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We reproduce and review some of the main results of three of our earlierpapers, utilizing in doing so a considerably more transparent formalism thanoriginally utilized. The most fundamental result to which we pay especialattention in this paper, is that the exact Fermi surface (FS) of the uniformmetallic ground state (GS) of any single-band Hamiltonian, describing fermions,is a subset of the FS within the framework of the exact Hartree-Fock theory. Wealso review some of the physical implications of the latter result. Ourconsiderations reveal that the interacting FS of a uniform metallic GS cannotbe calculated exactly to order \nu (\nu \ge 2) in the coupling constant \lambdaof the interaction potential in terms of the self-energy calculated to order\nu in a non-self-consistent fashion. We show this to be interlinked with thefailure of the Luttinger-Ward identity, and thus of the Luttinger theorem, fora self-energy that is not appropriately related to the single-particle Greenfunction from which the FS is deduced. We further show that the same mechanismthat embodies the Luttinger theorem within the framework of the exact theory,accounts for a non-trivial dependence of the exact self-energy on \lambda thatcannot be captured within a non-self-consistent framework. We thus establishthat the extant calculations that purportedly prove deformation of theinteracting FS of the metallic GS of the single-band Hubbard Hamiltonian withrespect to its Hartree-Fock counterpart at the second order in the on-siteinteraction energy U, are fundamentally deficient. In an appendix we show thatthe number-density distribution function, to be distinguished from thesite-occupation distribution function, corresponding to the GS of the HubbardHamiltonian is not non-interacting v-representable, a fact established earliernumerically. [Abridged Abstract]

Some rigorous results concerning the uniform metallic ground states of single-band Hamiltonians in arbitrary dimensions