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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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