  It was recently pointed out that topological liquid phases arising in the
fractional quantum Hall effect (FQHE) are not required to be rotationally
invariant, as most variational wavefunctions proposed to date have been.
Instead, they possess a geometric degree of freedom corresponding to a shear
deformation that acts like an intrinsic metric. We apply this idea to a system
with an anisotropic band mass, as is intrinsically the case in many-valley
semiconductors such as AlAs and Si, or in isotropic systems like GaAs in the
presence of a tilted magnetic field, which breaks the rotational invariance. We
perform exact diagonalization calculations with periodic boundary conditions
(torus geometry) for various filling fractions in the lowest, first and second
Landau levels. In the lowest Landau level, we demonstrate that FQHE states
generally survive the breakdown of rotational invariance by moderate values of
the band mass anisotropy. At 1/3 filling, we generate a variational family of
Laughlin wavefunctions parametrized by the metric degree of freedom. We show
that the intrinsic metric of the Laughlin state adjusts as the band mass
anisotropy or the dielectric tensor are varied, while the phase remains robust.
In the n=1 Landau level, mass anisotropy drives transitions between
incompressible liquids and compressible states with charge density wave
ordering. In n>=2 Landau levels, mass anisotropy selects and enhances stripe
ordering with compatible wave vectors at partial 1/3 and 1/2 fillings.
