Abstract
The aim of this paper is to classify cohomogeneity one isometric actions on the 4-dimensional Minkowski space \(\mathbb {R}^{3,1}\), up to orbit equivalence. Representations, up to conjugacy, of the acting groups in \(O(3,1) < imes \mathbb {R}^{3,1}\) are given in both cases, proper and non-proper actions. When the action is proper, the orbits and the orbit spaces are determined.
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References
Adams, S.: Dynamics on Lorentz Manifolds. World Sci, 2001
Alekseevsky, A.V., Alekseevsky, D.V.: \(G\)-manifolds with one dimensional orbit space. Adv. Sov. Math. 8, 1–31 (1992)
Alekseevsky, D.V.: On a proper action of a Lie group. Uspekhi Math. Nauk 34, 219–220 (1979)
Alekseevsky, A.V., Alekseevsky, D.V.: Riemannian \(G\)-manifolds with one dimensional orbit space. Ann. Global Anal. Geom. 11, 197–211 (1993)
Ahmadi, P., Kashani, S.M.B.: Cohomogeneity one de Sitter space \(S^n_1\). Acta Math. Sin. 26(10), 1915–1926 (2010)
Ahmadi, P., Kashani, S.M.B.: Cohomogeneity one Minkowski space \(\mathbb{R}^n_1\). Publ. Math. Debr. 78(1), 49–59 (2011)
Ahmadi, P.: Cohomogeneity one three dimensional anti-de Sitter space, proper and nonproper actions. Differ. Geom. Appl. 39, 93–112 (2015)
Ahmadi, P.: Cohomogeneity One Dynamics on Three Dimensional Minkowski Space. Zh. Mat. Fiz. Anal. Geom. 15(2), 155–169 (2019)
Ahmadi, P., Safari, S.: On Cohomogeneity one linear actions on Pseudo-Euclidean Space \(\mathbb{R}^{p, q}\). Differ. Geom. Appl. 68, 1–17 (2020)
Berard-Bergery, L.: Sur de nouvells vari\(\acute{e}\)t\(\acute{e}\) riemanniennes d’Einstein, Inst.\(\acute{E}\)lie Cartan 6 (1982), 1-60
Di Scala, A.J., Olmos, C.: The geometry of homogeneous submanifolds of hyperbolic space. Math. Z. 237(1), 199–219 (2001)
Gorbatsevich, V.V., Onishik, A.L., Vinberg, E.B.: Lie Groups and Lie Algebras III. Springer-Verlag, New York (1994)
Grove, K., Wilking, B., Ziller, W.: Positively curved cohomogeneity one manifolds and 3-Sasakian geometry. J. Differ. Geom. 78(1), 33–111 (2008)
Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. 152(1), 331–367 (2000)
Grove, K., Ziller, W.: Cohomogeneity one manifolds with positive Ricci curvature. Invent. Math. 149(3), 619–646 (2002)
Mirzaie, R., Kashani, S.M.B.: On cohomogeneity one flat Riemannian manifolds. Glasgow Math. J. 44, 185–190 (2002)
Mostert, P.S.: On a compact Lie group acting on a manifold. Ann. Math. 65(3), 447–455 (1957)
Palais, R.S., Terng, C.H.L.: A general theory of canonical forms. Trans. Am. Math. Soc. 300, 771–789 (1987)
Palais, R.S., Terng, CH.L.: Critical Point Theory and Submanifold Geometry, Lectture Notes in Mathematics, Springer-Verlag, 1988
Podesta, F., Spiro, A.: Some topological properties of chomogeneity one manifolds with negative curvature. Ann. Global Anal. Geom. 14, 69–79 (1996)
Searle, C.: Cohomogeneity and positive curvature in low dimension. Math. Z. 214, 491–498 (1993)
Verdiani, L.: Cohomogeneity one Riemannian manifolds of even dimension with strictly positive sectional curvature. I. Math. Z. 241(2), 329–339 (2002)
Verdiani, L.: Cohomogeneity one manifolds of even dimension with strictly positive sectional curvature. J. Differ. Geom. 68(1), 31–72 (2004)
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The authors are grateful to the referee for invaluable suggestions leading to the improvement of the paper.
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Communicated by Mohammad Koushesh.
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Ahmadi, P., Safari, S. & Hassani, M. A Classification of Cohomogeneity One Actions on the Minkowski Space \(\mathbb {R}^{3,1}\). Bull. Iran. Math. Soc. 47, 1905–1924 (2021). https://doi.org/10.1007/s41980-020-00479-2
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DOI: https://doi.org/10.1007/s41980-020-00479-2