On Submanifolds in a Riemannian Manifold with Golden Structure

Year 2019, Volume: 11 Issue: 1, 8 - 23, 30.06.2019

Mobin Ahmad , Mohammad Aamir Qayyoom

Abstract

A golden Riemannian structure $(J,g)$ on a Riemannian  manifold is given by a tensor field $J$ of type $(1,1)$ satisfying the golden section relation $J^{2}=J+I,$ and a pure Riemannian metric $g$, that is  a metric satisfying $g(JX,Y)=g(X,JY).$ We investigate  some fundamental properties of the induced structure on submanifolds immersed in golden Riemannian manifolds. We obtain effective relations for some induced structures on submanifolds of codimension 2. We also construct an example on submanifold of a golden Riemannian manifold.

Keywords

Golden structure, Riemannian manifold, totally geodesics, normal induced structure, killing vector fields

References

• Ahmad M., Jun, J. B., Haseeb A., Submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric semi-metric connection, Tensor, N. S., 70(2008), 311--321.
• Ahmad M., Jun, J. B., Haseeb A., Submanifolds of an almost r-paracontact Riemannian manifold endowed with a quarter symmetric non-metric connection, J. Chungcheong Math. Soc., 24(1)(2011), 91--104.
• Ahmad M., Al-Hossain A.Y., Ojha, J.P., On submanifolds of an almost r-paracontact Riemannian manifold endowed with a quarter symmetric metric connection, Theoretical Mathematics and Application, 4(4)(2014), 1--17.
• Bahadir, O., Uddin, S., Slant submanifolds of golden Riemannian manifolds, arXiv:1804.11126v1.
• Blaga, A.M., Hretcanu, C. E., Invariant, anti-invariant and slant submanifolds of a metallic Riemannian manifold, arXiv: 1803.01415.
• Crasmareanu, M., Hretcanu, C.E., Golden differential geometry, Chaos Solitons Fractals, 38(5)(2008), 1229--1238.
• Crasmareanu, M., Hretcanu, C.E., On some invariant submanifolds in a Riemannian manifold with golden structure, Analele Stintifice Universitatis "Al. I. Cuza" DIN IASI (S.N.) Matematica, Tomul LIII (2007), 199--211.
• Das L. S., Nivas R., Ali S., Ahmad M., Study of submanifolds in a manifold with quarter symmetric semi-metric connection, Tensor N. S., 65(2004), 250--260.
• Etayo, F., Santamaria, R., Upadhyay, A., On the geometry of almost golden Riemannian manifolds, Mediterr. J. Math. 187(14)(2017), 991--1007.
• Gezer, A., Cengiz, N., Salimov, A., On integrability of golden Riemannian structures, Turk.J.Math., 37(2013), 366--703.
• Gherici, B., Induced structure on golden Riemannian manifolds, Beitr Algebra Geom., 18(8)(2018), 366--392.
• Goldberg, S.I., Yano, K., Polynomial structures on manifolds, Kodai Maths.Sem. Rep., 22(2)(1970), 199--218.
• Hretcanu, C.E., Submanifolds in Riemannian manifold with golden structure, Workshop on Finsler geometry and its applications, Hungary (2007).
• Hretcanu, C.E., Induced structure on submanifolds in almost product Riemannian manifolds, arXiv:math/0608533.
• Hretcanu, C.E., Crasmareanu, M., Metallic structure on Riemannian manifolds, Revista De La Union Mathematica Argentina, 54(2)(2013), 15--27.
• Hretcanu, C.E., Blaga, A.M., Submanifolds in metallic Riemannian manifolds, arXiv:1803.02184.
• Ozkan, M., Prolongations of golden structures to tangent bundles, Differential Geometry - Dynamical Systems, 16(2014), 227--238.
• Ozgur, C., Ahmad, M., Haseeb, A., CR-submanifolds of LP-Sasakian manifolds with semi-symmetric metric connection, Hacettepe J. Math. And Stat., 39(4)(2010), 489--496.
• Poyraz, N.O., Erol, Y., Lightlike hypersurfaces of a golden semi-Riemannian manifold, Mediterr. J. Math., (2017) 14:204, DOI 10.1007/s00009-017-0999-2.