Smooth embedded surface

A smooth embedded surface in ${\mathbb{R}}^3$ is a two-dimensional smooth manifold locally represented by an injective regular parametrized surface $\sigma :U\subset {\mathbb{R}}^2\to {\mathbb{R}}^3$, with ${\sigma }_u\times {\sigma }_v\ne 0$. The first fundamental form measures lengths and angles, and a choice of unit normal determines the second fundamental form, which measures bending. A surface with boundary additionally uses local half-plane charts along its boundary.