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Mat. Tr., 2008, Volume 11, Number 2, Pages 115–147 (Mi mt127)  

This article is cited in 19 scientific papers (total in 19 papers)

The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The unimodular case

A. G. Kremlyov, Yu. G. Nikonorov

Rubtsovsk Industrial Intitute, Branch of Altai State Technical University

Abstract: This paper gives a complete classification of the possible values of the signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional unimodular Lie groups.

Key words: homogeneous Riemannian manifold, Lie algebra and group, left-invariant Riemannian metric, Ricci curvature.

Full text: PDF file (305 kB)
References: PDF file   HTML file

English version:
Siberian Advances in Mathematics, 2009, 19:4, 245–267

Bibliographic databases:

UDC: 514.765
Received: 14.05.2008

Citation: A. G. Kremlyov, Yu. G. Nikonorov, “The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The unimodular case”, Mat. Tr., 11:2 (2008), 115–147; Siberian Adv. Math., 19:4 (2009), 245–267

Citation in format AMSBIB
\Bibitem{KreNik08}
\by A.~G.~Kremlyov, Yu.~G.~Nikonorov
\paper The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The unimodular case
\jour Mat. Tr.
\yr 2008
\vol 11
\issue 2
\pages 115--147
\mathnet{http://mi.mathnet.ru/mt127}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2500127}
\elib{http://elibrary.ru/item.asp?id=15307346}
\transl
\jour Siberian Adv. Math.
\yr 2009
\vol 19
\issue 4
\pages 245--267
\crossref{https://doi.org/10.3103/S1055134409040038}


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    This publication is cited in the following articles:
    1. A. G. Kremlyov, Yu. G. Nikonorov, “The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The nonunimodular case”, Siberian Adv. Math., 20:1 (2010), 1–57  mathnet  crossref  mathscinet  elib
    2. M. S. Chebarykov, “On the Ricci curvature of nonunimodular solvable metric Lie algebras of small dimension”, Siberian Adv. Math., 21:2 (2011), 81–99  mathnet  crossref  mathscinet  elib
    3. Gladunova O.P., Slavskii V.V., “Left-invariant Riemannian metrics on four-dimensional unimodular Lie groups with zero-divergence Weyl tensor”, Dokl. Math., 81:2 (2010), 298–300  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. Gladunova O.P., Rodionov E.D., Slavskii V.V., “Ob operatore krivizny na chetyrekhmernykh gruppakh Li s levoinvariantnoi rimanovoi metrikoi”, Izv. Altaiskogo gos. un-ta, 2010, no. 1-2, 29–33  elib
    5. O. P. Gladunova, V. V. Slavskii, “O garmonichnosti tenzora Veilya levoinvariantnykh rimanovykh metrik na chetyrekhmernykh unimodulyarnykh gruppakh Li”, Matem. tr., 14:1 (2011), 50–69  mathnet  mathscinet  elib
    6. O. P. Gladunova, E. D. Rodionov, V. V. Slavskii, “O konformno poluploskikh 4-mernykh gruppakh Li”, Vladikavk. matem. zhurn., 13:3 (2011), 3–16  mathnet
    7. Gladunova O.P., Rodionov E.D., Slavskii V.V., “Invariantnye tenzornye polya na gruppakh li maloi razmernosti”, Vestnik Kemerovskogo gosudarstvennogo universiteta, 2011, no. 3-1, 119–133  mathscinet  elib
    8. D. S. Voronov, O. P. Gladunova, E. D. Rodionov, V. V. Slavskii, “Ob invariantnykh tenzornykh polyakh na gruppakh Li malykh razmernostei”, Vladikavk. matem. zhurn., 14:2 (2012), 3–30  mathnet
    9. Yu. G. Nikonorov, M. S. Chebarykov, “The Ricci operator of completely solvable metric Lie algebras”, Siberian Adv. Math., 24:1 (2014), 18–25  mathnet  crossref  mathscinet  elib
    10. Gladunova O.P., Rodionov E.D., Slavskii V.V., “On Half Conformally Flat Four-Dimensional Lie Algebras”, Dokl. Math., 85:1 (2012), 48–50  crossref  mathscinet  zmath  zmath  isi  elib  elib  scopus
    11. N. A. Abiev, “On the Ricci curvature of solvable metric lie algebras with two-step nilpotent derived algebras”, Siberian Adv. Math., 24:1 (2014), 1–11  mathnet  crossref  mathscinet
    12. Cairns G., Galic A.H., Nikolayevsky Yu., “Totally Geodesic Subalgebras of Nilpotent Lie Algebras”, J. Lie Theory, 23:4 (2013), 1023–1049  mathscinet  zmath  isi
    13. M. S. Chebarykov, “O krivizne Richchi trekhmernykh metricheskikh algebr Li”, Vladikavk. matem. zhurn., 16:1 (2014), 57–67  mathnet
    14. Nikonorov Yu.G., “Negative Eigenvalues of the Ricci Operator of Solvable Metric Lie Algebras”, Geod. Dedic., 170:1 (2014), 119–133  crossref  mathscinet  zmath  isi  scopus
    15. Kubo A., Onda K., Taketomi Yu., Tamaru H., “on the Moduli Spaces of Left-Invariant Pseudo-Riemannian Metrics on Lie Groups”, Hiroshima Math. J., 46:3 (2016), 357–374  mathscinet  zmath  isi
    16. Ngaha M.B.D., Boucetta M., Kamga J.W., “the Signature of the Ricci Curvature of Left-Invariant Riemannian Metrics on Nilpotent Lie Groups”, Differ. Geom. Appl., 47 (2016), 26–42  crossref  mathscinet  zmath  isi  scopus
    17. Hashinaga T., Tamaru H., Terada K., “Milnor-Type Theorems For Left-Invariant Riemannian Metrics on Lie Groups”, J. Math. Soc. Jpn., 68:2 (2016), 669–684  crossref  mathscinet  zmath  isi  scopus
    18. Tamaru H., “the Space of Left-Invariant Riemannian Metrics”, Geometry and Topology of Manifolds, Springer Proceedings in Mathematics & Statistics, 154, eds. Futaki A., Miyaoka R., Tang Z., Zhang W., Springer Japan, 2016, 315–326  crossref  mathscinet  zmath  isi  scopus
    19. Cairns G., Galic A.H., Nikolayevsky Yu., “Curvature Properties of Metric Nilpotent Lie Algebras Which Are Independent of Metric”, Ann. Glob. Anal. Geom., 51:3 (2017), 305–325  crossref  mathscinet  zmath  isi  scopus
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