  The theoretical and computational description of materials properties is a
task of utmost scientific and technological importance. A first-principles
description of electron-electron interactions poses an immense challenge that
is usually approached by converting the many-electron problem to an effective
one-electron problem. There are different ways to obtain an exact one-electron
theory for a many-electron system. An emergent method is the exact electron
factorization (EEF) -- one of the branches of the Exact Factorization approach
to many-body systems. In the EEF, the Schr\"odinger equation for one electron,
in the environment of all other electrons, is formulated. The influence of the
environment is reflected in the potential $v^{\rm H}$, which represents the
energy of the environment, and in a potential $v^{\rm G}$, which has a
geometrical meaning. In this paper, we focus on $v^{\rm G}$ and study its
properties in detail. We investigate the geometric origin of $v^{\rm G}$ as a
metric measuring the change of the environment, exemplify how translation and
scaling of the state of the environment are reflected in $v^{\rm G}$, and
explain its shape for homo- and heteronuclear diatomic model systems. Based on
the close connection between the EEF and density functional theory, we also use
$v^{\rm G}$ to provide an alternative interpretation to the Pauli potential in
orbital-free density functional theory.
