Upper Bound Inequalities for δ-Casorati Curvatures of Submanifolds in Generalized Sasakian Space Forms Admitting a Semi-Symmetric Metric Connection

Year 2018, Volume: 11 Issue: 1, 57 - 67, 30.04.2018

Aliya Naaz Siddiqui

Abstract

In this paper, we establish two sharp inequalities, which involve the generalized normalized

δ-Casorati curvatures and the generalized normalized scalar curvature of any submanifold in

generalized Sasakian space forms with semi-symmetric metric connection by using T Oprea’s

technique. Afterwards, we examine that the equality holds if and only if the submanifold is

invariantly quasi-umbilical in both inequalities. We also develop these inequalities for invariant,

anti-invariant, CR, slant, semi-slant, hemi-slant and bi-slant submanifolds in the same ambient

space form with SSMC.

Keywords

δ-Casorati Curvatures, bi-slant submanifold, generalized Sasakian space forms, semi-symmetric metric connection, quasi-umbilical submanifold

References

• [1] Alegre, P., Blair, D.E. and Carriazo, A., Generalized Sasakian-space-forms, Israel J. Math., 141 (2004), 157–183.
• [2] Cabrerizo, J.L., Carriazo, A., Fernandez, L.M. and Fernandez, M., Semi-slant submanifolds of a Sasakian manifold, Geom. Dedicata, 78 (1999) 183–199.
• [3] Cabrerizo J.L., Carriazo A., Fernandez L.M. and Fernandez M., Slant submanifolds in Sasakian manifolds, Glasgow Math. J. 42 (2000), 125–138.
• [4] Chen, B.-Y., Relationship between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow. Math. J., 41 (1999), 33–41.
• [5] Chen, B.-Y., Some pinching and classification theorems for minimal submanifolds, Arch. Math. 60 (1993), 568–578.
• [6] Chen, B.-Y. and Uddin, S., Warped Product Pointwise Bi-slant Submanifolds of Kaehler Manifolds, Publ. Math. Debrecen 92(1-2) (2018), 183–199.
• [7] Decu, S., Haesen, S. and Verstralelen, L., Optimal inequalities involving Casorati curvatures, Bull. Transylv. Univ. Brasov, Ser B, 14 (2007), 85–93.
• [8] Decu, S., Haesen, S. and Verstralelen, L., Optimal inequalities characterizing quasi-umbilical submanifolds, J. Inequalities Pure. Appl. Math., 9 (2008), Article ID 79, 7pp.
• [9] Friedmann, A., Schouten, J.A., Uber die Geometrie der halbsymmetrischen Ubertragungen, Mathematische Zeitschrift, 21 (1924), 211–223.
• [10] Ghisoiu, V., Inequalities for thr Casorati curvatures of the slant submanifolds in complex space forms, Riemannian geometry and applications. Proc. RIGA 2011, ed. Univ. Bucuresti, Bucharest, (2011), 145–150.
• [11] Hayden, H.A. Subspace of a space with torsion, Proc. London Math. Soc., 34 (1932), 27–50.
• [12] Khan, V.A. and Khan, M.A., Pseudo-slant submanifolds of a Sasakian manifold, Indian J. Pure Appl. Math., 38 (2007), 31–42.
• [13] Kowalczyk, D., Casorati curvatures, Bull. Transilvania Univ. Brasov Ser. III, 50(1) (2008), 2009–2013.
• [14] Lee, C.W., Lee, J.W. and Vilcu, G.E., Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms, Advances in Geometry, 17 (2017), doi:10.1515/advgeom-2017–0008.
• [15] Lee, C.W., Lee, J.W., Vilcu, G.E. and Yoon, D.W. Optimal inequalities for the Casorati curvatures of the submanifolds of generalized space form endowed with semi-symmetric metric connections, Bull. Korean Math. Soc., 52 (2015), 1631–1647.
• [16] Lee, C.W., Yoon, D.W. and Lee, J.W., Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections, J. Inequal. Appl., 2014(327) (2014), pp. 9.
• [17] Lee, J.W. and Vilcu, G.E., Inequalities for generalized normalized -Casorati curvatures of slant submanifolds in quaternion space forms, Taiwanese J. Math. 19 (2015), 691–702.
• [18] Mihai, A. and Özgür, C., Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection, Taiwanese J. Math., 14(4) (2010), 1465-1477.
• [19] Mihai, A. and Özgür, C., Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semisymmetric metric connections, Rocky Mountain J. Math., 41(5) (2011), 1653-1673.
• [20] Oprea, T., Optimization methods on Riemannian submanifolds, An. Univ Bucur. Mat., 54 (2005), 127-136.
• [21] Özgür, C. and Murathan, C., Chen inequalities for submanifolds of a cosymplectic space form with a semi-symmetric metric connection, An. St. Univ. Al.I. Cuza Iasi, 58 (2012), 395-408.
• [22] Siddiqui, A.N. and Shahid, M.H., A Lower Bound of Normalized Scalar Curvature for Bi-slant Submanifolds in Generalized Sasakian Space Forms using Casorati Curvatures, Acta Math. Univ. Comenianae, LXXXVII(1), 127-140 (2018).
• [23] Su, M., Zhang, L. and Zhang, P., Some inequalities for submanifolds in a Riemannian manifold of nearly quasi-constant curvature, Filomat, 31(8) (2017), 2467–2475.
• [24] Sular, S. and Özgür, C., Generalized Sasakian space forms with semi-symmetric metric connections, An. St. Univ. Al.I. Cuza Iasi, 60 (2014), 145-155.
• [25] Tripathi, M.M., Inequalities for algebraic Casorati curvatures and their applications, arXiv:1607.05828v1 [math.DG] 20 Jul 2016.
• [26] Uddin, S., Chen, B.-Y. and Al-Solamy, F.R., Warped product bi-slant immersions in Kaehler manifolds, Mediterr. J. Math., 14(95) (2017) doi:10.1007/s00009-017-0896-8.
• [27] Verstralelen, L., Geometry of submanifolds I, The first Casorati curvature indicatrices, Kragujevac J. Math., 37 (2013), 5–23.
• [28] Verstralelen, L., The geometry of eye and brain, Soochow J. Math., 30 (2004), 367–376.
• [29] Yano, K. and Kon, M., Differential geometry of CR-submanifolds, Geom. Dedicata, 10 (1981), 369–391.
• [30] Yano, K. and Kon, M., Structures on manifolds, Worlds Scientific, Singapore, 1984.
• [31] Yano, K. On semi-symmetric metric connection, Rev. Roumaine Math. Pures Appl., 15 (1970), 1579–1586.
• [32] Zhang, P. and Zhang, L., Casorati inequalities for submanifolds in a Riemannian manifold of quasi-constant curvature with a semisymmetric metric connection, Symmetry, 8(4) (2016), 19; doi:10.3390/sym8040019.