On the Schouten-Weyl tensor of 3-dimensional metric Lie groups

  • Олеся Павловна Хромова Алтайский государственный университет
  • Павел Николаевич Клепиков Алтайский государственный университет
  • Светлана Владимировна Клепикова Алтайский государственный университет
  • Евгений Дмитриевич Родионов Алтайский государственный университет

Аннотация

The main purpose of this paper is to investigate the Schouten–Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten–Weyl tensor are studied. Moreover, the three-dimensional metric Lie groups with almost harmonic (i.e. with zero curl and divergence) Schouten–Weyl tensor are investigated. In addition, the question about the harmonicity of contraction of the Schouten–Weyl tensor is considered.

Литература

1. Besse A. Einstein manifolds. – Springer-Verlag, Berlin-Heidelberg, 1987.

2. Rodionov E.D., Slavskii V.V., Chibrikova L.N. Left-invariant Lorentz metrics on three-dimensional lie groups with a Schouten–Weyl tensor of squared length zero // Doklady Mathematics. – 2005. – Vol. 71, no. 2. – P. 238–240.

3. Milnor J. Curvature of left invariant metric on Lie groups // Advances in mathematics. – 1976. – Vol. 21. – P. 293–329.

4. Gladunova O.P., Rodionov E.D., Slavskii V.V. On harmonic tensors on three-dimensional Lie groups with left-invariant Riemannian metric // Doklady Mathematics. – 2008. – Vol. 77, no. 2. – P. 306–309.

5. Yano K., Bochner S. Curvature and Betti Numbers. – Princeton, NJ : Princeton Univ. Press, 1953.

6. Pastukhova S.V., Khromova O.P. On the signature of the operator of the Ricci curvature tensor on three-dimensional Lie groups with left-invariant Lorentzian metrics [in Russian] // The news of ASU. – 2015. – no. 1/2. – P. 141–146.

7. Calvaruso G. Homogeneous structures on three-dimensional Lorentzian manifolds //
J. Geom. Phys. – 2007. – Vol. 57. – P. 1279–1291.

8. Cordero L.A., Parker P.E. Left-invariant Lorentzian metrics on 3-dimensional Lie groups // Rend. Mat. – 1997. – no. 17, Serie VII. – P. 129–155.

9. Gladunova O.P., Rodionov E.D., Slavskii V.V. Harmonic tensors on three-dimensional Lie groups with left-invariant Lorentz metric // Doklady Mathematics. – 2009. – Vol. 80, no. 2. – P. 755–758.
Опубликован
2017-12-18