Realizing a singular first fundamental form as a nonimmersed surface in Euclidean 3-space

Abstract.

We describe local properties of the first fundamental form for a C^∞ surface in Euclidean 3-space $\mathbb{E}^{3}$ which fails to immerse on a small set D⁰. We then show that (subject to the regularity assumption KdA ≠ 0, on D⁰) given such a locally defined analytic tensor on an open subset of the plane there exists an analytic mapping into $\mathbb{E}^{3}$ whose first fundamental form is the given tensor. We then describe several global consequences of the condition KdA ≠ 0 on D⁰.