Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$
embedded in $\mathbb{R}^3$.
You do not know anything about $M$.
You shoot off a geodesic $\gamma$ in some direction $u$,
and learn back the shape of the full curve $\gamma$ as it sits in $\mathbb{R}^3$.
(One could imagine a vehicle traveling along $\gamma$, sending back $xyz$-coordinates at regular time intervals; assume
$t \rightarrow \infty$.)
For example, if the geodesic happens to be closed, your probe might
return the blue curve left below:

_{(Based on an image created by
Mark Irons.)
}

I would like to know what information one could learn about $M$
from such *geodesic probes*.
I am interested in the best case rather than the worst case.
For example, you might learn that $M$ is unbounded, if you are
lucky enough to shoot a geodesic to infinity.
In particular,

Are there circumstances (a manifold $M$, a point $p$, directions $u$) that permit one to definitively conclude that the genus of $M$ is nonzero, by shooting (perhaps many) geodesics from one fixed (well-chosen) point $p$?

I believe that, if one knew all the geodesics through every point,
then there are natural circumstances under which the metric
is determined
[e.g., "Metric with Ergodic Geodesic Flow is Completely Determined by Unparameterized Geodesics."
Vladimir Matveev and Petar Topalov.
*Electronic Research Announcements of the AMS*.
Volume 6, Pages 98-101, 2000].
But I am more interested what can be determined from a single point $p$ (and many directions $u$).
Thanks for thoughts/pointers!

(Tangentially related MO question: Shortest-path Distances Determining the Metric?.)