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

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.