Rank of locally symmetric spaces in terms of flat immersions

Question

Let $M$ be a complete locally symmetric space of finite volume and noncompact type. The rank of $M$ is usually defined as the rank of the symmetric space $\tilde{M}$ universally covering $M$, that is, it is the maximal dimension of a flat totally geodesic embedded submanifold of $\tilde{M}$.

Does the following alternative characterization hold?$$ \text{rank}(M) = \max\left\{ n \in \mathbb{N} \,\Big|\, M \text{ has a totally geodesic flat immersed $n$-submanifold } \right\}. $$ It's clear that every flat totally geodesic embedded submanifold of $\tilde{M}$ descends to an immersion in $M$. But must every such immersion come from $\tilde{M}$?

To make sure I understand the issue, any immersed submanifold in $M$ has a cover which lifts to an immersion in $\tilde{M}$, but you want it to lift to something embedded. Is that right? — Jason DeVito - on hiatus, Jul 25 '19 at 13:30

score 3 · Accepted Answer · answered Jul 25 '19 at 19:44

Every immersion $E^n\to M$ lifts to an immersion $E^n\to \tilde{M}$ by the basic covering theory (since $E^n$ is simply-connected). Each totally geodesic isometric immersion $E^n\to \tilde{M}$ is an embedding by Cartan-Hadamard theorem (you just need the target to be a complete simply connected manifold $X$ of nonpositive curvature). C-H theorem tells you is that the exponential map $exp_p: T_pX\to X$ is a diffeomorphism (for every base-point $p$). In particular, every nonconstant geodesic map ${\mathbb R}\to X$ is a (proper) embedding. Injectivity of $E^n\to X$ is now immediate.

Rank of locally symmetric spaces in terms of flat immersions

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