Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection
Abstract
:1. Introduction
2. Preliminaries
3. Main Results
- (i)
- The normalized δ-Casorati curvature satisfies
- (ii)
- The normalized δ-Casorati curvature satisfies
Acknowledgments
Author Contributions
Conflicts of Interest
References
- Chen, B.Y. Mean curvature and shape operator of isometric immersions in real space forms. Glasg. Math. J. 1996, 38, 87–97. [Google Scholar] [CrossRef]
- Chen, B.Y. Geometric and topological obstructions to various immersions in submanifold theory and some related open problems. Kragujev. J. Math. 2015, 39, 93–109. [Google Scholar] [CrossRef]
- Casorati, F. Nuova definitione della curvatura delle superficie e suo confronto con quella di Gauss. Rend. Inst. Matem. Accad. Lomb. 1889, 22, 1867–1868. (In Italian) [Google Scholar]
- Decu, S.; Haesen, S.; Verstraelen, L. Optimal inequalities involving Casorati curvatures. Bull. Transylv. Univ. Brasov, Ser. B 2007, 14, 85–93. [Google Scholar]
- Decu, S.; Haesen, S.; Verstraelen, L. Optimal inequalities characterising quasi-umbilical submanifolds. J. Inequalities Pure Appl. Math. 2008, 9, 1–7. [Google Scholar]
- Ghişoiu, V. Inequalities for the Casorati curvatures of slant submanifolds in complex space forms. In Proceedings of the RIGA 2011 Riemannian Geometry and Applications, Bucharest, Romania, 10–14 May 2011; pp. 145–150.
- Lee, C.W.; Yoon, D.W.; Lee, J.W. Optimal inequalities for the Casorati curvatures of submanifolds of real space forms endowed with semi-symmetric metric connections. J. Inequalities Appl. 2014, 2014, 327. [Google Scholar] [CrossRef]
- Lee, C.W.; Yoon, D.W.; Vîlcu, G.E.; Lee, J.W. Optimal inequalities for the Casorati curvatures of submanifolds of generalized space forms endowed with semi-symmetric metric connections. Bull. Korean Math. Soc. 2015, 52, 1631–1647. [Google Scholar] [CrossRef]
- Lee, J.W.; Vîlcu, G.E. Inequalities for generalized normalized δ-Casorati curvatures of slant submanifolds in quaternionic space forms. Taiwan. J. Math. 2015, 19, 691–702. [Google Scholar] [CrossRef]
- Slesar, V.; Sahin, B.; Vîlcu, G.E. Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms. J. Inequalities Appl. 2014, 2014, 123. [Google Scholar] [CrossRef]
- Lee, C.W.; Lee, J.W.; Vîlcu, G.E. A new proof for some optimal inequalities involving generalized normalized δ-Casorati curvatures. J. Inequalities Appl. 2015, 2015, 310. [Google Scholar] [CrossRef]
- Zhang, P.; Zhang, L. Inequalities for Casorati curvatures of submanifolds in real space forms. 2014; arXiv:1408.4996 [math.DG]. [Google Scholar]
- Zhang, P.; Zhang, L. Remarks on inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms. J. Inequalities Appl. 2014, 2014, 452. [Google Scholar] [CrossRef]
- Oprea, T. Optimization methods on Riemannian submanifolds. An. Univ. Bucur. Mat. 2005, 54, 127–136. [Google Scholar]
- Mihai, A.; Özgür, C. Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection. Taiwan. J. Math. 2010, 14, 1465–1477. [Google Scholar]
- Mihai, A.; Özgür, C. Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections. J. Math. 2011, 41, 1653–1673. [Google Scholar] [CrossRef]
- Özgür, C.; Murathan, C. Chen inequalities for submanifolds of a locally conformal almost cosymplectic manifold with a semi-symmetric metric connection. An. St. Univ. Ovidius Constunta 2010, 18, 239–254. [Google Scholar]
- Özgür, C.; Murathan, C. Chen inequalities for submanifolds of a cosymplectic space form with a semi-symmetric metric connection. Ann. Alexandru Ioan Cuza Univ. Math. 2012, 58, 395–408. [Google Scholar] [CrossRef]
- Özgür, C. BY Chen inequalities for submanifolds a Riemannian manifold of a quasi-constant curvature. Turk. J. Math. 2011, 35, 501–509. [Google Scholar]
- Zhang, P.; Zhang, L.; Song, W. Chen’s inequalities for submanifolds of a Riemannian manifold of quasi-constatnt curvature with a semi-symmetric metric connection. Taiwan. J. Math. 2014, 18, 1841–1862. [Google Scholar] [CrossRef]
- Yano, K. On semi-symmetric metric connection. Rev. Roum. Math. Pures Appl. 1970, 15, 1579–1586. [Google Scholar]
- Nakao, Z. Submanifolds of a Riemannian manifold with semi-symmetric metric connections. Proc. Am. Math. Soc. 1976, 54, 261–266. [Google Scholar] [CrossRef]
- Imai, T. Notes on semi-symmetric metric connections. Tensor (N.S.) 1972, 24, 293–296. [Google Scholar]
- Chen, B.Y.; Yano, K. Hypersurfaces of a conformally flat space. Tensor (N.S.) 1972, 26, 318–322. [Google Scholar]
- Chen, B.Y. Geometry of Submanifolds; Marcel Dekker, Inc.: New York, NY, USA, 1973. [Google Scholar]
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Zhang, P.; Zhang, L. Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection. Symmetry 2016, 8, 19. https://doi.org/10.3390/sym8040019
Zhang P, Zhang L. Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection. Symmetry. 2016; 8(4):19. https://doi.org/10.3390/sym8040019
Chicago/Turabian StyleZhang, Pan, and Liang Zhang. 2016. "Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection" Symmetry 8, no. 4: 19. https://doi.org/10.3390/sym8040019
APA StyleZhang, P., & Zhang, L. (2016). Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection. Symmetry, 8(4), 19. https://doi.org/10.3390/sym8040019