Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the ergodicity of geodesic flows on surfaces of nonpositive curvature   Let $M$ be a smooth compact surface of nonpositive curvature, with genus
$\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent
bundle of $M$ with respect to the Liouville measure under the condition that
the set of points with negative curvature on $M$ has finitely many connected
components. Under the same condition, we prove that a non closed "flat"
geodesic doesn't exist, and moreover, there are at most finitely many flat
strips, and at most finitely many isolated closed "flat" geodesics.

34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
17
The categories of this paper are: q-fin