Periodic Hartree-Fock calculations have been performed on the bcc lattices of Li, Na, K, V, and on the fee ones of Al, Ca, Sc, Cu in order to investigate the topological properties of the electron charge density and of the electron localization function, ELF. All systems are calculated to be conductors. It is found that the existence of nonnuclear attractors of the electron charge density gradient field first evidenced in lithium clusters is not a prerequisite for metallic behavior. They are missing the V and Cu cells. The topology of ELF is characterized by di- or polysynaptic valence basins. The value of ELF at the valence basin attractors eta(r(a)) is rather low: typically less than 0.6, which is consistent with an antiparallel pairing close to that of a homogeneous electron gas. At the index I saddle points located on the separatrices between valence basins, the ELF value eta(r(s)) is very close to that at the valence attractors eta(r(a)). The isosurface eta(r) = eta(r(s)) - epsilon defines a reducible valence domain which is spread all over the crystal and which forms a tridimensional network of channels. Except for Al, the valence basins have a synaptic order larger than 2. The different topologies of the ELF gradient fields calculated for different metals can be explained by the relative sizes of their core basins. The metallic bond appears to be a partial covalent bond which is often multicentric and is characterized by a low population of the valence basins (less than 1.0 e(-)) and by synaptic orders as large as 6.