  There are different ways to obtain an exact one-electron theory for a
many-electron system, and the exact electron factorization (EEF) is one of
them. In the EEF, the Schr\"odinger equation for one electron in the
environment of other electrons is constructed. The environment provides the
potentials that appear in this equation: A scalar potential $v^{\rm H}$
representing the energy of the environment and another scalar potential $v^{\rm
G}$ as well as a vector potential that have geometric meaning. By replacing the
interacting many-electron system with the non-interacting Kohn-Sham (KS)
system, we show how the EEF is related to density functional theory (DFT) and
we interpret the Hartree-exchange-correlation potential as well as the Pauli
potential in terms of the EEF. In particular, we show that from the EEF
viewpoint, the Pauli potential does not represent the difference between a
fermionic and a bosonic non-interacting system, but that it corresponds to
$v^{\rm G}$ and partly to $v^{\rm H}$ for the (fermionic) KS system. We then
study the meaning of $v^{\rm G}$ in detail: Its geometric origin as a metric
measuring the change of the environment is presented. Additionally, its
behavior for a simple model of a homo- and heteronucler diatomic is
investigated and interpreted with the help of a two-state model. In this way,
we provide a physical interpretation for the one-electron potentials that
appear in the EEF and in DFT.
