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 mightreturn the blue curve left below:
(Based on an image created byMark 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 arelucky 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 metricis 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?.)
