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Classification of \(\big (\widetilde{\mathrm {Sp}}(n,\mathbb {R})\times \widetilde{\mathrm {Sp}}(1,\mathbb {R})\big )\)-manifolds

  • Gestur Ólafsson
  • Eli Roblero-Méndez
Original Article
  • 15 Downloads

Abstract

Let M be an analytic complete finite volume pseudo-Riemannian manifold. We characterize the structure of the manifold M assuming that the Lie group \(\widetilde{\mathrm {Sp}}(n,\mathbb {R})\times \widetilde{\mathrm {Sp}}(1,\mathbb {R})\) acts isometrically with a dense orbit on M, where the \(\widetilde{\mathrm {Sp}}(1,\mathbb {R})\)-orbits are non-degenerated and its dimension satisfies \(\dim (M)\le (n+1)(2n+3)\).

Keywords

Semisimple Lie groups Rigidity results Pseudo-Riemannian manifolds Isometric actions 

Mathematics Subject Classification

57S20 53C24 53C50 

References

  1. 1.
    Berger, M.: Les espaces symmétriques noncompacts. Ann. Sci. Ec. Norm. Super. 74, 85–177 (1957)CrossRefGoogle Scholar
  2. 2.
    Candel, A., Quiroga-Barranco, R.: Gromov’s centralizer theorem. Geom. Dedicata 100, 123–155 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Goodman, R., Wallach, N.R.: Symmetry, Representations and Invariants. Graduate Texts in Mathematics, vol. 255. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Müller, D.: Isometries of bi-invariant pseudo-Riemannian metrics on Lie groups. Geom. Dedicata 29(1), 65–96 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Nevo, A., Zimmer, R.J.: Invariant rigid geometric structures and smooth projective factors. Geom. Funct. Anal. 19, 520–535 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Pure and Applied Mathematics, vol. 103. Academic Press, Inc., New York (1983)zbMATHGoogle Scholar
  7. 7.
    Onishchik, A.L.: Lectures on Real Semisimple Lie Algebras and their Representations. ESI Lectures in Mathematics and Physics, European Mathematical Society. European Mathematical Society (EMS), Zürich (2004)CrossRefGoogle Scholar
  8. 8.
    Ólafsson, G., Quiroga-Barranco, R.: On low dimensional manifolds with isometric \(SO_{0}(p,q)\)-actions. Transform. Groups 17(3), 835–860 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ólafsson, G., Quiroga-Barranco, R.: On Low Dimensional Manifolds with Isometric \(\widetilde{U}(p, q)\)-Actions. Asian. J. Math. 21(5), 873–908 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Quiroga-Barranco, R.: Isometric actions of simple Lie groups and transverse structures: the integrable normal case. In: Farb, B., Fisher, D. (eds.) Geometry, Rigidity and Group Actions. Chicago Lectures in Mathematics. pp. 229–261. University of Chicago Press, Chicago (2011)Google Scholar
  11. 11.
    Quiroga-Barranco, R.: Pseudo-Riemannian \(G_{2(2)}\)-manifolds with dimension at most \(21\). Math Nachr 291(8–9), 1390–1399 (2018)CrossRefGoogle Scholar
  12. 12.
    Szaro, J.: Isotropy of semisimple group actions on manifolds with geometric structure. Am. J. Math. 120, 129–158 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Warner, G.: Harmonic Analysis on Semi-simple Lie Groups I. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  14. 14.
    Morris, D.W., Zimmer, R.J.: Ergodic theory, groups and geometry. In: CBMS regional conference series in mathematics, vol. 109, University of Minnesota (1998)Google Scholar
  15. 15.
    Zimmer, R.J.: Actions of semisimple groups and discrete subgroups. In: Proceedings of the international congress of mathematicians, Berkeley, California, USA (1986)Google Scholar
  16. 16.
    Zimmer, R.J.: Entropy and arithmetic quotients for simple automorphism groups of geometric manifolds. Geom. Dedicata 107, 47–56 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2018

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.CONACYT-Facultad de Ciencias en Física y MatemáticasUniversidad Autónoma de ChiapasTuxtla GutiérrezMexico

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