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Proceedings of the American Mathematical Society
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.


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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
PII: S 0002-9939(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