Isometric Embedding of 2-dim Riemannian Manifolds in Euclidean 3-Space

It is a classical problem to study whether any 2-dimensional Riemannian manifolds admit isometric embedding in the 3-dimensional Euclidean space. There are two versions of this problem, the local version and the global version. The local version was presented by Schlaefli in 1873 and is still open. In this talk, we review both versions of the problem and present some recent results.