On the Ricci curvature of three-dimensional metric lie algebras

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 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. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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