Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms
Abstract
:1. Introduction
2. Preliminaries
3. Main Inequalities
4. Examples
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Chen, B.-Y. Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension. Glasgow Math. J. 1999, 41, 33–41. [Google Scholar] [CrossRef] [Green Version]
- Chen, B.-Y. Pseudo-Riemannian Geometry, δ-Invariants and Applications; World Scientific: Hackensack, NJ, USA, 2011. [Google Scholar]
- Decu, S.; Haesen, S.; Verstraelen, L.; Vîlcu, G.E. Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature. Entropy 2018, 20, 529. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Decu, S.; Haesen, S.; Verstraelen, L. Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature. Mathematics 2020, 8, 251. [Google Scholar] [CrossRef] [Green Version]
- Chen, B.-Y. Recent developments in δ-Casorati curvature invariants. Turk. J. Math. 2021, 45, 1–46. [Google Scholar] [CrossRef]
- Chen, B.-Y. Recent developments in Wintgen inequality and Wintgen ideal submanifolds. Int. Electron. J. Geom. 2021, 20, 6–45. [Google Scholar] [CrossRef]
- Chen, B.-Y. Some pinching and classification theorems for minimal submanifolds. Arch. Math. 1993, 60, 568–578. [Google Scholar] [CrossRef]
- Van der Veken, J.; Carriazo, A.; Dimitrić, I.; Oh, Y.M.; Suceavă, B.; Vrancken, L. Reflections on some research work of Bang-Yen Chen. In Geometry of Submanifolds: AMS Special Session in Honor of Bang-Yen Chen’s 75th Birthday, University of Michigan, Ann Arbor, MI, USA, 20–21 October 2018; Van der Veken, J., Carriazo, A., Dimitrić, I., Oh, Y.M., Suceavă, B., Vrancken, L., Eds.; Contemporary Mathematics; American Mathematical Society: Providence, RI, USA, 2020; Volume 756. [Google Scholar]
- Verstraelen, L. Submanifold theory—A contemplation of submanifolds. In Geometry of Submanifolds: AMS Special Session in Honor of Bang-Yen Chen’s 75th Birthday, University of Michigan, Ann Arbor, MI, USA, 20–21 October 2018; Van der Veken, J., Carriazo, A., Dimitrić, I., Oh, Y.M., Suceavă, B., Vrancken, L., Eds.; Contemporary Mathematics; American Mathematical Society: Providence, RI, USA, 2020; Volume 756. [Google Scholar]
- Mihai, I. Statistical manifolds and their submanifolds. Results on Chen-like invariants. In Geometry of Submanifolds: AMS Special Session in Honor of Bang-Yen Chen’s 75th Birthday, University of Michigan, Ann Arbor, MI, USA, 20–21 October 2018; Van der Veken, J., Carriazo, A., Dimitrić, I., Oh, Y.M., Suceavă, B., Vrancken, L., Eds.; Contemporary Mathematics; American Mathematical Society: Providence, RI, USA, 2020; Volume 756. [Google Scholar]
- Amari, S. Differential-Geometrical Methods in Statistics. In Lecture Notes in Statistics; Berger, J., Fienberg, S., Gani, J., Krickeberg, K., Olkin, I., Singer, B., Eds.; Springer: Berlin, Germany, 1985; Volume 28. [Google Scholar]
- Chen, B.-Y.; Mihai, A.; Mihai, I. A Chen first inequality for statistical submanifolds in Hessian submanifolds of constant Hessian curvature. Results Math. 2019, 74, 165. [Google Scholar] [CrossRef]
- Aytimur, H.; Kon, M.; Mihai, A.; Özgur, C.; Takano, K. Chen inequalities for statistical submanifolds of Kähler-like statistical manifolds. Mathematics 2019, 7, 1202. [Google Scholar] [CrossRef] [Green Version]
- Graves, J.T. On a connection between the general theory of normal couples and the theory of complete quadratic functions of two variables. Philos. Mag. 1845, 26, 315–320. [Google Scholar] [CrossRef]
- Rashevskij, P.K. The scalar field in stratified space. Trudy Sem. Vektor. Tenzor. Anal. 1948, 6, 225–248. [Google Scholar]
- Ruse, H.S. On parallel fields of planes in a Riemannian manifold. Quart. J. Math. 1949, 20, 218–234. [Google Scholar] [CrossRef]
- Rozenfeld, B.A. On unitary and stratified spaces. Trudy Sem. Vektor. Tenzor. Anal. 1949, 7, 260–275. [Google Scholar]
- Cruceanu, V.; Fortuny, P.; Gadea, P.M. A survey on paracomplex geometry. Rocky Mt. J. Math. 1996, 26, 83–115. [Google Scholar] [CrossRef]
- Defever, F.; Deszcz, R.; Verstraelen, L. On pseudosymmetric para-Kähler manifolds. Colloq. Math. 1997, 74, 253–260. [Google Scholar] [CrossRef] [Green Version]
- Mihai, A.; Rosca, R. Skew-symmetric vector fields on a CR-submanifold of a para-Kähler manifold. Int. J. Math. Math. Sci. 2004, 10, 535–540. [Google Scholar] [CrossRef] [Green Version]
- Fei, T.; Zhang, J. Interaction of Codazzi couplings with (para)-Kähler geometry. Results Math. 2017, 72, 2037–2056. [Google Scholar] [CrossRef]
- Vîlcu, G.E. Almost products structures on statistical manifolds and para-Kähler-like statistical submersions. Bull. Sci. Math. 2021, 171, 103018. [Google Scholar] [CrossRef]
- Chen, B.-Y.; Decu, S.; Vîlcu, G.E. Inequalities for the Casorati Curvature of Totally Real Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms. Entropy 2021, 23, 1399. [Google Scholar] [CrossRef]
- Chen, B.-Y.; Ogiue, K. On totally real submanifolds. Trans. Am. Math. Soc. 1974, 193, 257–266. [Google Scholar] [CrossRef]
- Chen, B.-Y.; Dillen, F.; Verstraelen, L.; Vrancken, L. Totally real submanifolds of CPn satisfying a basic equality. Arch. Math. 1994, 63, 553–564. [Google Scholar] [CrossRef]
- Chen, B.-Y. Riemannian geometry of Lagrangian submanifolds. Taiwanese J. Math. 2001, 5, 681–723. [Google Scholar] [CrossRef]
- Chen, B.-Y.; Dillen, F.; Van der Veken, J.; Vrancken, L. Classification of δ(2,n − 2)-ideal Lagrangian submanifolds in n-dimensional complex space forms. J. Math. Anal. Appl. 2018, 458, 1456–1485. [Google Scholar] [CrossRef]
- Chen, B.-Y. Lagrangian submanifolds in para-Kähler manifolds. Nonlinear Anal. 2010, 73, 3561–3571. [Google Scholar] [CrossRef]
- Chen, B.-Y. Lagrangian H-umbilical submanifolds of para-Kähler manifolds. Taiwanese J. Math. 2011, 15, 2483–2502. [Google Scholar] [CrossRef]
- Chen, B.-Y. Classification of flat Lagrangian H-umbilical submanifolds in para-Kähler n-plane. Int. Electron. J. Geom. 2011, 4, 1–14. [Google Scholar]
- Furuhata, H.; Hasegawa, I. Submanifold theory in holomorphic statistical manifolds. In Geometry of Cauchy-Riemann Submanifolds; Dragomir, S., Shahid, M.H., Al-Solamy, F.R., Eds.; Springer Science+Business Media: Singapore, 2016; pp. 179–214. [Google Scholar]
- Vos, P. Fundamental equations for statistical submanifolds with applications to the Barlett correction. Ann. Inst. Statist. Math. 1989, 41, 429–450. [Google Scholar] [CrossRef]
- Opozda, B. Bochners technique for statistical manifolds. Ann. Glob. Anal. Geom. 2015, 48, 357–395. [Google Scholar] [CrossRef] [Green Version]
- Oprea, T. Constrained Extremum Problems in Riemannian Geometry; University of Bucharest Publishing House: Bucharest, Romania, 2006. [Google Scholar]
- Gadea, P.M.; Montesinos-Amilibia, A.M. Spaces of constant paraholomorphic sectional curvature. Pac. J. Math. 1989, 136, 85–101. [Google Scholar] [CrossRef] [Green Version]
- Gadea, P.M.; Muñoz Masqué, J. Classification of homogeneous parakählerian space forms. Nova J. Algebra Geom. 1992, 1, 111–124. [Google Scholar]
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Decu, S.; Haesen, S. Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms. Mathematics 2022, 10, 330. https://doi.org/10.3390/math10030330
Decu S, Haesen S. Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms. Mathematics. 2022; 10(3):330. https://doi.org/10.3390/math10030330
Chicago/Turabian StyleDecu, Simona, and Stefan Haesen. 2022. "Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms" Mathematics 10, no. 3: 330. https://doi.org/10.3390/math10030330
APA StyleDecu, S., & Haesen, S. (2022). Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms. Mathematics, 10(3), 330. https://doi.org/10.3390/math10030330