Closures of totally geodesic immersions into locally symmetric spaces of noncompact type

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.