Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Length product of homologically independent loops for tori   We prove that any Riemannian torus of dimension $m$ with unit volume admits
$m$ homologically independent closed geodesics whose length product is bounded
from above by $m^m$.

18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys