| Title:
|
Projective Curvature Tensorin 3-dimensional Connected Trans-Sasakian Manifolds (English) |
| Author:
|
De, Krishnendu |
| Author:
|
De, Uday Chand |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
55 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
29-40 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The object of the present paper is to study $\xi $-projectively flat and $\phi $-projectively flat 3-dimensional connected trans-Sasakian manifolds. Also we study the geometric properties of connected trans-Sasakian manifolds when it is projectively semi-symmetric. Finally, we give some examples of a 3-dimensional trans-Sasakian manifold which verifies our result. (English) |
| Keyword:
|
Trans-Sasakian manifold |
| Keyword:
|
$\xi $-projectively flat |
| Keyword:
|
$\phi $-projectively flat |
| Keyword:
|
Einstein manifold |
| MSC:
|
53C15 |
| MSC:
|
53C40 |
| idZBL:
|
Zbl 1365.53030 |
| . |
| Date available:
|
2017-03-16T12:38:40Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146059 |
| . |
| Reference:
|
[1] Bagewadi, C. S., Venkatesha, A.: Some curvature tensors on a trans-Sasakian manifold.. Turk. J. Math. 31 (2007), 111–121. Zbl 1138.53028, MR 2335656 |
| Reference:
|
[2] Blair, D. E.: Contact Manifolds in Riemannian Geometry.. Lecture Note in Mathematics 509, Springer-Verlag, Berlin–New York, 1976. Zbl 0319.53026, MR 0467588 |
| Reference:
|
[3] Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds.. Progress in Mathematics 203, Birkhäuser, Boston, 2002 Zbl 1011.53001, MR 1874240 |
| Reference:
|
[4] Blair, D. E., Oubina, J. A.: Conformal and related changes of metric on the product of two almost contact metric manifolds.. Publ. Mat. 34, 1 (1990), 199–207. Zbl 0721.53035, MR 1059874, 10.5565/PUBLMAT_34190_15 |
| Reference:
|
[5] Cabrerizo, J. L., Fernandez, L. M., Fernandez, M., Zhen, G.: The structure of a class of K-contact manifolds.. Acta Math. Hungar. 82, 4 (1999), 331–340. Zbl 0924.53024, MR 1675621, 10.1023/A:1006696410826 |
| Reference:
|
[6] Chinea, D., Gonzales, C.: A classification of almost contact metric manifolds.. Ann. Mat. Pura Appl. 156, 4 (1990), 15–36. MR 1080209, 10.1007/BF01766972 |
| Reference:
|
[7] Chinea, D., Gonzales, C.: Curvature relations in trans-sasakian manifolds.. In: Proceedings of the XIIth Portuguese–Spanish Conference on Mathematics II, Braga, 1987, Univ. Minho, Braga, 1987, 564–571. MR 1139218 |
| Reference:
|
[8] De, U. C., De, K.: On a class of three- dimensional Trans-Sasakian manifold.. Commun. Korean Math. Soc. 27 (2012), 795–808. MR 3025885, 10.4134/CKMS.2012.27.4.795 |
| Reference:
|
[9] De, U. C., Sarkar, A.: On three-dimensional trans-Sasakian manifolds.. Extracta Mathematicae 23, 3 (2008), 265–277. Zbl 1175.53058, MR 2524542 |
| Reference:
|
[10] De, U. C., Tripathi, M. M.: Ricci tensor in 3-dimensional trans-Sasakian manifolds.. Kyungpook Math. J. 43, 2 (2003), 247–255. Zbl 1073.53060, MR 1982228 |
| Reference:
|
[11] Gray, A., Hervella, L. M.: The sixteen classes of almost hermitian manifolds and their linear invariants.. Ann. Mat. Pura Appl. 123, 4 (1980), 35–58. Zbl 0444.53032, MR 0581924, 10.1007/BF01796539 |
| Reference:
|
[12] Janssens, D., Vanhecke, L.: Almost contact structures and curvature tensors.. Kodai Math. J. 4 (1981), 1–27. Zbl 0472.53043, MR 0615665, 10.2996/kmj/1138036310 |
| Reference:
|
[13] Kim, J. S., Prasad, R., Tripathi, M. M.: On generalized ricci- recurrent trans- sasakian manifolds.. J. Korean Math. Soc. 39 (2002), 953–961. Zbl 1025.53023, MR 1932790, 10.4134/JKMS.2002.39.6.953 |
| Reference:
|
[14] Kowalski, O.: An explicit classification of 3-dimensional Riemannian spaces satisfying $R(X, Y ).R = 0$.. Czechoslovak Math. J. 46(121) (1996), 427–474. MR 1408298 |
| Reference:
|
[15] Marrero, J. C.: The local structure of trans-sasakian manifolds.. Ann. Mat. Pura Appl. 162, 4 (1992), 77–86. Zbl 0772.53036, MR 1199647, 10.1007/BF01760000 |
| Reference:
|
[16] Marrero, J. C., Chinea, D.: On trans-sasakian manifolds.. In: Proceedings of the XIVth Spanish–Portuguese Conference on Mathematics I-III, Puerto de la Cruz, 1989, Univ.La Laguna, La Laguna, 1990, 655–659. MR 1112951 |
| Reference:
|
[17] Mikeš, J.: On Sasaki spaces and equidistant Kähler spaces.. Sov. Math., Dokl. 34 (1987), 428–431. Zbl 0631.53018, MR 0819428 |
| Reference:
|
[18] Mikeš, J.: Differential Geometry of Special Mappings.. Palacky Univ. Press, Olomouc, 2015. Zbl 1337.53001, MR 3442960 |
| Reference:
|
[19] Mikeš, J., Starko, G. A.: On hyperbolically Sasakian and equidistant hyperbolically Kählerian spaces.. Ukr. Geom. Sb. 32 (1989), 92–98. Zbl 0711.53042, MR 1049372 |
| Reference:
|
[20] Mishra, R. S.: Structures on Differentiable Manifold and Their Applications.. Chandrama Prakasana, Allahabad, 1984. |
| Reference:
|
[21] Oubina, J. A.: New classes of almost contact metric structures.. Publ. Math. Debrecen 32, 3-4 (1985), 187–193. Zbl 0611.53032, MR 0834769 |
| Reference:
|
[22] Ozgur, C.: $\phi $-conformally flat Lorentzian Para-Sasakian manifolds.. Radovi Matematicki 12 (2003), 99–106. MR 2022248 |
| Reference:
|
[23] Shukla, S. S., Singh, D. D.: On $\epsilon $-trans-sasakian manifolds.. Int. J. Math. Anal. 49 (2010), 2401–2414. MR 2770033 |
| Reference:
|
[24] Sinyukov, N. S.: Geodesic Mappings of Riemannian Spaces.. Nauka, Moscow, 1979. Zbl 0637.53020 |
| Reference:
|
[25] Szabo, Z. I.: Structure theorems on Riemannian spaces satisfying $R(X, Y ).R = 0$.. J. Diff. Geom. 17 (1982), 531–582. Zbl 0508.53025, MR 0683165, 10.4310/jdg/1214437486 |
| Reference:
|
[26] Yano, K., Bochner, S.: Curvature and Betti Numbers.. Annals of Math. Studies 32, Princeton Univ. Press, Princeton, 1953. Zbl 0051.39402, MR 0062505 |
| Reference:
|
[27] Yano, K., Kon, M.: Structure on Manifolds.. Series in Math. 3, World Scientific, Singapore, 1984. MR 0794310 |
| Reference:
|
[28] Zhen, G., Cabrerizo, J. L., Fernandez, L. M., Fernandez, M.: On $\xi $-conformally flat contact metric manifolds.. Indian J. Pure Appl. Math. 28 (1997), 725–734. Zbl 0882.53031, MR 1461184 |
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