. and Erken I. K. Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian submersions, Filomat. 29 (7), 1429-1444, 2015.
• Erken, I. K. and Cengizhan, M. Anti-invariant Riemannian submersions from Sasakian manifolds, arxiv:1302.4906.
• Chinea, D. Harmonicity on maps between almost contact metric manifolds. Acta Math. Hungar. 126 (4), 352-365, 2010.
• Chinea, D. Harmonicity of holomorphic maps between almost Hermitian manifolds. Canad. Math. Bull. 52 (1), 18-27, 2009.
• Chinea, D. On horizontally conformal $(\varphi,\varphi')$-holomorphic submersions. Houston J. Math. 34 (3), 721-737, 2008.
• Fuglede, B. Harmonic Morphisms Between Riemannian Manifolds, Ann. Inst. Fourier (Grenoble) 28, 107-144, 1978.
• Falcitelli, M., Ianus, S. and Pastore, A. M.Riemannian submersions and Related Topics. World Scientic, River Edge, NJ, 2004.
• Gundmundsson, S. The Geometry of Harmonic Morphisms, Ph.D. Thesis, University of Leeds, 1992.
• Gundmundsson, S. and Wood, J. C. Harmonic Morphisms between almost Hermitian man- ifolds. Boll. Un. Mat. Ital. B. 11 (2), 185-197, 1997.
• 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.
• Gray, A. Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16, 715-737, 1967.
• Ishihara, T. A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. kyoto Univ. 19, 215-229, 1979.
• 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.
• Lee, J. W. Anti-invariant $\xi^\perp$-Riemannian submersions from almost contact manifolds, Hacettepe Journal of Mathematics and Statistic, 42 (3), 231-241, 2013.
• Ludden, G. D. Submanifolds of cosymplectic manifolds, J. Differential Geom. 4, 237-244, 1970.
• Olzsak, Z. On almost cosymplectic manifolds, Kodai Math J. 4, 239-250, 1981.
• O'Neill, B. The fundamental equations of a submersion. Mich. Math. J. 13, 458-469, 1966.
• 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.
• Ponge, R. and Reckziegel, H. Twisted products in pseudo-Riemannian geometry, Geom. Dedicata. 48 (1), 15-25, 1993.
• Shahid, A. and Tanveer, F. Anti-invariant Riemannian submersions from nearly Kählerian manifolds, Filomat, 27 (7), 1219-1235, 2013.
• Şahin, B. Anti-invariant Riemannian submersions from almost Hermitian manifolds, Central European J. Math, No. 3, 437-447, 2010.
• Ş“ahin, B. Semi-invariant Riemannian submersions from almost Hermitian manifolds, Canad. Math. Bull, 56 (1), 173-182, 2013.
• Şahin, B. Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie, 54 (102), No. 1, 93-105, 2011.
• Şahin, B. Riemannian submersions from almost Hermitian manifolds, Taiwanese J. Math. 17, 629-659, 2012.
• Ş“ahin, B. Riemannian submersions, Riemannian maps in Hermitian Geometry and Their Applications, (London, Elsevier, Academic Press, 2017).
• Taştan, H. M. On Lagrangian submersions, Hacet. J. Math. Stat. 43 (6), 993-1000, 2014.
• Watson, B. Almost Hermitian submersions, J. Differential Geometry, 11 (1), 147-165, 1976.