Lower bounds of generalised normalised δ- Casorati curvature for real hypersurfaces in complex quadric endowed with semi-symmetric metric connection
The main intention of the present paper is to develop two extremal inequalities involving normalized δ-Casorati curvature and extrinsic generalised normalised δ-Casorati curvature for real hypersurfaces in complex quadric Qm admitting semi-symmetric metric connection. Further, we derive the necessary and sufficient condition for the equality in both cases
B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math., 60,(1993),
B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR ±-invariants,
Internat. J.Math., 23, No. 3, (2012), 1250045(17 pages).
B. Y. Chen, F. Dillen, J. Van der Veken and L. Vrancken, Curvature inequalities for Lagrangian submanifolds:The final solution, Differential Geom. Appl., 31 (2013), 808–819.
C.W. Lee, J.W. Lee and G. E. Vilcu, Optimal inequalities for the normalised ±-Casorati curvatures of submanifolds
in Kenmotsu space forms, Adv. Geom., 17 (2017), 355–362.
C.W. Lee, J.W. Lee andG. E. Vilcu, A newproof for some optimal inequalities involving generalized normalized
±-Casorati curvatures, Journal of Inequalities and Applications, (2015) 2015:310 DOI 10.1186/s13660-015-
D. E. Blair, Contact manifolds in Riemannian Geometry, Lecture Notes inMath, 509, Springer-Verlag, Berlin,
F. Casorati,Mesure de la courbure des surfaces suivant l’idée commune, Ses rapports avec lesmesures de courbure
gaussienne etmoyenne, Acta.Math., 14 (1890), 95–110.
H. Reckziegel, On the geometry of the complex quadric, in :Geometry and Topology of Submanifolds VIII
(Brussels/Nordfjordeid 1995), World Sci. Publ., River Edge, NJ, 1995, 302–315.
J.W. Lee and G. E. Vilcu, Inequalities for generalized normalized ±-Casorati curvatures of slant submanifolds
in quaternionic space forms, Taiwan. J.Math., 19 (2015), 691–702.
K. S. Park, Inequalities for the Casorati curvatures of real hypersurfaces in some Grassmannians, Taiwanese J.
Math., 22 (2018), 63–77.
P. Bansal, Hopf real hypersurfaces in the complex quadric Qm with recurrent Jacobi operator, Advances in
Intelligent Systems and Computing series, Springer, 816, (2019).
P. Bansal andM. H. Shahid, Optimization Approach for Bounds Involving Generalized Normalized ±-Casorati
Curvatures, Advances in Intelligent Systems and Computing series, Springer, 741 (2019), 227–237.
P. Bansal, S. Uddin and M. H. Shahid, Extremities involving B. Y. Chen’s invariants for real hypersurfaces in
complex quadric, Int. Electron. J. Geom., 11 (2018), No. 2, 89–101.
P. Zhang and L. Zhang, Inequalities for Casorati curvatures of submanifolds in real space forms, Adv. Geom.,16 (2016), 329–335.
S.Decu, S. Haesen and L. Verstraelen,Optimal inequalities involving Casorati curvatures, Bull. Transilv.Univ. Brasov Ser. B (N. S.), 14 (2007), No. 49, Supp. 1, 85–93.
V. Ghisoiu, Inequalities for the Casorati curvatures of slant submanifolds in complex space forms, Riemannian
Geometry and Applications, Proceedings RIGA 2011, ed. Univ. Bucuresti, Bucharest, 2011, 145–150.
V. Slesar, B. Sahin and G. E. Vilcu, Inequalities for the Casorati curvatures of slant submanifolds in quaternionic
space forms, J. Inequal. Appl., 2014, 2014:123.
Y. J. Suh, Real hypersurfaces in the complex quadric with parallel Ricci tensor, Advances inMathematics, 281
Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb parallel shape operator, Internat. J. Math., 25
(2014), 1450059, 17pp.
J. Berndt and Y. J. Suh, Real hypersurfaceswith isometric Reeb flow in complex quadrics, Internat. J.Math., 24(2013), No. 7, 1350050, 18pp.
H. A. Hayden, Subspaces of a space with torsion, Proc. Lond.Math. Soc., 34 (1932), No. 2, 27–50.
A. Mihai and C. Özgür, Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection, Taiwanese J.Math., 14 (2010), 1465–1477.
K. Yano, On semi-symmetricmetric connection, Rev. RoumaineMath. Pures Appl., 15 (1970), 1579–1586.