Aliya Naaz SIDDIQUI

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

In this paper, we establish two sharp inequalities, which involve the generalized normalizedδ− Casorati curvatures and the generalized normalized scalar curvature of any submanifold ingeneralized Sasakian space forms with semi-symmetric metric connection by using T Oprea’stechnique. Afterwards, we examine that the equality holds if and only if the submanifold isinvariantly 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 ambientspace form with SSMC.

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Alegre, P., Blair, D.E. and Carriazo, A., Generalized Sasakian-space-forms, Israel J. Math., 141 (2004), 157–183.
Cabrerizo, J.L., Carriazo, A., Fernandez, L.M. and Fernandez, M., Semi-slant submanifolds of a Sasakian manifold, Geom. Dedicata, 78 (1999) 183–199.
Cabrerizo J.L., Carriazo A., Fernandez L.M. and Fernandez M., Slant submanifolds in Sasakian manifolds, Glasgow Math. J. 42 (2000), 125–138.
Chen, B.-Y., Relationship between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow. Math. J., 41 (1999), 33–41.
Chen, B.-Y., Some pinching and classification theorems for minimal submanifolds, Arch. Math. 60 (1993), 568–578.
Chen, B.-Y. and Uddin, S., Warped Product Pointwise Bi-slant Submanifolds of Kaehler Manifolds, Publ. Math. Debrecen 92(1-2) (2018), 183–199.
Decu, S., Haesen, S. and Verstralelen, L., Optimal inequalities involving Casorati curvatures, Bull. Transylv. Univ. Brasov, Ser B, 14 (2007), 85–93.
Decu, S., Haesen, S. and Verstralelen, L., Optimal inequalities characterizing quasi-umbilical submanifolds, J. Inequalities Pure. Appl. Math., 9 (2008), Article ID 79, 7pp.
Friedmann, A., Schouten, J.A., Uber die Geometrie der halbsymmetrischen Ubertragungen, Mathematische Zeitschrift, 21 (1924), 211–223.
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.
Hayden, H.A. Subspace of a space with torsion, Proc. London Math. Soc., 34 (1932), 27–50.
Khan, V.A. and Khan, M.A., Pseudo-slant submanifolds of a Sasakian manifold, Indian J. Pure Appl. Math., 38 (2007), 31–42.
Kowalczyk, D., Casorati curvatures, Bull. Transilvania Univ. Brasov Ser. III, 50(1) (2008), 2009–2013.
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.
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.
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.
Lee, J.W. and Vilcu, G.E., Inequalities for generalized normalized δ-Casorati curvatures of slant submanifolds in quaternion space forms, Taiwanese J. Math. 19 (2015), 691–702.
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.
Mihai, A. and Özgür, C., Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi- symmetric metric connections, Rocky Mountain J. Math., 41(5) (2011), 1653-1673.
Oprea, T., Optimization methods on Riemannian submanifolds, An. Univ Bucur. Mat., 54 (2005), 127-136.
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.
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).
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.
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.
Tripathi, M.M., Inequalities for algebraic Casorati curvatures and their applications, arXiv:1607.05828v1 [math.DG] 20 Jul 2016.
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.
Verstralelen, L., Geometry of submanifolds I, The first Casorati curvature indicatrices, Kragujevac J. Math., 37 (2013), 5–23.
Verstralelen, L., The geometry of eye and brain, Soochow J. Math., 30 (2004), 367–376.
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.
Yano, K. On semi-symmetric metric connection, Rev. Roumaine Math. Pures Appl., 15 (1970), 1579–1586.
Zhang, P. and Zhang, L., Casorati inequalities for submanifolds in a Riemannian manifold of quasi-constant curvature with a semi- symmetric metric connection, Symmetry, 8(4) (2016), 19; doi:10.3390/sym8040019.