Abstract: In 1950’s, Nash-Kuiper built up the C1 isometric embedding for any surface into R3, this can be viewed as analysis side of isometric embedding. On the other hand, there is obstruction for the existence of C2 isometric embedding of surface into R3 known since Hilbert, which reflects the geometry flavor of isometric embedding. What’s happening from C1 to C2 (from analysis to geometry)? We will present our partial progress along this direction. The talk will be accessible to audience with basic knowledge of analysis and differential geometry.