Clairaut Pointwise Slant Submersion from Locally Product Riemannian Manifolds

Clairaut Pointwise Slant Submersion from Locally Product Riemannian Manifolds

The goal of the present paper is to analyze some geometric features of Clairaut pointwise slant submersions whose total manifold is a locally product Riemannian manifold. We describe Clairaut pointwise slant submersions from locally product Riemannian manifold onto a Riemannian manifold. We study pointwise slant submersions by providing a consequent which defines the geodesics on the total space of this type submersions. We also give a non-trivial example of the Clairaut pointwise slant submersions whose total manifolds are locally product Riemannian.

___

  • [1] M.A. Akyol, Generic Riemannian submersions from almost product Riemannian manifolds, GUJ Sci. 30, no. 3, 89-100, 2017.
  • [2] M.A. Akyol, Conformal semi-slant submersions, Int. J. Geom. Methods Mod. Phys.14, no.7, 1750114, 2017.
  • [3] S.A. Aykurt and M. Ergut, Pointwise slant submersions from cosymplectic manifolds, Turk. J. Math. 40, no. 3, 582-593, 2016.
  • [4] P. Baird and J.C.Wood, Harmonic morphism between Riemannian manifolds, Oxford science publications, Oxford, 2003.
  • [5] A. Beri, E.I. Kupeli and C. Murathan, Anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds, Turk. J. Math. 40, no. 3, 540-552, 2016.
  • [6] R. L. Bishop, Clairaut submersions, Differential Geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 21-31, 1972.
  • [7] B.Y. Chen, geometry of real submanifolds in Kahler manifold, Monatsh. Math., 91, 257-274, 1981.
  • [8] B.Y. Chen and O. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turk. J. Math. 364, 630-640, 2012.
  • [9] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16, 715-737, 1967.
  • [10] Y. Gunduzalp, Anti-invariant Riemannian submersions from almost product Riemannian manifolds, Math. Sci. Appl. E Notes 1, 58-66, 2013.
  • [11] Y. Gündüzalp, Anti-invariant Pseudo-Riemannian Submersions and Clairaut Submersions from Paracosymplectic Manifolds. Mediterr. J. Math. 16, 94, 2019.
  • [12] Y. Gunduzalp, Anti-invariant submersions from almost paracontact Riemannian manifolds, Honam Mathematical Journal, 41(4), 769-780, 2019.
  • [13] Y. Gündüzalp, Slant submersions from almost product Riemannian manifolds, Turkish Journal of Mathematics, 37(5), 863-873, 2013.
  • [14] Y. Gündüzalp and M. Polat, Some inequalities of anti-invariant Riemannian submersions in complex space forms, Miskolc Mathematical Notes, accepted, 2021.
  • [15] Y. Gündüzalp and M. Polat, Chen-Ricci inequalities in slant submersions for complex space forms, F˙ILOMAT, accepted, 2021.
  • [16] I. Kupeli Erken and C. Murathan, On slant submersions for cosymplectic manifolds, Bull. Korean Math. Soc. 51, no. 6, 1749-1771, 2014.
  • [17] J.W. Lee, B. Şahin, Pointwise slant submersions, Bull. Korean Math. Soc. 51, 1115-1126, 2014.
  • [18] J. Lee, J.H. Park, B. Şahin and D.Y. Song, Einstein conditions for the base of anti-invariant Riemannian submersions and Clairaut submersions, Taiwanse J. Math., 19, no. 4, 1145-1160, 2015.
  • [19] B. O‘Neill, The fundamental equations of a submersion, Michigan Math. J. 13, 459-469, 1966.
  • [20] F. ¨ Ozdemir, C. Sayar and H.M. Tas.tan, Semi-invariant submersions whose total manifolds are locally product Riemannian, Quaestiones Mathematicae, Vol. 49 No. 7, 2017.
  • [21] K.S. Park and R. Prasad, Semi-slant submersions, Bull. Korean Math. Soc. 50, no. 3, 951-962, 2013.
  • [22] G.B. Ronsse, Generic and skew CR-submanifolds of a Kahler manifold, Bull. Inst.Math. Acad. Sin. 18, 127-141, 1990.
  • [23] S. A.Sepet and M.Ergut, Pointwise slant submersions from almost product Riemannian manifolds, Journal of Interdisciplinary Mathematics, 23(3) (2020), 639-655.
  • [24] A. Shahid and F. Tanveer, Anti-invariant Riemannian submersions from nearly Kahler manifolds, Filomat, 27, 1219-1235, 2013.
  • [25] A. Shahid and F. Tanveer, Generic Riemannian submersions, Tamkang J. Math. 44, no. 4, 395-409, 2013.
  • [26] B. Sahin, Riemannian submersions, Riemannian maps in Hermitian Geometry, and their Applications, Elsevir, Academic, Amsterdam, 2017.
  • [27] B. Sahin, Semi-invariant submersions from almost Hermitian manifolds, Canad. Math. Bull. 56(1), 173-182, 2013.
  • [28] B. Şahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie Tome 54(102), 93–105, 2011.
  • [29] B. Şahin, Anti-invariant Riemannian submersion from almost Hermitian manifolds, Cent. Eur. J. Math., 8(3) , 437-447, 2010.
  • [30] H.M. Taştan, On Lagrangian submersions, Hacet. J. Math. Stat. 43, no. 6, 993-1000, 2014.
  • [31] H.M. Taştan, B. Sahin and S. Yanan, Hemi-slant submersions, Mediterr. J. Math. 13, no. 4, 2171-2184, 2016.
  • [32] H.M. Taştan, S. Gerdan, Clairaut anti-invariant submersions from Sasakian and Kenmotsu manifolds, Mediterr. J. Math. 14, no. 6, paper no. 235, 17 pp., 2017.
  • [33] H.M. Taştan and S.G. Aydin, Clairaut anti-invariant submersions from cosymplectic manifolds, Honam Math. J. 41, no. 4, 707-724, 2019.
  • [34] K. Yano and M. Kon, Structures on Manifolds, Singapore: World Scientific, 1984.
  • [35] B. Watson, Almost Hermitian submersions, J. Differential Geom. 11, 147-165, 1976.