Abstract
The present paper is concerned with obtaining a classification regarding to four-dimensional semi-symmetric neutral Lie groups. Moreover, we discuss some geometric properties of these spaces. We exhibit a rich class of non-Einstein Ricci soliton examples.
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T. Arias-Marco, O. Kowalski: Classification of 4-dimensional homogeneous D’Atri spaces. Czech. Math. J. 133 (2008), 203–239.
L. Bérard-Bérgery: Les espaces homogenes Riemanniens de dimension 4. Géométrie Riemannienne en Dimension 4. Séminaire Arthur Besse. Cedic, Paris, 1981, pp. 40–60. (In French.)
E. Boeckx: Einstein-like semi-symmetric spaces. Arch. Math., Brno 29 (1993), 235–240.
E. Boeckx, G. Calvaruso: When is the unit tangent sphere bundle semi-symmetric? Tohoku Math. J., II. Ser. 56 (2004), 357–366.
E. Boeckx, O. Kowalski, L. Vanhecke: Riemannian Manifolds of Conullity Two. World Scientific, Singapore, 1996.
G. Calvaruso: Three-dimensional semi-symmetric homogeneous Lorentzian manifolds. Acta Math. Hung. 121 (2008), 157–170.
G. Calvaruso: Semi-symmetric Lorentzian metrics and three-dimensional curvature homogeneity of order one. Abh. Math. Semin. Univ. Hamb. 79 (2009), 1–10.
G. Calvaruso, B. De Leo: Semi-symmetric Lorentzian three-manifolds admitting a parallel degenerate line field. Mediterr. J. Math. 7 (2010), 89–100.
G. Calvaruso, A. Fino: Ricci solitons and geometry of four-dimensional non-reductive homogeneous spaces. Can. J. Math. 64 (2012), 778–804.
G. Calvaruso, A. Fino: Four-dimensional pseudo-Riemannian homogeneous Ricci solitons. Int. J. Geom. Methods Mod. Phys. 12 (2015), Article ID 1550056, 21 pages.
G. Calvaruso, L. Vanhecke: Special ball-homogeneous spaces. Z. Anal. Anwend. 16 (1997), 789–800.
G. Calvaruso, A. Zaeim: Neutral metrics on four-dimensional Lie groups. J. Lie Theory 25 (2015), 1023–1044.
H.-D. Cao: Recent progress on Ricci solitons. Recent advances in geometric analysis (Y.-I. Lee et al., eds.). Advanced Lectures in Mathematics (ALM) 11, International Press, Somerville, 2010, pp. 1–38.
A. Haji-Badali, R. Karami: Ricci solitons on four-dimensional neutral Lie groups. J. Lie Theory 27 (2017), 943–967.
G. R. Jensen: Homogeneous Einstein spaces of dimension four. J. Differ. Geom. 3 (1969), 309–349.
R. Karami, A. Zaeim, A. Haji-Badali: Ricci solitons and geometry of four dimensional Einstein-like neutral Lie groups. Period. Math. Hung. 78 (2019), 58–78.
B. O’Neill: Semi-Riemannian Geometry: With Applications to Relativity. Pure and Applied Mathematics 103, Academic Press, New York, 1983.
S. Rahmani: Métriques de Lorentz sur les groupes de Lie unimodulaires de dimension trois. J. Geom. Phys. 9 (1992), 295–302. (In French.)
K. Sekigawa: On some 3-dimensional curvature homogeneous spaces. Tensor, New Ser. 31 (1977), 87–97.
Z. I. Szabo: Structure theorems on Riemannian spaces satsfying R(X, Y) · R = 0 I: The local version. J. Differ. Geom. 17 (1982), 531–582.
H. Takagi: An example of Riemannian manifold satisfying R(X, Y) · R but not ∇R = 0. Tohoku Math. J. 24 (1972), 105–108.
A. Zaeim, R. Karami: Geometric consequences of four dimensional neutral Lie groups. Bull. Braz. Math. Soc. (N.S.) 50 (2019), 167–186.
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Haji-Badali, A., Zaeim, A. Semi-Symmetric Four Dimensional Neutral Lie Groups. Czech Math J 70, 393–410 (2020). https://doi.org/10.21136/CMJ.2019.0342-18
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DOI: https://doi.org/10.21136/CMJ.2019.0342-18