Abstract
The main purpose of this paper is to give a complete description of isometry groups on the 4-dimensional simply connected nilpotent Lie groups. We distinguish between two geometrically distinct cases of degenerate and nondegenerate center of the group. Since Walker metrics appear as the underlying structure of neutral signature metrics on the nilpotent Lie groups with degenerate center, we find necessary and sufficient condition for them to locally admit the nilpotent group of isometries.
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V. V. Balashchenko, Yu. G. Nikonorov, E. D. Rodionov, and V. V. Slavskiĭ, Homogeneous Spaces: Theory and Applications [in Russian], Poligrafist, Hanty-Mansiĭsk (2008).
V. del Barco and G. P. Ovando, “Isometric actions on pseudo-Riemannian nilmanifolds,” Ann. Global Anal. Geom., 45, No. 2, 95–110 (2014).
A. Bejancu and H. R. Farran, Foliations and Geometric Structures, Vol. 580, Springer (2006).
M. Blau and M. O’Loughlin, “Homogeneous plane waves,” Nucl. Phys. B, 654, 135–176 (2003).
N. Bokan, T. Šukilović, and S. Vukmirović, “Lorentz geometry of 4-dimensional nilpotent Lie groups,” Geom. Dedicata, 177, 83–102 (2015).
L. A. Cordero and P. E. Parker, “Left-invariant Lorentz metrics on 3-dimensional Lie groups,” Rend. Mat. Appl., 17, 129–155 (1997).
L. A. Cordero and P. E. Parker, “Isometry groups of pseudoriemannian 2-step nilpotent Lie groups,” Houston J. Math., 35, No. 1, 49–72 (2009).
P. Eberlein, “Geometry of 2-step nilpotent groups with a left invariant metric,” Ann. Sci. École Norm. Sup. (4), 27, No. 5, 611–660 (1994).
M. Guediri, “Sur la complétude des pseudo-métriques invariantes à gauche sur les groupes de Lie nilpotents,” Rend. Sem. Mat. Univ. Pol. Torino, 52, 371–376 (1994).
Sz. Homolya and O. Kowalski, “Simply connected two-step homogeneous nilmanifolds of dimension 5,” Note Mat., 26, No. 1, 69–77 (2006).
A. Kaplan, “Riemannian nilmanifolds attached to Clifford modules,” Geom. Dedicata, 11, No. 2, 127–136 (1981).
A. J. Keane and B. O. Tupper, “Killing tensors in pp-wave spacetimes,” Classical Quantum Gravity, 27, No. 24, 245011 (2010).
J. Lauret, “Homogeneous nilmanifolds of dimension 3 and 4,” Geom. Dedicata, 68, 145–155 (1997).
J. Milnor, “Curvatures of left invariant metrics on Lie groups,” Adv. Math., 21, No. 3, 293–329 (1976).
J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, “Invariants of real low dimension Lie algebras, J. Geom. Phys., 17, No. 6, 986–994 (1976).
R. Penrose, “Any space-time has a plane wave as a limit,” in: Differential Geometry and Relativity, Math. Phys. Appl. Math., Vol. 3, Springer Netherlands (1976), pp. 271–275.
A. Z. Petrov, “Classification of spaces defining gravitational fields,” Uchen. Zap. Kazan. Gos. Univ., 114, No. 8, 55–69 (1954).
V. V. Trofimov and A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras,” Russ. Math. Surv., 39, No. 2, 1–67 (1984).
V. V. Trofimov and A. T. Fomenko, “Geometric and algebraic mechanisms of the integrability of Hamiltonian systems on homogeneous spaces and Lie algebras,” in: Dynamical systems. VII, Encycl. Math. Sci., Vol. 16, Springer, Berlin (1994), pp. 261–333.
F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott Foresman, Glenview (1971).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 3, pp. 257–271, 2015.
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Šukilović, T. Isometry Groups of 4-Dimensional Nilpotent Lie Groups. J Math Sci 225, 711–721 (2017). https://doi.org/10.1007/s10958-017-3488-z
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DOI: https://doi.org/10.1007/s10958-017-3488-z