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Isometric actions on pseudo-Riemannian nilmanifolds. (English) Zbl 1295.53081

Authors’ abstract: This work deals with the structure of the isometry group of pseudo-Riemannian \(2\)-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannian situation; for instance, the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with a left-invariant pseudo-Riemannian metric, we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-\(H\)-type Lie groups.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
22E25 Nilpotent and solvable Lie groups
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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[1] Chen, Z., Wolf, J.: Pseudo-Riemannian weakly symmetric manifolds. Ann. Glob. Anal. Geom. 41, 381-390 (2012) · Zbl 1237.53071 · doi:10.1007/s10455-011-9291-z
[2] Ciatti, P.: Scalar products on Clifford modules and pseudo-\[HH\]-type Lie algebras. Ann. Mat. Pura Appl. IV. Ser. 178, 1-31 (2000) · Zbl 1027.17008
[3] Cordero, L., Parker, P.: Isometry groups of pseudoriemannian 2-step nilpotent Lie groups. Houston J. Math. 35(1), 49-72 (2009) · Zbl 1170.53048
[4] Cordero, L., Parker, P.: Lattices and periodic geodesics in pseudoriemannian 2-step nilpotent Lie groups. Int. J. Geom. Methods Phys. 5(1), 79-99 (2008) · Zbl 1162.53031 · doi:10.1142/S0219887808002667
[5] del Barco, V., Ovando, G.P.: Free nilpotent Lie algebras admitting ad-invariant metrics. J. Algebra 366, 205-216 (2012) · Zbl 1351.17013 · doi:10.1016/j.jalgebra.2012.05.016
[6] del Barco, V., Ovando, G.P., Vittone, F.: Naturally reductive pseudo-Riemannian Lie groups in low dimensions. To appear in Mediterr. J. Math. · Zbl 1318.53075
[7] Duncan, D., Ihrig, E.C.: Flat pseudo-Riemannian manifolds with a nilpotent transitive group of isometries. Ann. Global Anal. Geom. 10(1), 87-101 (1992) · Zbl 0810.53038 · doi:10.1007/BF00128341
[8] Dusek, Z.: Survey on homogeneous geodesics. Note Mat. 1(suppl. no. 1), 147-168 (2008) · Zbl 1198.53052
[9] Eberlein, P.: Geometry of \[22\]-step nilpotent groups with a left invariant metric. Ann. Sci. École Norm. Sup. (4) 27(5), 611-660 (1994) · Zbl 0820.53047
[10] Figueroa-O’Farill, J.: Lorentzian symmetric spaces in supergravity. In: Alekseevsky, Dmitri V. (ed.) et al., Recent developments in pseudo-Riemannian geometry. ESI Lectures in, Mathematics and Physics, pp. 419-494 (2008) · Zbl 1151.53338
[11] Gordon, C.: Transitive Riemannian isometry groups with nilpotent radicals. Ann. Inst. Fourier 31(2), 193-204 (1981) · Zbl 0441.53034 · doi:10.5802/aif.835
[12] Jang, C., Parker, P., Park, K.: Pseudo \[HH\]-type 2-step nilpotent Lie groups. Houston J. Math. 31(3), 765-786 (2005) · Zbl 1083.53066
[13] Kaplan, A.: Riemannian nilmanifolds attached to Clifford modules. Geom. Dedicata 11(2), 127-136 (1981) · Zbl 0495.53046 · doi:10.1007/BF00147615
[14] Jensen, G.: The scalar curvature of left-invariant Riemannian metrics. Indiana U. Math. J. 20, 1125-1144 (1971) · Zbl 0206.31801 · doi:10.1512/iumj.1971.20.20104
[15] Müller, D.: Isometries of bi-invariant pseudo-Riemannian metrics on Lie groups. Geom. Dedicata 29(1), 65-96 (1989) · Zbl 0681.53027 · doi:10.1007/BF00147471
[16] Nappi, C.R., Witten, E.: Wess-Zumino-Witten model based on a nonsemisimple group. Phys. Rev. Lett. 71(23), 3751-3753 (1993) · Zbl 0972.81635 · doi:10.1103/PhysRevLett.71.3751
[17] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press, London (1983) · Zbl 0531.53051
[18] Ovando, G.: Naturally reductive pseudo Riemannian 2-step nilpotent Lie groups. Houston J. Math. 39(1), 147-168 (2013) · Zbl 1277.53071
[19] Streater, R.F.: The representations of the oscillator group. Comm. Math. Phys. 4(3), 217-236 (1967) · Zbl 0155.32503 · doi:10.1007/BF01645431
[20] Wilson, E.: Isometry groups on homogeneous nilmanifolds. Geom. Dedicata 12(2), 337-345 (1982) · Zbl 0489.53045
[21] Wolf, J.: On Locally Symmetric Spaces of Non-negative Curvature and certain other Locally Homogeneous Spaces. Comment. Math. Helv. 37, 266-295 (1962-1963) · Zbl 0113.37101
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