We unify three approaches within the vast body of gauge-theory research that have independently developed distinct representations of a geometrical surfacelike 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.