Clairaut semi invariant submersions from locally product Riemannian manifolds

The goal of the present paper is to analyze some geometric features of Clairaut semi invariant Riemannian submersions whose total manifold is a locally product Riemannian manifold and investigate fundamental results on such submersion. We also ensure an explicit example of Clairaut semi invariant Riemannian submersion.

Keywords:

Riemannian submersion, Clairaut semi invariant submersion locally product Riemannian manifold,

PDF

___

[1] M.A. Akyol, Generic Riemannian submersions from almost product Riemannian manifolds, GUJ Sci. 30, no. 3, 89-100, 2017.
[2] D. Allison, (1996). Lorentzian Clairaut submersions, Geometriae Dedicata, 63(3), 309-319.
[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 and O. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turk. J. Math. 364, 630-640, 2012.
[8] M. Falcitelli, S. Ianus and A. M. Pastore, Riemannian Submersions and Related Topics,World Scientific, 2004.
[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. G¨und¨uzalp, 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. G¨und¨uzalp, Slant submersions from almost product Riemannian manifolds, Turkish Journal of Mathematics, 37(5), 863-873, 2013.
[14] Y. G¨und¨uzalp and M. Polat, Some inequalities of anti-invariant Riemannian submersions in complex space forms, Miskolc Mathematical Notes, accepted, 2021.
[15] Y. G¨und¨uzalp 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. Sahin, Pointwise slant submersions, Bull. Korean Math. Soc. 51, 1115-1126, 2014.
[18] J. Lee, J.H. Park, B. Shahin 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] C.Sayar, F. Ozdemir, H.M. Tastan, (2018). Pointwise semi-slant submersions whose total manifolds are locally product Riemannian manifolds. International Journal of Maps in Mathematics, 1(1), 91-115.
[24] .C. Sayar, H.M Ta1e63tan, F. ¨ Ozdemir and M.M. Tripathi, (2020). Generic submersions from Kaehler manifolds. Bulletin of the Malaysian Mathematical Sciences Society, 43(1), 809-831.
[25] A. Shahid and F. Tanveer, Anti-invariant Riemannian submersions from nearly Kahler manifolds, Filomat, 27, 1219-1235, 2013.
[26] A. Shahid and F. Tanveer, Generic Riemannian submersions, Tamkang J. Math. 44, no. 4, 395-409, 2013.
[27] B. Şahin, Riemannian submersions, Riemannian maps in Hermitian Geometry, and their Applications, Elsevir, Academic, Amsterdam, 2017.
[28] B. Şahin, Semi-invariant submersions from almost Hermitian manifolds, Canad. Math. Bull. 56(1), 173-182, 2013.
[29] B. Şahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie Tome 54(102), 93–105, 2011.
[30] B. Şahin, Invariant and anti-invariant Riemannian maps to Kahler manifolds, International Journal of Geometric Methods in Modern Physics, 7(3),337-355, 2010.
[31] B. Şahin, Anti-invariant Riemannian submersion from almost Hermitian manifolds, Cent. Eur. J. Math., 8(3) , 437-447, 2010.
[32] H.M. Taştan, On Lagrangian submersions, Hacet. J. Math. Stat. 43, no. 6, 993-1000, 2014.
[33] H.M. Taştan, B. Sahin and S. Yanan, Hemi-slant submersions, Mediterr. J. Math. 13, no. 4, 2171-2184, 2016.
[34] 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.
[35] H.M. Taştan and S.G. Aydin, Clairaut anti-invariant submersions from cosymplectic manifolds, Honam Math. J. 41, no. 4, 707-724, 2019.
[36] K. Yano and M. Kon, Structures on Manifolds, Singapore: World Scientific, 1984.
[37] D. W. Yoon, Inequality for Ricci curvature of slant submanifolds in cosymplectic space forms, Turk. J. Math., 30(2006), 43-56.
[38] B. Watson, Almost Hermitian submersions, J. Differential Geom. 11, 147-165, 1976.