Closures of totally geodesic immersions into locally symmetric spaces of noncompact type
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- by Tracy L. Payne PDF
- Proc. Amer. Math. Soc. 127 (1999), 829-833 Request permission
Abstract:
It is established that if $\mathcal {M}_1$ and $\mathcal {M}_2$ are connected locally symmetric spaces of noncompact type where $\mathcal {M}_2$ has finite volume, and $\phi :\mathcal {M}_1 \to \mathcal {M}_2$ is a totally geodesic immersion, then the closure of $\phi (\mathcal {M}_1)$ in $\mathcal {M}_2$ is an immersed “algebraic” submanifold. It is also shown that if in addition, the real ranks of $\mathcal {M}_1$ and $\mathcal {M}_2$ are equal, then the the closure of $\phi (\mathcal {M}_1)$ in $\mathcal {M}_2$ is a totally geodesic submanifold of $\mathcal {M}_2.$ The proof is a straightforward application of Ratner’s Theorem combined with the structure theory of symmetric spaces.References
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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
- Received by editor(s): November 4, 1996
- Received by editor(s) in revised form: June 10, 1997
- Communicated by: Christopher Croke
- © Copyright 1999 American Mathematical Society
- 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