  Skyrmions are topologically stable and energetically balanced spin
configurations appearing under the presence of ferromagnetic interaction (FMI)
and Dzyaloshinskii-Moriya interaction (DMI). Much of the current interest has
focused on the effects of magneto-elastic coupling (MEC) on these interactions
under mechanical stimuli, such as uniaxial stresses for future applications in
spintronics devices. Recent studies suggest that skyrmion shape deformations in
thin films are attributed to an anisotropy in the coefficient of DMI, such that
$D_{x}\!\not=\!D_{y}$. This anisotropy is naturally understood as an effect of
MEC, however, the relationship between MEC and anisotropy in DMI remains to be
clarified. In this paper, we study this problem using a new modeling technique
constructed based on Finsler geometry (FG). In the FG model, an MEC is
implemented purely geometrically, and the implemented MEC dynamically deforms
the coefficients of FMI and DMI to be direction-dependent. This modeling
technique is in sharp contrast to the standard model of MEC, in which
anisotropic constants are explicitly assumed as an input parameter. Two
possible FG models are examined: In the first (second) model, the FG modeling
prescription is applied to the FMI (DMI) Hamiltonian. We find that these two
different FG models' results are consistent with the reported experimental data
for skyrmion deformation. We also study responses of spins under lattice
deformations corresponding to uniaxial extension/compression and find a clear
difference between these two models in the stripe phase, elucidating which
interaction of FMI and DMI is deformed to be anisotropic by uniaxial stresses.
