The shape of the Fermi surface

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