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 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 whose first fundamental form is the given tensor. We then describe several global consequences of the condition KdA 0 on D⁰.