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

Original Article

First Online:

## 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.Berger, M.: Les espaces symmétriques noncompacts. Ann. Sci. Ec. Norm. Super.
**74**, 85–177 (1957)CrossRefGoogle Scholar - 2.Candel, A., Quiroga-Barranco, R.: Gromov’s centralizer theorem. Geom. Dedicata
**100**, 123–155 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Goodman, R., Wallach, N.R.: Symmetry, Representations and Invariants. Graduate Texts in Mathematics, vol. 255. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
- 4.Müller, D.: Isometries of bi-invariant pseudo-Riemannian metrics on Lie groups. Geom. Dedicata
**29**(1), 65–96 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Nevo, A., Zimmer, R.J.: Invariant rigid geometric structures and smooth projective factors. Geom. Funct. Anal.
**19**, 520–535 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.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.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.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.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.Szaro, J.: Isotropy of semisimple group actions on manifolds with geometric structure. Am. J. Math.
**120**, 129–158 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Warner, G.: Harmonic Analysis on Semi-simple Lie Groups I. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
- 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.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.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