  Although rare, spontaneous breakdown of inversion symmetry sometimes occurs
in a material which is metallic: these are commonly known as polar metals or
ferroelectric metals. Their 'polarization', however, cannot be switched via an
electric field, which limits the experimental control over band topology. Here
we shall investigate, via first-principles theory, flexoelectricity as a
possible way around this obstacle with the well known polar metal LiOsO$_3$.
The flexocoupling coefficients are computed for this metal with high accuracy
with a completely new approach based on real-space sums of the inter-atomic
force constants. A Landau-Ginzburg-Devonshire-type first-principles Hamiltonian
is built and a critical bending radius to switch the material is estimated,
whose order of magnitude is comparable to that of BaTiO$_3$.
