Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.
We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by $GS$), if every $2n$, $n\geq1$, points on surface can be separate by some geodesic to two distinct subsets $V_1$ and $V_2$, where $|V_1|=|V_2|=n$. Also, if a surface $S$ is not good, we say it is bad an denote it by $BS$.
For example, it is not difficult to show that plane is a $GS$. Also, a sphere is $GS$.
1) Do we have some $BS$ examples(class of examples)?
2) Can we characterize the $GS$ and $BS$ surfaces?
I can't find any $BS$ examples and also I can't prove that they are $GS$.
For example, is Klein Bottle $GS$ or $BS$?
Is there any related works and questions about this post?
Best Answer
Assuming that "geodesic" in this question means "simple closed geodesic", then every complete hyperbolic surface $S$ of finite area is "bad": You cannot even separate an arbitrary pair of points. The reason is that the union of simple closed geodesics on $S$ is nowhere dense (even more, its closure has Hausdorff dimension 1) by the result of Birman and Series, "Geodesics with bounded intersection numbers on surfaces are sparsely distributed", Topology 24 (1985). The paper is available at: http://www.math.columbia.edu/~jb/bdd-int.no-sparce.pdf
In view of this theorem, there exists an open disk $D\subset S$ which is disjoint from all simple closed geodesics in $S$. Now, take two points from this disk. I did not check it, but it is quite likely that Birman-Series result also holds in the case of negatively pinched variable curvature.
Hyperbolic surfaces are probably still "bad" if you allow non-simple closed geodesics, but pairs of points no longer suffice; one could try to use Hausdorff dimension arguments in the products of hyperbolic surface by itself to get a contradiction.