FAST electric field data and ion drift moments are combined to allow full DC E$perp$ (electric field perpendicular to the geomagnetic field) studies of auroral return current regions. Statistical comparison of 71 return current potential structures showed several differences between sheetlike structured perpendicular E$perp$ field events, where the ratio of the two E$perp$ components remains constant during the spacecraft crossing, and curved structures where the ratio varies. Sheetlike structures can be interpreted as straight arcs, but curved structures require gradients in another dimension. We define a parameter η, which is a proxy for the ratio of the potential at the spacecraft and the upgoing electron characteristic energy. Thus η is a measure of the extent to which the potential contours are closed below the spacecraft. Statistical comparison shows that U-shaped closed-potential models are mostly consistent with curved events and ionospheric effects are dominant in sheetlike structures. This result implies that the spatial structure of the events, as indicated by the ratio of the E$perp$ components, allows us to distinguish ionospheric fields and U-shaped potentials. Statistical studies of scale sizes, magnitudes of electric fields and magnetic perturbations, and downward current density, sorted by the parameter η, reveal various interesting features. We attempt to explain these properties on the basis of different potential closure models for sheetlike and curved structures, which have important implications for models of the formation and evolution of potential structures for downward current regions.