  The observations on galaxy rotation curves show significant discrepancies
from the Newtonian theory. This issue could be explained by the effect of the
anisotropy of the spacetime. Conversely, the spacetime anisotropy could also be
constrained by the galaxy rotation curves. Finsler geometry is a kind of
intrinsically anisotropic geometry. In this paper, we study the effect of the
spacetime anisotropy at the galactic scales in the Finsler spacetime. It is
found that the Finslerian model has close relations with the Milgrom's MOND. By
performing the best-fit procedure to the galaxy rotation curves, we find that
the anisotropic effects of the spacetime become significant when the Newtonian
acceleration \(GM/r^2\) is smaller than the critical acceleration \(a_0\).
Interestingly, the critical acceleration \(a_0\), although varies between
different galaxies, is in the order of magnitude \(cH_0/2\pi\sim 10^{-10}
\rm{m\,\, s^{-2}}\).
