This question is driven entirely by idle speculation.
Suppose we have a complete Riemannian manifold $M$, a point $p \in M$, and some lattice $L \subset T_p M$.
By Hopf-Rinow, there exists an exponential map $\exp_p : T_pM \to M$. (In general, if $M$ weren't complete, the exponential map might only be defined on an open neighborhood of the origin in $T_p M$.)
Depending on $M$, there are different possibilities for the image $X := \exp_p(L)$, such as:
- If $M = \mathbb{R}^n$, then $X$ is always a discrete subset of $M$.
- If $M = S^1$, then $X$ could either be discrete or dense in $M$ depending on $L$.
- If $M = S^1 \times \mathbb{R}$ then $X$ might be discrete, or it might be dense in a one-dimensional submanifold.
- I imagine that for "general" choices of $M$, $p$, and $L$ we have $X$ dense in $M$.
Questions:
- What are all the possibilities for the nature of $X$? (Or just for $\overline{X}$?)
- Can we (partially?) characterize the situations in which each can occur?
Edit:
Per the comments, some more specific questions:
- For which $M$ can $X$ be discrete? Dense?
- For which $M$ is $X$ necessarily discrete? Dense?
- The above with "$M$" replaced with "pairs $(M,p)$"
- Do any well-known topological or geometric properties of $M$ translate to statements about the nature of $X$?