  We unify three approaches within the vast body of gauge-theory research that
have independently developed distinct representations of a geometrical
surface-like structure underlying the vector-potential. The three approaches
that we unify are: those who use the compactified dimensions of Kaluza-Klein
theory, those who use Grassmannian models (also called gauge theory embedding
or $CP^{N-1}$ models) to represent gauge fields, and those who use a hidden
spatial metric to replace the gauge fields. In this paper we identify a
correspondence between the geometrical representations of the three
schools.Each school was mostly independently developed, does not compete with
other schools, and attempts to isolate the gauge-invariant geometrical
surface-like structures that are responsible for the resulting physics. By
providing a mapping between geometrical representations, we hope physicists can
now isolate representation-dependent physics from gauge-invariant physical
results and share results between each school. We provide visual examples of
the geometrical relationships between each school for $U(1)$ electric and
magnetic fields. We highlight a first new result: in all three representations
a static electric field (electric field from a fixed ring of charge or a sphere
of charge) has a hidden gauge-invariant time dependent surface that is
underlying the vector potential.
