Browse > Article
http://dx.doi.org/10.4134/JKMS.j170689

MINIMAL AND HARMONIC REEB VECTOR FIELDS ON TRANS-SASAKIAN 3-MANIFOLDS  

Wang, Yaning (Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control School of Mathematics and Information Sciences Henan Normal University)
Publication Information
Journal of the Korean Mathematical Society / v.55, no.6, 2018 , pp. 1321-1336 More about this Journal
Abstract
In this paper, we obtain some necessary and sufficient conditions for the Reeb vector field of a trans-Sasakian 3-manifold to be minimal or harmonic. We construct some examples to illustrate main results. As applications of the above results, we obtain some new characteristic conditions under which a compact trans-Sasakian 3-manifold is homothetic to either a Sasakian or cosymplectic 3-manifold.
Keywords
trans-Sasakian manifold; Reeb vector field; minimal vector field; harmonic vector field;
Citations & Related Records
연도 인용수 순위
  • Reference
1 L. Vanhecke and D. Janssens, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), no. 1, 1-27.   DOI
2 Y. Wang, Minimal Reeb vector fields on almost Kenmotsu manifolds, Czechoslovak Math. J. 67(142) (2017), no. 1, 73-86.
3 G. Wiegmink, Total bending of vector fields on Riemannian manifolds, Math. Ann. 303 (1995), no. 2, 325-344.
4 C. M. Wood, On the energy of a unit vector field, Geom. Dedicata 64 (1997), no. 3, 319-330.   DOI
5 A. Yildiz, U. C. De, and M. Turan, On 3-dimensional f-Kenmotsu manifolds and Ricci solitons, Ukrainian Math. J. 65 (2013), no. 5, 684-693.   DOI
6 S. Deshmukh, U. C. De, and F. Al-Solamy, Trans-Sasakian manifolds homothetic to Sasakian manifolds, Publ. Math. Debrecen 88 (2016), no. 3-4, 439-448.   DOI
7 S. Deshmukh and M. M. Tripathi, A note on trans-Sasakian manifolds, Math. Slovaca 63 (2013), no. 6, 1361-1370.   DOI
8 O. Gil-Medrano, Relationship between volume and energy of vector fields, Differential Geom. Appl. 15 (2001), no. 2, 137-152.
9 O. Gil-Medrano and E. Llinares-Fuster, Minimal unit vector fields, Tohoku Math. J. (2) 54 (2002), no. 1, 71-84.   DOI
10 J. C. Gonzalez-Davila and L. Vanhecke, Examples of minimal unit vector fields, Ann. Global Anal. Geom. 18 (2000), no. 3-4, 385-404.   DOI
11 J. C. Gonzalez-Davila and L. Vanhecke, Minimal and harmonic characteristic vector fields on three-dimensional contact metric manifolds, J. Geom. 72 (2001), no. 1-2, 65-76.   DOI
12 A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35-58.
13 J. C. Marrero, The local structure of trans-Sasakian manifolds, Ann. Mat. Pura Appl. (4) 162 (1992), 77-86.
14 Z. Olszak, Normal almost contact metric manifolds of dimension three, Ann. Polon. Math. 47 (1986), no. 1, 41-50.   DOI
15 Z. Olszak, Locally conformal almost cosymplectic manifolds, Colloq. Math. 57 (1989), no. 1, 73-87.   DOI
16 D. Perrone, Minimal Reeb vector fields on almost cosymplectic manifolds, Kodai Math. J. 36 (2013), no. 2, 258-274.   DOI
17 Z. Olszak and R. Rosca, Normal locally conformal almost cosymplectic manifolds, Publ. Math. Debrecen 39 (1991), no. 3-4, 315-323.
18 A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), no. 3-4, 187-193.
19 D. Perrone, Almost contact metric manifolds whose Reeb vector field is a harmonic section, Acta Math. Hungar. 138 (2013), no. 1-2, 102-126.
20 M. D. Siddiqi, A. Haseeb, and M. Ahmad, On generalized Ricci-recurrent (${\epsilon},\;{\delta}$)-trans-Sasakian manifolds, Palest. J. Math. 4 (2015), no. 1, 156-163.
21 N. Aktan, M. Yildirim, and C. Murathan, Almost f-cosymplectic manifolds, Mediterr. J. Math. 11 (2014), no. 2, 775-787.   DOI
22 D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, second edition, Progress in Mathematics, 203, Birkhauser Boston, Inc., Boston, MA, 2010.
23 D. E. Blair and J. A. Oubina, Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Mat. 34 (1990), no. 1, 199-207.
24 X. Chen, Notes on Ricci solitons in f-cosymplectic manifolds, Zh. Mat. Fiz. Anal. Geom. 13 (2017), no. 3, 242-253.
25 D. Chinea and C. Gonzalez, A classification of almost contact metric manifolds, Ann. Mat. Pura Appl. (4) 156 (1990), 15-36.   DOI
26 U. C. De and K. De, On a class of three-dimensional trans-Sasakian manifolds, Commun. Korean Math. Soc. 27 (2012), no. 4, 795-808.   DOI
27 K. De and U. C. De, Projective curvature tensorin 3-dimensional connected trans-Sasakian manifolds, Acta Univ. Palackianae Olomucensis, Facultas Rerum Naturalium, Math. 55 (2016), 29-40.
28 D. Debnath and A. Bhattacharyya, On generalized $\varphi$-recurrent trans-Sasakian manifolds, Acta Univ. Apulensis Math. Inform. No. 36 (2013), 253-265.
29 U. C. De and A. Sarkar, On three-dimensional trans-Sasakian manifolds, Extracta Math. 23 (2008), no. 3, 265-277.
30 U. C. De and M. M. Tripathi, Ricci tensor in 3-dimensional trans-Sasakian manifolds, Kyungpook Math. J. 43 (2003), no. 2, 247-255.
31 S. Deshmukh, Trans-Sasakian manifolds homothetic to Sasakian manifolds, Mediterr. J. Math. 13 (2016), no. 5, 2951-2958.
32 S. Deshmukh, Geometry of 3-dimensional trans-Sasakaian manifolds, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 62 (2016), no. 1, 183-192.
33 S. Deshmukh and F. Al-Solamy, A note on compact trans-Sasakian manifolds, Mediterr. J. Math. 13 (2016), no. 4, 2099-2104.