Paracontact Metric Structures on Five-dimensional Unsolvable Lie Algebras
УДК 514.76
Abstract
This paper studies the problem of the existence of para-contact metric structures and parasasakian structures on five-dimensional unsolvable Lie algebras. According to the classification by A. Diatta, there are three such Lie algebras. These are the decomposable aff(K)xsl(2,K), aff(K)xso(3) ) Lie algebras and the indecomposable sl(2,K)xp К2 Lie algebra, where p is the usual action of sl(2,K) on К2. Decomposable Lie algebras are direct products of the exact symplectic aff(K) Lie algebra and three-dimensional contact Lie algebras.
For an indecomposable sl(2,K)xp К2 Lie algebra, Diatta gives a procedure for constructing a contact struc-ture. Each Lie algebra is considered in detail. It is shown that para-sasakian structures exist only on aff(K)xsl(2,K). Their explicit expressions, Ricci tensors, and scalar cur-vatures are obtained. It is shown that there are paracontact metric structures for other aff(K)xso(3) and sl(2,K)xp К2 Lie algebras, but none of them have the K-paracontact property. Examples of paracontact metric structures are provided for the last two Lie algebras indicated above.
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References
Blair D.E. Riemannian Geometry of Contact and Symplectic Manifolds, Second Edition. Progress in Mathematics, 203, Birkhauser, Boston Inc., Boston, MA, 2010. 343 p. DOI: 10.1007/978-0-8176-4959-3
Calvaruso G. Perrone A. Five-Dimensional ParaContact Lie Algebras // Differential Geometry and its Applications. 2016. Vol. 45. P. 115–129. DOI: 10.1016/j.difgeo.2016.01.001
Смоленцев Н.К. Левоинвариантные парасасакиевы структуры на группах Ли // Вестник Томского государственного университета. Математика и механика. 2019. Т. 62. С. 27–37. DOI: 10.17223/19988621/62/3
Diatta A. Left Invariant Contact Structures on Lie Groups // Differential Geometry and its Applications. 2008. Vol. 26. P. 544-552. DOI: 10.1016/j.difgeo.2008.04.001
Goze M., Khakimdjanov Y., Medina A. Symplectic or Contact Structures on Lie Groups // Differential Geometry and its Applications. 2004. Vol. 21. No 1. P. 41-54. DOI: 10.1016/j.dif-geo.2003.12.006
Goze M., Remm E. Contact and Frobeniusian Forms on Lie Groups // Differential Geometry and its Applications. 2014. Vol. 25. P. 74-94. DOI: 10.1016/j.difgeo.2014.05.008
Славолюбова Я.В. Применение систем компьютерной математики к исследованию левоинвариантных контактных метрических структур на пятимерных группах Ли : дисс. ... канд. физ.-мат. наук. Кемеровский государственный университет. Кемерово, 2011. 202 с.
Loiudice E. Lotta A. On Five Dimensional sasakian Lie Algebras with Trivial Center // Osaka Journal of Mathematics. 2018. Vol. 55. P. 39-49.
Calvaruso G., Fino A. Five-Dimensional K-Contact Lie Algebras. // Monatshefte fur Mathematik. 2012. Vol. 167. P. 35-59. DOI: 10.1007/s00605-011-0308-2
Смоленцев Н.К., Шагабудинова И.Ю. О парасасакиевых структурах на пятимерных алгебрах Ли // Вестник Томского государственного университета. Математика и механика. 2021. Т. 69. С. 37-51. DOI: 10.17223/19988621/69/4
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