On the Ricci curvature of three-dimensional metric lie algebras

		ISSN 1683-3414 (Print) • ISSN 1814-0807 (Online)

Home Editorial board Publication ethics Peer review guidelines Current Archive Rules for authors Contacts Address: Markusa st. 22, Vladikavkaz, 362027, RNO-A, Russia Phone: (8672)50-18-06 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2014.1.7424 On the Ricci curvature of three-dimensional metric lie algebras Chebarikov M. S. Vladikavkaz Mathematical Journal 2014. Vol. 16. Issue 1. Abstract: In this paper, we get a classification of possible values of the Ricci curvature signature of left invariant Riemannian metrics on three-dimensional Lie groups, that is a specification of some results of J. Milnor. As an auxiliary problem, we classify three-dimensional nonunimodular metric Lie algebras. Keywords: homogeneous Riemannian manifold, Lie group, metric Lie algebra, three-dimensional Lie group, Ricci operator, eigenvalues of the Ricci operator, Ricci curvature Language: Russian Download the full text For citation: Chebarikov M. S. On the Ricci curvature of three-dimensional metric lie algebras. Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 16, no. 1, pp.57-67. DOI 10.23671/VNC.2014.1.7424 + References 1. Alekseevskij D. V. Homogeneous Riemannian spaces of negative curvature. Mathematics of the USSR-Sbornik, 1975, vol. 25, no. 1, pp. 87109. 2. Alekseevskij D. V., Kimel'fel'd B. N. Structure of homogeneous Riemann spaces with zero Ricci curvature. Functional Analysis and Its Applications, 1975, vol. 9, no. 2, pp. 97-102. 3. Balashhenko V. V., Nikonorov Ju. G., Rodionov E. D., Slavskij V. V. Odnorodnye Prostranstva: Teorija i Prilozhenija, Hanty-Mansijsk, Poligrafist, 2008, 280 p. (Russian). 4. Besse A. L. Mnogoobrazija Jejnshtejna [Einstein Manifolds], Moskva, Mir, 1990, 704 p. (Russian). 5. Gladunova O. P., Rodionov E. D., Slavskij V. V. On half conformally flat 4-dimensional Lie groups. Vladikavk. Mat. Zhurn. [Vladikavk. Math. J.], 2011, vol. 13, no. 3, pp. 3-16 (Russian). 6. Kremlev A. G., Nikonorov Ju. G. The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional lie groups. The unimodular case. Siberian Advances in Mathematics, 2009, vol. 19, no. 4, pp. 245-267. 7. Kremlev A. G., Nikonorov Ju. G. The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The nonunimodular case. Siberian Advances in Mathematics, 2010, vol. 20, no. 1, pp. 1-57. 8. Nikitenko E. V., Nikonorov Ju. G. Six-dimensional Einstein solvmanifolds. Siberian Advances in Mathematics, 2006, vol. 16, no. 1, pp. 66-112. 9. Nikonorov Ju. G., Rodionov E. D., Slavskij V. V. Geometrija odnorodnyh rimanovyh mnogoobrazij. J. Math. Sci., 2007, vol. 146, no. 6, pp. 6313-6390. 10. Chebarykov M. S. On the Ricci curvature of nonunimodular solvable metric Lie algebras of small dimension. Siberian Advances in Mathematics, 2011, vol. 21, no. 2, pp. 81-99. 11. Dotti Miatello I. Ricci curvature of left-invariant metrics on solvable unimodular Lie groups. Math. Zeit., 1982, vol. 180, pp. 257-263. 12. Jensen G. The scalar curvature of left invariant Riemannian metrics. Indiana Univ. Math. J., 1971, vol. 20, pp. 1125-1143. 13. Milnor J. Curvature of left invariant metrics on Lie groups. Adv. Math., 1976, vol. 21, pp. 293-329. ← Contents of issue


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