Remarks on conformal anti-invariant Riemannian maps to cosymplectic manifolds

Yıl 2021, Cilt: 50 Sayı: 4, 1131 - 1139, 06.08.2021

Yılmaz Gündüzalp Mehmet Akif Akyol

https://doi.org/10.15672/hujms.677910

Cited By: 2

Öz

M.A. Akyol and B. Şahin [Conformal anti-invariant Riemannian maps to Kaehler manifolds, U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 4, 2018] defined and studied the notion of conformal anti-invariant Riemannian maps to Kaehler manifolds. In this paper, as a generalization of totally real submanifolds and anti-invariant Riemannian maps, we extend this notion to almost contact metric manifolds. In this manner, we introduce conformal anti-invariant Riemannian maps from Riemannian manifolds to cosymplectic manifolds. In order to guarantee the existence of this notion, we give a non-trivial example, investigate the geometry of foliations which are arisen from the definition of a conformal Riemannian map and obtain decomposition theorems by using the existence of conformal Riemannian maps. Moreover, we investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal anti-invariant Riemannian maps to be totally geodesic. Finally, we study weakly umbilical conformal Riemannian maps and obtain a classification theorem for conformal anti-invariant Riemannian maps.

Anahtar Kelimeler

Cosymplectic manifold, anti-invariant Riemannian map, conformal anti-invariant Riemannian map

Kaynakça

• [1] R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications, Appl. Math. Sci. Vol. 75, Springer, New York, 1988.
• [2] M.A. Akyol, Conformal anti-invariant Riemannian submersions from cosymplectic manifolds, Hacet. J. Math. Stat. 46 (2), 177–192, 2017.
• [3] M.A. Akyol and B. Şahin, Conformal anti-invariant Riemannian maps to Kahler manifolds, U.P.B. Sci. Bull., Series A. 80 (4), 187–198, 2018.
• [4] M.A. Akyol and B. Şahin, Conformal slant Riemannian maps to Kahler manifolds, Tokyo J. Math. 42 (1), 225–237, 2019.
• [5] P. Baird and J.C. Wood, Harmonic Morphisms Between Riemannian Manifolds, Clarendon Press, Oxford, 2003.
• [6] D.E. Blair, The theory of quasi-Sasakian structure, J. Differential Geom. 1, 331–345, 1967.
• [7] D.E. Blair, Contact Manifolds in Riemannian Geometry, Lectures Notes in Mathematics 509, Springer-Verlag, Berlin, 1976.
• [8] A.E. Fischer, Riemannian maps between Riemannian manifolds, Contemp. Math. 132, 331–366, 1992.
• [9] J.P. Jaiswal, Harmonic maps on Sasakian manifolds, J. Geom. 104 (2), 309–315, 2013.
• [10] G.D. Ludden, Submanifolds of cosymplectic manifolds, J. Differential Geom. 4, 237– 244, 1970.
• [11] T. Nore, Second fundamental form of a map, Ann. Mat. Pura Appl. 146, 281–310, 1987.
• [12] B. Pandey, J.P. Jaiswal and R.H. Ojha, Necessary and Sufficient Conditions for the Riemannian Map to be a Harmonic Map on Cosymplectic Manifolds, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 85 (2), 265–268, 2015.
• [13] K.S. Park, H-conformal anti-invariant submersions from almost quaternionic Hermitian manifolds, Czechoslovak Math. J. 70, 631–656, 2020.
• [14] R. Prasad and S. Pandey, Slant Riemannian maps from an almost contact manifold, Filomat, 31 (13), 3999–4007, 2017.
• [15] B. Şahin, Conformal Riemannian maps between Riemannian manifolds, their harmonicity and decomposition theorems, Acta Appl. Math. 109 (3), 829–847, 2010.
• [16] B. Şahin, Invariant and anti-invariant Riemannian maps to Kahler manifolds, Int. J. Geom. Methods Mod. Phys. 7 (3), 1–19, 2010.
• [17] B. Şahin, Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications, Elsevier, Academic Press, 2017.
• [18] B.Şahin and Ş. Yanan, Conformal Riemannian maps from almost Hermitian manifolds, Turkish J. Math. 42 (5), 2436–2451, 2018.
• [19] K. Yano and M. Kon, Anti-invariant submanifolds, Lect. Notes Pure Appl. Math. Vol. 21, Marcel Dekker Inc., 1976.