Abstract
Blair et al. [7] introduced the notion of bicontact manifold in the context of Hermitian geometry. Bande and Hadjar [1] studied on this notion under the name of contact pairs. These type of structures have important properties and their geometry is some different from classical contact structures. In this paper, we study on some semi-symmetry properties of the normal contact pair manifolds. We prove that a Ricci semi-symmetric (or concircularly Ricci semi-symmetric) normal metric contact pair manifold is a generalized quasi-Einstein manifold. Also, we classify normal metric contact pair manifolds as a generalized quasi-Einstein manifold with certain semi-symmetry conditions and for the concircular curvature tensor , the Riemannian curvature tensor , and an arbitrary vector field .
References
- [1] Bande, G. and Hadjar, A. 2005. “Contact pairs” Tohoku Mathematical Journal, Second Series, 57(2), 247-260.
- [2] Bande, G. and Hadjar, A. 2009. “Contact pair structures and associated metrics” In Differential Geometry (pp. 266-275).
- [3] Bande, G. and Hadjar, A. 2010. “On normal contact pairs” International Journal of Mathematics, 21(06), 737-754.
- [4] Bande, G., Blair, D. E. and Hadjar, A. 2013. “On the curvature of metric contact pairs” Mediterranean journal of mathematics, 10(2), 989-1009.
- [5] Bande, G., Blair, D.E.: Symmetry in the geometry of metric contact pairs. Math. Nachr. 286, 1701–1709(2013)
- [6] Bande, G., Blair, D. E. and Hadjar, A. 2015. “Bochner and conformal flatness of normal metric contact pairs” Annals of Global Analysis and Geometry, 48(1), 47-56.
- [7] Blair, D. E., Ludden and G. D., Yano, K. 1974. “Geometry of complex manifolds similar to the Calabi-Eckmann manifolds” Journal of Differential Geometry, 9(2), 263-274.
- [8] Blair, D. E., Kim, J. S., & Tripathi, M. M. (2005). On the Concircular Curvature Tensor of a contact metric manifold. J. Korean Math. Soc, 42(5), 883-892.
- [9] Blair, D. E. (2010). “Riemannian geometry of contact and symplectic manifolds” Springer Science Business Media.
- [10] De, U. C. and Ghosh, G. C., 2004. “On generalized quasi–Einstein manifolds”, Kyungpook Math. J. 44 , 607–615.
- [11] Szabo, Y. I., Structure theorems on Riemannian manifolds satisfying R(X.Y ) R = 0, I, the local version, J. Differential Geom. 17 (1982), 531-582.
- [12] Turgut Vanli, A., and Unal, I. Conformal, concircular, quasi-conformal and conhar- monic flatness on normal complex contact metric manifolds. International Journal of Geometric Methods in Modern Physics, 14(05), 1750067. (2017).
- [13] Ünal, İ. 2019. “Some flatness conditions on normal.metric.contact Pairs” arXiv preprint arXiv:1902.05327.
- [14] Ünal, İ. (2020). Generalized Quasi-Einstein Manifolds in Contact Geometry. arXiv preprint arXiv:2006.07462.
- [15] Ünal, İ., & Altin, M. (2020). contact Metric Manifolds with Generalized Tanaka-Webster Connection. arXiv preprint arXiv:2004.02536.
- [16] Yano, K. (1940). Concircular geometry I. Concircular transformations. Proceedings of the Imperial Academy, 16(6), 195-200.
- [17] Yano K. and Kon M., Structure on manifolds, Series in Pure Math.Vol. 3, World Scien- tific, Singapore, (1984).
Abstract
Blair et al. [7] introduced the notion of bicontact manifold in the context of Hermitian geometry. Bande and Hadjar [1] studied on this notion under the name of contact pairs. These type of structures have important properties and their geometry is some different from classical contact structures. In this paper, we study on some semi-symmetry properties of the normal contact pair manifolds. We prove that a Ricci semi-symmetric (or concircularly Ricci semi-symmetric) normal metric contact pair manifold is a generalized quasi-Einstein manifold. Also, we classify normal metric contact pair manifolds as a generalized quasi-Einstein manifold with certain semi-symmetry conditions and for the concircular curvature tensor , the Riemannian curvature tensor , and an arbitrary vector field .
References
- [1] Bande, G. and Hadjar, A. 2005. “Contact pairs” Tohoku Mathematical Journal, Second Series, 57(2), 247-260.
- [2] Bande, G. and Hadjar, A. 2009. “Contact pair structures and associated metrics” In Differential Geometry (pp. 266-275).
- [3] Bande, G. and Hadjar, A. 2010. “On normal contact pairs” International Journal of Mathematics, 21(06), 737-754.
- [4] Bande, G., Blair, D. E. and Hadjar, A. 2013. “On the curvature of metric contact pairs” Mediterranean journal of mathematics, 10(2), 989-1009.
- [5] Bande, G., Blair, D.E.: Symmetry in the geometry of metric contact pairs. Math. Nachr. 286, 1701–1709(2013)
- [6] Bande, G., Blair, D. E. and Hadjar, A. 2015. “Bochner and conformal flatness of normal metric contact pairs” Annals of Global Analysis and Geometry, 48(1), 47-56.
- [7] Blair, D. E., Ludden and G. D., Yano, K. 1974. “Geometry of complex manifolds similar to the Calabi-Eckmann manifolds” Journal of Differential Geometry, 9(2), 263-274.
- [8] Blair, D. E., Kim, J. S., & Tripathi, M. M. (2005). On the Concircular Curvature Tensor of a contact metric manifold. J. Korean Math. Soc, 42(5), 883-892.
- [9] Blair, D. E. (2010). “Riemannian geometry of contact and symplectic manifolds” Springer Science Business Media.
- [10] De, U. C. and Ghosh, G. C., 2004. “On generalized quasi–Einstein manifolds”, Kyungpook Math. J. 44 , 607–615.
- [11] Szabo, Y. I., Structure theorems on Riemannian manifolds satisfying R(X.Y ) R = 0, I, the local version, J. Differential Geom. 17 (1982), 531-582.
- [12] Turgut Vanli, A., and Unal, I. Conformal, concircular, quasi-conformal and conhar- monic flatness on normal complex contact metric manifolds. International Journal of Geometric Methods in Modern Physics, 14(05), 1750067. (2017).
- [13] Ünal, İ. 2019. “Some flatness conditions on normal.metric.contact Pairs” arXiv preprint arXiv:1902.05327.
- [14] Ünal, İ. (2020). Generalized Quasi-Einstein Manifolds in Contact Geometry. arXiv preprint arXiv:2006.07462.
- [15] Ünal, İ., & Altin, M. (2020). contact Metric Manifolds with Generalized Tanaka-Webster Connection. arXiv preprint arXiv:2004.02536.
- [16] Yano, K. (1940). Concircular geometry I. Concircular transformations. Proceedings of the Imperial Academy, 16(6), 195-200.
- [17] Yano K. and Kon M., Structure on manifolds, Series in Pure Math.Vol. 3, World Scien- tific, Singapore, (1984).
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Research Article |
| Authors | |
| Publication Date | March 1, 2021 |
| Submission Date | July 14, 2020 |
| Published in Issue | Year 2021 Volume: 24 Issue: 1 |
Cite
Cited By
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https://doi.org/10.1142/S0219887823501761
Generalized Quasi-Conformal Curvature Tensor On Normal Metric Contact Pairs
International Journal of Pure and Applied Sciences
İ̇nan ÜNAL
https://doi.org/10.29132/ijpas.803809
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