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Closures of Totally Geodesic Immersions into Locally Symmetric Spaces of Noncompact Type

Tracy L. Payne
Proceedings of the American Mathematical Society
Vol. 127, No. 3 (Mar., 1999), pp. 829-833
Stable URL: http://www.jstor.org/stable/119017
Page Count: 5
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Closures of Totally Geodesic Immersions into Locally Symmetric Spaces of Noncompact Type
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Abstract

It is established that if M1 and M2 are connected locally symmetric spaces of noncompact type where M2 has finite volume, and φ : M1→ M2 is a totally geodesic immersion, then the closure of φ ( M1) in M2 is an immersed "algebraic" submanifold. It is also shown that if in addition, the real ranks of M1 and M2 are equal, then the the closure of φ ( M1) in M2 is a totally geodesic submanifold of M2. The proof is a straightforward application of Ratner's Theorem combined with the structure theory of symmetric spaces.

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