Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry. (English) Zbl 1312.53049

In this paper, the authors are looking for three-dimensional Ricci-solitons with a Lorentz metric (signature \((2, 1)\)).
They start with an almost paracontact structure \((\varphi , \psi , \eta , g)\) at the beginning of the second section. Here, \(\eta\) is the almost-contact differential 1-form, \(\psi\) is a vector field (the dual of \(\eta\)), \(\varphi\) is a complex structure on the kernel \({\mathcal D}\) of \(\eta\) (actually a product structure, i.e., \(\varphi ^2 =I\)), and \(g\) satisfies \((\varphi \;, \varphi \;)=-g+\eta \otimes \eta\).
\(M\) is normal (Definition 2.1) if \(\varphi\) extends to an integrable product structure on \(M\times {\mathbb R}\) with the natural product of \({\mathcal D}\times {\mathbb R}^2\). This is the same as (2.4) according to [J. Wełyczko, Result. Math. 54, No. 3–4, 377–387 (2009; Zbl 1180.53080)].
There is a serious “typo” in the page 119, line 7. “(1.4a)” should possibly be “the first equation of (2.4)”.
Definition 2.2 says that 1. \(M\) is quasi-para-Sasakian if \(\alpha =0\) and \(\beta \neq 0\) (in (2.4)); and 2. para-Kenmotsu if \(\beta=0\) and \(\alpha \neq 0\).
One of the major result is Theorem 3.1: If \(\rho\) is a parallel symmetric 2-tensor which is \(\varphi\) skew-symmetric, then it is a multiple of the metric \(g\).
By applying Theorem 3.1 to the \(\rho\) in (4.11), the authors obtains all kind of Ricci-solitons.


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics


Zbl 1180.53080
Full Text: DOI


[1] Bejan, CL; Crasmareanu, M, Ricci solitons in manifolds with quasi-constant curvature, Publ. Math. Debrecen, 78, 235-243, (2011) · Zbl 1274.53097
[2] Brînzănescu, V; Slobodeanu, R, Holomorphicity and walczak formula on Sasakian manifolds, J. Geom. Phys., 57, 193-207, (2006) · Zbl 1160.53359
[3] Brozos-Vázquez, M; Calvaruso, G; García-Río, E; Gavino-Fernández, S, Three-dimensional Lorentzian homogeneous Ricci solitons, Israel J. Math., 188, 385-403, (2012) · Zbl 1264.53052
[4] Calvaruso, G, Homogeneous paracontact metric three-manifolds, Illinois J. Math., 55, 697-718, (2012) · Zbl 1273.53020
[5] Calvaruso, G; Kowalski, O, On the Ricci operator of locally homogeneous Lorentzian 3-manifolds, Cent. Eur. J. Math., 7, 124-139, (2009) · Zbl 1180.53070
[6] Calvaruso, G., Perrone, D.: Geometry of \(H\)-paracontact metric manifolds. arXiv:1307.7662 · Zbl 1374.53111
[7] Cappelletti Montano, B, Bi-paracontact structures and Legendre foliations, Kodai Math. J., 33, 473-512, (2010) · Zbl 1215.53074
[8] Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, American Mathematical Society, Providence, RI; Science Press, New York, (2006). MR2274812 (2008a:53068) · Zbl 1118.53001
[9] Craioveanu, M., Slesar, V.: A Weitzenböck formula for a closed Riemannian manifold with two orthogonal complementary distributions, Bull. Math. Soc. Sci. Math. Roumanie (NS), 52(100), 3, 271-279 (2009). MR2554486 (2010j:58078) · Zbl 1199.58003
[10] De, UC; Tripathi, MM, Ricci tensor in \(3\)-dimensional trans-Sasakian manifolds, Kyungpook Math. J., 43, 247-255, (2003) · Zbl 1073.53060
[11] Ivanov, S; Vassilev, D; Zamkovoy, S, Conformal paracontact curvature and the local flatness theorem, Geom. Dedicata, 144, 79-100, (2010) · Zbl 1195.53048
[12] Kaneyuki, S; Williams, FL, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99, 173-187, (1985) · Zbl 0576.53024
[13] Leistner, T, On the classification of Lorentzian holonomy groups, J. Differ. Geom., 76, 423-484, (2007) · Zbl 1129.53029
[14] Onda, K, Lorentz Ricci solitons on 3-dimensional Lie groups, Geom. Dedicata, 147, 313-322, (2010) · Zbl 1203.53044
[15] Petersen, P.: Riemannian geometry, 2nd edn. Graduate Texts in Mathematics, vol. 171, Springer, New York (2006). MR2243772 (2007a:53001) · Zbl 1220.53002
[16] Sharma, R, Second order parallel tensor in real and complex space forms, Internat. J. Math. Math. Sci., 12, 787-790, (1989) · Zbl 0696.53012
[17] Welyczko, J, On Legendre curves in 3-dimensional normal almost contact metric manifolds, Soochow J. Math., 33, 929-937, (2007) · Zbl 1144.53041
[18] Welyczko, J.: On Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Results Math. 54(3-4) 377-387 (2009). MR2534454 (2010g:53153) · Zbl 1180.53080
[19] Welyczko, J.: Slant curves in 3-dimensional normal almost paracontact metric manifolds. Mediterr. J. Math. (2013). doi:10.1007/s00009-013-0361-2, arXiv:1212.5839 · Zbl 1300.53021
[20] Zamkovoy, S, Canonical connections on paracontact manifolds, Ann. Global Anal. Geom., 36, 37-60, (2008) · Zbl 1177.53031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.