TITLE: Probing a manifold with geodesics
QUESTION [27 upvotes]: 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?.)

REPLY [21 votes]: There's a different kind of answer to this that you might be interested in:  Suppose that, when you fire off a probe along a unit speed geodesic starting at $p\in M$, you record the direction $\theta$ in which you sent it, and the probe reports the inertial forces it is experiencing, i.e., it sends back a running report on the curvature and torsion of the curve it is traveling along.  Thus, you get to record these two data as functions $\kappa(t,\theta)$ and $\tau(t,\theta)$ of $t$, the time since the probe was launched, and $\theta$, the direction in which it was sent.  
The question, then, is "Can you recover the metric on the surface $M$ from the data $\kappa$ and $\tau$?"   
Not surprisingly, the answer is yes in the generic situation.  If, for example, you are in the situation in which $\tau\not=0$, you find (by computing with the structure equations) that the induced metric must be of the form
$$
g = dt^2 + f(t,\theta)^2\ d\theta^2,
$$
where
$$
f(t,\theta) 
= \frac{\sqrt{|\tau(t,\theta)|}}{2\ \tau(t,\theta)}
\int_0^t\frac{\kappa_\theta(\rho,\theta)}{\sqrt{|\tau(\rho,\theta)|}}\ d\rho\ .
$$
(Obviously, there will be some singularity issues at places where $\tau$ vanishes, but, generically, this isn't a problem.  The case in which $\tau$ vanishes identically, such as when $p$ is a pole of rotational symmetry of the surface $M$, has to be treated separately.)
Since you can recover the curvature from $f$ by the formula $K = -f_{tt}/f$, you could detect when, for example, the surface $M$ is locally convex, and so forth. 
As for computing the Euler characteristic, since $K\ dA = -f_{tt}\ dt\wedge d\theta
= -d\left(\ f_t\ d\theta\ \right)$, it follows that, if you could figure out the star-shaped domain (in good cases, of the form $0\le t < T(\theta)$ for some piecewise differentiable $2\pi$-periodic function $T$ ) that maps one-to-one and onto the complement of the cut locus of $p$, then you could compute the Euler characteristic as
$$
\chi(M) = \frac{-1}{2\pi}\int_0^{2\pi} f_t\bigl(T(\theta),\theta\bigr)\ d\theta.
$$
How numerically stable all these calculations are, I don't know.  (I also don't know how hard it would be to figure out or approximate $T$.)  Since you know that the result is an integer, though, you might be able to tolerate a reasonable amount of numerical error.