A Lower Bound of normalized scalar curvature for bi-slant submanifolds in generalized Sasakian space forms using Casorati curvatures
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Abstract
In this paper, we prove two optimal inequalities between the generalized normalized $\delta-$Casorati curvatures and the normalized scalar curvature for bi-slant submanifolds in a generalized Sasakian space form. Moreover, we show that the equality at all points characterizes the invariantly quasi-umbilical submanifolds in both cases.
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Siddiqui, A., & Shahid, M.
(2018).
A Lower Bound of normalized scalar curvature for bi-slant submanifolds in generalized Sasakian space forms using Casorati curvatures.
Acta Mathematica Universitatis Comenianae, 87(1), 127-140.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/623/572
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References
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\bibitem{bihai1} Cabrerizo J. L., Carriazo A., Fernandez L. M. and Fernandez M., {\em Slant submanifolds in Sasakian manifolds}, Glasgow Math. J., {\bf42} (2000), 125--138.
\bibitem{chen3} Chen B. Y., {\em Relationship between Ricci curvature and shape operator for submanifolds with arbitrary codimensions}, Glasgow. Math. J., {\bf41} (1999), 33--41.
\bibitem{chen4} Chen B. Y., {\em Some pinching and classification theorems for minimal submanifolds}, Arch. math., {\bf60} (1993), 568--578.
\bibitem{decu1} Decu S., Haesen S. and Verstralelen L., {\em Optimal inequalities involving Casorati curvatures}, Bull. Transylv. Univ. Brasov, Ser B, {\bf14} (2007), 85--93.
\bibitem{decu2} Decu S., Haesen S; Verstralelen L., {\em Optimal inequalities characterizing quasi-umbilical submanifolds}, J. Inequalities Pure. Appl. Math, {\bf9} (2008), Article ID 79, 7pp.
\bibitem{ghisoiu} Ghisoiu V., {\em Inequalities for thr Casorati curvatures of the slant submanifolds in complex space forms}, Riemannian geometry and applications. Proceedings RIGA 2011, ed. Univ. Bucuresti, Bucharest, (2011), 145--150.
\bibitem{bihai3} Khan V. A. and Khan M. A., {\em Pseudo-slant submanifolds of a Sasakian manifold}, Indian J. Pure Appl. Math., {\bf38} (2007), 31--42.
\bibitem{kowalczyk} Kowalczyk D., {\em Casorati curvatures}, Bull. Transilvania Univ. Brasov Ser. III, {\bf50(1)} (2008), 2009--2013.
\bibitem{lee1} Lee C. W., Lee J. W., Vilcu G. E. and Yoon D. W., {\em Optimal inequalities for the Casorati curvatures of the submanifolds of generalized space form endowed with semi-symmetric metric connections}, Bull. Korean Math. Soc. {\bf52} (2015), 1631--1647.
\bibitem{lee3} Lee C. W., Yoon D. W. and Lee J. W., {\em Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections}, J. Inequal. Appl. (2014), 2014:327, 9 pp. MR 3344114.
\bibitem{lee2} Lee J. W. and Vilcu G. E., {\em Inequalities for generalized normalized $\delta$-Casorati curvatures of slant submanifolds in quaternion space forms}, Taiwanese J. Math., {\bf19} (2015), 691--702.
\bibitem{tripathi} Tripathi M. M., {\em Inequalities for algebraic Casorati curvatures and their applications}, arXiv:1607.05828v1 [math.DG] 20 Jul 2016.
\bibitem{verstraelen1} Verstralelen L., {\em Geometry of submanifolds I, The first Casorati curvature indicatrices}, Kragujevac J. Math., {\bf37} (2013), 5--23.
\bibitem{verstraelen2} Verstralelen L., {\em The geometry of eye and brain}, Soochow J. Math., {\bf30} (2004), 367--376.
\bibitem{article.1} Yano K. and Kon M., {\em Differential geometry of CR-submanifolds}, Geometria Dedicata, {\bf10} (1981), 369--391.
\bibitem{article.9} Yano K. and Kon M., {\em Structures on manifolds}, Worlds Scientific, Singapore (1984).