Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the number of geodesic segments connecting two points on manifolds of
  non-positive curvature   In this paper we show that on a complete Riemannian manifold of negative
curvature and dimension $n>1$ every two points which realize a local maximum
for the distance function are connected by at least $2n+1$ geometrically
distinct geodesic segments (i.e. length minimizing). Using a similar method, we
obtain that in the case of non-positive curvature, for every two points with
the same property as above the number of connecting distinct geodesic segments
is at least $n+1$.

34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an