  The integer quantum Hall effect (IQHE) is usually modeled using
Galilean-invariant or rotationally-invariant Landau levels. However, these are
not generic symmetries of electrons moving in a crystalline background, even in
the low-density continuum limit. We present a treatment of the IQHE which
abandons the Galilean dispersion relation while keeping inversion symmetry. We
define an emergent metric $g^n_{ab}$ for each Landau level with a reformulation
of the Hall viscosity. The metric is then used to define a guiding-center
coherent state and the wavefunctions are holomorphic functions of $z^*$ times a
Gaussian. By numerically studying cases with quartic dispersion, we show that
the number of the zeroes of the wavefunction encircled by the semiclassical
orbit, denoted by $n$, defines a "topological spin" $s_n$ by
$s_n=n+\frac{1}{2}$, with its original definition as an "intrinsic angular
momentum" no longer valid without rotational symmetry. At the end of the paper
we show our results for the density and current responses which differentiate
between diagonal and Landau-level-mixing terms. In conclusion, this treatment
extracts topological information without resort to Galilean or rotational
symmetry, and also reveals more generic geometric structures.
