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

摘要

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 D 0. We then show that (subject to the regularity assumption KdA ≠ 0, on D 0) 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 D 0.