Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Closures of totally geodesic immersions
into locally symmetric spaces
of noncompact type


Author: Tracy L. Payne
Journal: Proc. Amer. Math. Soc. 127 (1999), 829-833
MSC (1991): Primary 53C42
MathSciNet review: 1468202
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Abstract: It is established that if ${\ensuremath{\mathcal M}} _1$ and ${\ensuremath{\mathcal M}} _2$ are connected locally symmetric spaces of noncompact type where ${\ensuremath{\mathcal M}} _2$ has finite volume, and $\phi:{\ensuremath{\mathcal M}} _1 \to {\ensuremath{\mathcal M}} _2$ is a totally geodesic immersion, then the closure of $\phi({\ensuremath{\mathcal M}} _1)$ in ${\ensuremath{\mathcal M}} _2$ is an immersed ``algebraic'' submanifold. It is also shown that if in addition, the real ranks of ${\ensuremath{\mathcal M}} _1$ and ${\ensuremath{\mathcal M}} _2$ are equal, then the the closure of $\phi({\ensuremath{\mathcal M}} _1)$ in ${\ensuremath{\mathcal M}} _2$ is a totally geodesic submanifold of ${\ensuremath{\mathcal M}} _2.$ The proof is a straightforward application of Ratner's Theorem combined with the structure theory of symmetric spaces.


References [Enhancements On Off] (What's this?)

  • 1. P. Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. CMP 97:10.
  • 2. Étienne Ghys, Dynamique des flots unipotents sur les espaces homogènes, Astérisque 206 (1992), Exp. No. 747, 3, 93–136 (French, with French summary). Séminaire Bourbaki, Vol. 1991/92. MR 1206065
  • 3. Marina Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), no. 3-4, 229–309. MR 1075042, 10.1007/BF02391906
  • 4. Marina Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990), no. 2, 449–482. MR 1062971, 10.1007/BF01231511
  • 5. Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607. MR 1135878, 10.2307/2944357
  • 6. Marina Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280. MR 1106945, 10.1215/S0012-7094-91-06311-8
  • 7. M. Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4 (1994), no. 2, 236–257. MR 1262705, 10.1007/BF01895839
  • 8. Nimish A. Shah, Closures of totally geodesic immersions in manifolds of constant negative curvature, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 718–732. MR 1170382
  • 9. Nimish A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann. 289 (1991), no. 2, 315–334. MR 1092178, 10.1007/BF01446574
  • 10. A. Zeghib, Laminations et hypersurfaces géodésiques des variétés hyperboliques, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 2, 171–188 (French). MR 1097690

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Additional Information

Tracy L. Payne
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130-4899
Address at time of publication: École Normale Supérieure de Lyon, 46, Allée d’Italie, 69364 Lyon Cedex 07, France
Email: tpayne@math.wustl.edu, tpayne@umpa.ens-lyon.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04552-9
Keywords: Totally geodesic, symmetric space, Ratner's Theorem
Received by editor(s): November 4, 1996
Received by editor(s) in revised form: June 10, 1997
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society