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

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.