  On increasing the coupling strength ($\lambda$) of a non-Abelian gauge field
that induces a generalized Rashba spin-orbit interaction, the topology of the
Fermi surface of a homogeneous gas of noninteracting fermions of density $\rho
\sim \kf^3$ undergoes a change at a critical value, $\lambda_T \approx \kf$
[Phys. Rev. B {\bf 84}, 014512 (2011)]. In this paper we analyze how this
phenomenon affects the size and shape of a cloud of spin-$\half$ fermions
trapped in a harmonic potential such as those used in cold atom experiments. We
develop an adiabatic formulation, including the concomitant Pancharatnam-Berry
phase effects, for the one particle states in the presence of a trapping
potential and the gauge field, obtaining approximate analytical formulae for
the energy levels for some high symmetry gauge field configurations of
interest. An analysis based on the local density approximation reveals that,
for a given number of particles, the cloud shrinks in a {\em characteristic
fashion with increasing $\lambda$}. For an isotropic harmonic trap, the local
density approximation predicts a spherical cloud for all gauge field
configurations, which are anisotropic in general. We show, via a calculation of
the cloud shape using exact eigenstates, that for certain gauge field
configurations there is systematic and observable anisotropy in the cloud shape
that increases with increasing gauge coupling $\lambda$. These results should
be useful in the design of cold atom experiments with fermions in non-Abelian
gauge fields. An important spin-off of our adiabatic formulation is that it
reveals exciting possibilities for the cold-atom realization of interesting
condensed matter Hamiltonians (eg. quantum hall spherical geometry) by using a
non-Abelian gauge field in conjunction with another potential.
