  Many-body dipolar effects in Fermi gases are quite subtle as they
energetically compete with the large kinetic energy at and below the Fermi
surface (FS). Recently it was experimentally observed that the FS is deformed
from a sphere to an ellipsoid due to the presence of the anisotropic and
long-range dipole-dipole interaction. Moreover, it was suggested that, when the
dipoles are rotated by means of an external field, the FS follows their
rotation, thereby keeping the major axis of the momentum-space ellipsoid
parallel to the dipoles. Here we generalise a previous Hartree-Fock mean-field
theory to systems confined in an elongated triaxial trap with an arbitrary
orientation of the dipoles relative to the trap. With this we study for the
first time the effects of the dipoles' arbitrary orientation on the
ground-state properties of the system. Furthermore, taking into account the
geometry of the system, we show how the ellipsoidal FS deformation can be
reconstructed, assuming ballistic expansion, from the experimentally measurable
real-space aspect ratio after a free expansion. We compare our theoretical
results with new experimental data measured with an erbium Fermi gas for
various trap parameters and dipole orientations. The observed remarkable
agreement demonstrates the ability of our model to capture the full angular
dependence of the FS deformation. Moreover, for systems with even higher dipole
moment, our theory predicts an additional unexpected effect: the FS does not
simply follow rigidly the orientation of the dipoles, but softens showing a
change in the aspect ratio depending on the dipoles' orientation relative to
the trap geometry, as well as on the trap anisotropy itself. Our theory
provides the basis for understanding and interpreting phenomena in which the
investigated physics depends on the underlying structure of the FS, such as
fermionic pairing and superfluidity.
