We show that in the surface region, where the matter density falls off
, the local momentum distribution of a many Fermion system becomes ani
sotropic. The local Fermi surface has a predominantly prolate shape, w
ith its long axis normal to the surface in both spherical and semi-inf
inite many Fermion systems, We determine the dependence of the smooth
and of the fluctuating contributions of the momentum anisotropy as fun
ctions of particle number.