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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
DOI: https://doi.org/10.1090/S0002-9939-99-04552-9
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. E. Ghys, Dynamique des flots unipotents sur les espaces homogènes, Astérisque 206 (1992), 93-136. MR 94e:58101
  • 3. M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), 229-309. MR 91m:57031
  • 4. -, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990), 449-482. MR 92h:22015
  • 5. -, On Raghunathan's measure conjecture, Ann. of Math. 134 (1991), 545-607. MR 93a:22009
  • 6. -, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235-280. MR 93f:22012
  • 7. -, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4 (1994), no. 2, 236-257. MR 95c:22018
  • 8. N. Shah, Closures of totally geodesic immersions in manifolds of constant negative curvature, Group Theory from a Geometrical Viewpoint, World Sci. Pub., 1991, pp. 718-732. MR 93d:53078
  • 9. -, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann. 289 (1991), 315-334. MR 93d:22010
  • 10. A. Zeghib, Laminations et hypersurfaces géodésiques des variétés hyperboliques, Ann. scient. Éc. Norm Sup. 24 (1991), 171-188. MR 92b:53098

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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: https://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