References
- L. Albertazzi, Handbook of Experimental Phenomenology: Visual Perception of Shape, Space and Apearance, Wiley, Chichester, 2013.
- P. Alegre, D. E. Blair, and A. Carriazo, Generalized Sasakian space form, Israel J. Math. 141 (2004), no. 1, 157-183. https://doi.org/10.1007/BF02772217
- P. Alegre, A. Carriazo, Y. H. Kim, and D. W. Yoon, B. Y. Chen's inequality for submanifolds of generalized space forms, Indian J. Pure Appl. Math. 38 (2007), no. 3, 185-201.
- D. E. Blair, The theory of quasi-Sasakian structures, J. Differential Geom. 1 (1967), 331-345. https://doi.org/10.4310/jdg/1214428097
- D. E. Blair and A. J. Ledger, Quasiumbilical, minimal submanifolds of Euclidean space, Simon Stevin 51 (1977), no. 1, 3-22.
- F. Casorati, Mesure de la courbure des surfaces suivant l'idee commune. Ses rapports avec les mesures de courbure gaussienne et moyenne, Acta Math. 14 (1890), no. 1, 95-110. https://doi.org/10.1007/BF02413317
- B.-Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arc. Math. (Basel) 60 (1993), no. 6, 568-578. https://doi.org/10.1007/BF01236084
- B.-Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasg. Math. J. 41 (1999), no. 1, 33-41. https://doi.org/10.1017/S0017089599970271
-
B.-Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR
${\delta}$ -invariant, Internat. J. Math. 23 (2012), no. 3, 1250045, 17 pp. https://doi.org/10.1142/S0129167X12500450 - B.-Y. Chen and F. Dillen, Optimal general inequalities for Lagrangian submanifolds in complex space forms, J. Math. Anal. Appl. 379 (2011), no. 1, 229-239. https://doi.org/10.1016/j.jmaa.2010.12.058
- 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), no. 6, 808-819. https://doi.org/10.1016/j.difgeo.2013.09.006
- S. Decu, S. Haesen, and L. Verstraelen, Optimal inequalities involving Casorati curvatures, Bull. Transylv. Univ. Brasov, Ser. B 14(49) (2007), Suppl., 85-93.
- S. Decu, S. Haesen, and L. Verstraelen, Optimal inequalities charaterising quasi-umbilical submanifolds, J. Inequal. Pure Appl. Math. 9 (2008), no. 3, Article ID 79, 7 pp.
- A. Friedmann and J. A. Schouten, Uber die Geometrie der halbsymmetrischen Ubertragungen, Math. Z. 21 (1924), no. 1, 211-223. https://doi.org/10.1007/BF01187468
- S. Haesen, D. Kowalczyk, and L. Verstraelen, On the extrinsic principal directions of Riemannian submanifolds, Note Math. 29 (2009), no. 2, 41-53.
- H. A. Hayden, Subspaces of a space with torsion, Proc. London Math. Soc. 34 (1932), 27-50.
- S. Hong and M. M. Tripathi, On Ricci curvature of submanifolds of generalized Sasakian space forms, Int. J. Pure Appl. Math. Sci. 2 (2005), no. 2, 173-201.
- T. Imai, Hypersurfaces of a Riemannian manifold with semi-symmetric metric connection, Tensor (N. S.) 23 (1972), 300-306.
- T. Imai, Notes on semi-symmetric metric connections, Tensor (N. S.) 24 (1972), 293-296.
- D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), no. 1, 1-27. https://doi.org/10.2996/kmj/1138036310
- K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. 24 (1972), 93-103. https://doi.org/10.2748/tmj/1178241594
- J. S. Kim, Y. M. Song, and M. M. Tripathi, B.-Y. Chen inequalities for submanifolds in generalized complex space forms, Bull. Korean Math. Soc. 40 (2003), no. 3, 411-423. https://doi.org/10.4134/BKMS.2003.40.3.411
-
J.W. Lee and G. E. Vilcu, Inequalities for generalized normalized
${\delta}$ -Casorati Curvatures of slant submanifolds in quaternionic space forms, Taiwanese J. Math. 19 (2015), no. 3, 691-702. https://doi.org/10.11650/tjm.19.2015.4832 - R. Lemence, On four-dimensional generalized complex space forms, Nihonkai Math. J. 15 (2004), no. 2, 169-176.
- G. D. Ludden, Submanifolds of cosymplectic manifolds, J. Differential Geometry 4 (1970), 237-244. https://doi.org/10.4310/jdg/1214429387
- J. C. Marrero, The local structure of trans-Sasakian manifolds, Ann. Mat. Pure Appl. 162 (1992), 77-86. https://doi.org/10.1007/BF01760000
- K. Matsumoto, I. Mihai, and Y. Tazawa, Ricci tensor of slant submanifolds in complex space forms, Kodai Math. J. 26 (2003), no. 1, 85-94. https://doi.org/10.2996/kmj/1050496650
- A. Mihai and C. Ozgur, Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection, Taiwanese J. Math. 14 (2010), no. 4, 1465-1477. https://doi.org/10.11650/twjm/1500405961
- A. Mihai and C. Ozgur, Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections, Rocky Mountain J. Math. 41 (2011), no. 5, 1653-1673. https://doi.org/10.1216/RMJ-2011-41-5-1653
- Z. Nakao, Submanifolds of a Riemannian manifold with semi-symmetric metric connections, Proc. Amer. Math. Soc. 54 (1976), 261-266. https://doi.org/10.1090/S0002-9939-1976-0445416-9
- A. Oiaga and I. Mihai, B.-Y. Chen inequalities for slant submanifolds in complex space forms, Demonstratio Math. 32 (1999), no. 4, 835-846.
- Z. Olszak, On the existence of generalized complex space forms, Israel J. Math. 65 (1989), no. 2, 214-218. https://doi.org/10.1007/BF02764861
- J. A. Oubi-na, New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), no. 3-4, 187-193.
- V. Slesar, B. Sahin, and G. E. Vlcu, Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms, J. Inequal. Appl. 2014 (2014), 123. https://doi.org/10.1186/1029-242X-2014-123
- F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), no. 2, 365-398. https://doi.org/10.1090/S0002-9947-1981-0626479-0
- M. M. Tripathi, Generic submanifolds of generalized complex space forms, Publ. Math. Debrecen 50 (1997), no. 3-4, 373-392.
- M. M. Tripathi, A new connection in a Riemannian manifold, Int. Electron. J. Geom. 1 (2008), no. 1, 15-24.
- M. M. Tripathi and S. S. Shukla, Ricci curvature and k-Ricci curvature of submanifolds of generalized complex space forms, Aligarh Bull. Math. 20 (2001), no. 1, 143-156.
- L. Verstraelen, The geometry of eye and brain, Soochow J. Math. 30 (2004), no. 3, 367-376.
- L. Verstraelen, Geometry of submanifolds I, The first Casorati curvature indicatrices, Kragujevac J. Math. 37 (2013), no. 1, 5-23.
- K. Yano, On semi-symmetric metric connection, Rev. Roumaine Math. Pure Appl. 15 (1970), 1579-1586.
- D. W. Yoon and K. S. Cho, Inequality for warped products in generalized Sasakian space forms, Int. Math. J. 5 (2004), no. 3, 225-235.
- P. Zhang, L. Zhang, and W. Song, Chen's inequalities for submanifolds of a Riemannian manifold of quasi-constatnt curvature with a semi-symmetric metric connection, Taiwanese J. Math. 18 (2014), no. 6, 1841-1862. https://doi.org/10.11650/tjm.18.2014.4045
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