Representation for a Smooth Isometric Mapping from a Connected Planar Domain to a Surface

Abstract

A representation theorem for a smooth isometric mapping of a flat, connected domain $\mathcal {D}$ in two-dimensional Euclidean point space $\mathcal {E}^{2}$ into a surface $\mathcal {S}$ in three-dimensional Euclidean point space $\mathcal {E}^{3}$ is presented. The form of the mapping is shown to be necessary and sufficient to describe any such smooth isometry. Importantly, this work is not based upon the hypothesis that the mapped surface is ruled. In general, a mapping from a flat planar domain into a ruled surface is far from being isometric, and the property of being ruled is a partial consequence of our representation theorem.