  In practical implementations of density-functional theory, the only term
where an orbital description is needed is the kinetic one. Even this term in
principle depends on the density only, but its explicit form is unknown. We
provide a novel solution of the N-representability problem for an extended
system, which implies an explicit form for the Kohn-Sham kinetic energy in
terms of the density. Our approach is based on a periodic coordinate mapping,
uniquely defined by the Fourier coefficients of the metric. The density
functional is thus expressed as an explicit functional of the metric tensor:
since N-representability is enforced, our constructive recipe provides a
variational approximation. Furthermore, we show that our geometric viewpoint is
quite naturally related to the electron localization function (ELF), which
provides a very informative analysis of the electron distribution. Studies of
ELF, as obtained from accurate Kohn-Sham orbitals in real materials, allow an
appraisal of the variational approximate density functional. We show that the
value of an approximate functional-either the present geometric-based one or
some previous ones based on different constructive recipes-strongly depends on
the nature of the chemical bonding in the material.
