Abstract
In this manuscript, we consider almost Hermitian manifolds and almost contact metric manifolds. We construct almost Hermitian manifolds from the product of almost contact metric manifolds with $\mathbb{R}$ by warped product. Depending on the function of warped product, we investigate the curvature properties of the almost Hermitian manifolds obtained in this way. In particular, we study Einstein almost Hermitian manifolds obtained from Einstein almost contact metric manifolds. In addition, we study the relationships between some classes of almost contact metric manifolds and almost Hermitian manifolds.
Keywords
almost contact metric structure almost Hermitian structure curvature Einstein manifold product manifold warped product
Ethical Statement
It is declared that during the preparation process of this study, scientific and ethical principles were followed and all the studies benefited from are stated in the bibliography.
Supporting Institution
No grants were received from any public, private or non-profit organizations for this research.
Thanks
The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.
References
- [1] Agricola, I., Friedrich, T. 3-Sasakian Manifolds in Dimension Seven, Their Spinors and G2-Structures, J. Geom. Phys. 60 (2), 326-332 (2010). https://doi.org/10.1016/j.geomphys.2009.10.003
- [2] Alexiev, V., Ganchev, G. On the Classification of Almost Contact Metric Manifolds. In: Mathematics and Education in Mathematics, Proceedings of 15th Spring Conference, Sunny Beach, pp. 155–161 (1986)
- [3] Attarchi, H. Warped Product Conformal Kaehler Manifolds and Kenmotsu Structures. arXiv:1206.2766v3[math.DG] 11 Sep 2021.
- [4] Blair, D.E. D-Homothetic Warping. Publ. Inst. Math. 94 (108), 47-54 (2013).
- [5] Bouzir, H., Beldjilali, G. Kaehlerian Structure on the Product of Two Trans-Sasakian Manifolds. Int. Electron. J. Geom. 13 (2), 135-143 (2020).
- [6] Beldjilali, G., Belkhelfa, M. Kaehlerian Structures and D-Homothetic Bi-Warping. J. Geom. Symmetry Phys. 42, 1-13 (2016).
- [7] Beldjilali, G., Cherif, A. M. and Zegga, K. From a Single Sasakian Manifold to a Family of Sasakian Manifolds. Beitr. Algebra Geom. 60, 445-458 (2019).
- [8] Chinea, D., Gonzalez, C. A Classification of Almost Contact Metric Manifolds. Ann. di Mat. Pura ed Appl. 156, 15–36 (1990). https://doi.org/10.1007/BF01766972
- [9] Chojnacka-Dulas, J., Deszcz, R., Glokowska, M. and Prvanovic, M. On Warped Product Manifolds Satisfying Some Curvature Conditions. J. Geom. Phys. 74, 328-341 (2013).
- [10] Ganchev, G., Mihova, V. Warped Product Kaehler Manifolds and Bochner-Kaehler metrics. J. Geom. Phys. 58, 803-824 (2008).
- [11] Gray, A., Hervella G.M. The Sixteen Classes of Almost Hermitian Manifolds and Their Linear Invariants. Annal. di Mat. Pura et Applicata. 123 (4), 35-38 (1980).
- [12] Kashiwada, T. A Note on a Riemannian Space with Sasakian 3-Structure. Nat. Sci. Reps. Ochanomizu Univ. 22, 1-2 (1971).
- [13] Oubina J.A. A Classification for Almost Contact Metric Structures, Preprint, 1980.
- [14] Özdemir, N., Aktay, S¸ ., Solgun, M. Almost Hermitian Structures on the Products of Two Almost Contact Metric Manifolds. Differ Geom Dyn Syst. 18, 102-109 (2016).
- [15] Özdemir, N., Erdogan, N. Some Relations Between Almost Paracontact Metric Manifolds and Almost Parahermitian Manifolds. Turk. J. Math. 46 (4), 1459-1477, (2022).
- [16] Sasaki S., Hatakeyama Y. On Differentiable Manifolds with Certain Structures Which Are Closely Related to Almost Contact Structure II. Tohoku Math. J. 13, 281-294 (1961).
- [17] Solgun, M, Karababa, Y. A Natural Way to Construct an Almost Complex B-metric Structure. Math. Meth. Appl. Sci. 44, 7607– 7613 (2021).
- https://doi.org/10.1002/mma.6430 [18] Tashiro, Y. On Contact Structures of Hypersurfaces in Complex Manifolds I. Tohoku Math. J. 15, 62-78 (1963).
- [19] Watanabe, Y. Almost Hermitian and Kaehler Structures on Product Manifolds. Proceedings of the 13th International Workshop on Diff. Geom. 13, 1-16 (2009).
- [20] Cabrera, F. M. On the classification of almost contact metric manifolds. Differential Geometry and its Applications 64, 13–28 (2019).
Abstract
References
- [1] Agricola, I., Friedrich, T. 3-Sasakian Manifolds in Dimension Seven, Their Spinors and G2-Structures, J. Geom. Phys. 60 (2), 326-332 (2010). https://doi.org/10.1016/j.geomphys.2009.10.003
- [2] Alexiev, V., Ganchev, G. On the Classification of Almost Contact Metric Manifolds. In: Mathematics and Education in Mathematics, Proceedings of 15th Spring Conference, Sunny Beach, pp. 155–161 (1986)
- [3] Attarchi, H. Warped Product Conformal Kaehler Manifolds and Kenmotsu Structures. arXiv:1206.2766v3[math.DG] 11 Sep 2021.
- [4] Blair, D.E. D-Homothetic Warping. Publ. Inst. Math. 94 (108), 47-54 (2013).
- [5] Bouzir, H., Beldjilali, G. Kaehlerian Structure on the Product of Two Trans-Sasakian Manifolds. Int. Electron. J. Geom. 13 (2), 135-143 (2020).
- [6] Beldjilali, G., Belkhelfa, M. Kaehlerian Structures and D-Homothetic Bi-Warping. J. Geom. Symmetry Phys. 42, 1-13 (2016).
- [7] Beldjilali, G., Cherif, A. M. and Zegga, K. From a Single Sasakian Manifold to a Family of Sasakian Manifolds. Beitr. Algebra Geom. 60, 445-458 (2019).
- [8] Chinea, D., Gonzalez, C. A Classification of Almost Contact Metric Manifolds. Ann. di Mat. Pura ed Appl. 156, 15–36 (1990). https://doi.org/10.1007/BF01766972
- [9] Chojnacka-Dulas, J., Deszcz, R., Glokowska, M. and Prvanovic, M. On Warped Product Manifolds Satisfying Some Curvature Conditions. J. Geom. Phys. 74, 328-341 (2013).
- [10] Ganchev, G., Mihova, V. Warped Product Kaehler Manifolds and Bochner-Kaehler metrics. J. Geom. Phys. 58, 803-824 (2008).
- [11] Gray, A., Hervella G.M. The Sixteen Classes of Almost Hermitian Manifolds and Their Linear Invariants. Annal. di Mat. Pura et Applicata. 123 (4), 35-38 (1980).
- [12] Kashiwada, T. A Note on a Riemannian Space with Sasakian 3-Structure. Nat. Sci. Reps. Ochanomizu Univ. 22, 1-2 (1971).
- [13] Oubina J.A. A Classification for Almost Contact Metric Structures, Preprint, 1980.
- [14] Özdemir, N., Aktay, S¸ ., Solgun, M. Almost Hermitian Structures on the Products of Two Almost Contact Metric Manifolds. Differ Geom Dyn Syst. 18, 102-109 (2016).
- [15] Özdemir, N., Erdogan, N. Some Relations Between Almost Paracontact Metric Manifolds and Almost Parahermitian Manifolds. Turk. J. Math. 46 (4), 1459-1477, (2022).
- [16] Sasaki S., Hatakeyama Y. On Differentiable Manifolds with Certain Structures Which Are Closely Related to Almost Contact Structure II. Tohoku Math. J. 13, 281-294 (1961).
- [17] Solgun, M, Karababa, Y. A Natural Way to Construct an Almost Complex B-metric Structure. Math. Meth. Appl. Sci. 44, 7607– 7613 (2021).
- https://doi.org/10.1002/mma.6430 [18] Tashiro, Y. On Contact Structures of Hypersurfaces in Complex Manifolds I. Tohoku Math. J. 15, 62-78 (1963).
- [19] Watanabe, Y. Almost Hermitian and Kaehler Structures on Product Manifolds. Proceedings of the 13th International Workshop on Diff. Geom. 13, 1-16 (2009).
- [20] Cabrera, F. M. On the classification of almost contact metric manifolds. Differential Geometry and its Applications 64, 13–28 (2019).
Details
| Primary Language | English |
|---|---|
| Subjects | Applied Mathematics (Other) |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | April 29, 2024 |
| Publication Date | April 30, 2024 |
| Submission Date | November 8, 2023 |
| Acceptance Date | January 5, 2024 |
| Published in Issue | Year 2024 Volume: 12 Issue: 1 |
Cite
The published articles in KJM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.