One way to describe a surface is as the level surface of some function, which can be taken to be a coordinate. In this section, we will work in three Euclidean dimensions with coordinates ($x^1$,$x^2$,$x^3$), and assume that our surface is given as $x^3=\hbox{constant}$. Thus, on our surface we have $dx^3=0$. More generally, in orthogonal coordinates, we can assume that $\sigma^3=0$ on our surface.

What is the curvature? Reasoning by analogy, we consider the curvature of coordinate curves in the surface. Introduce an orthonormal basis $\{\ee_1,\ee_2,\nn=\ee_3\}$, and look at how the normal vector $\nn$ changes along such curves. We have \begin{equation} -d\nn\cdot\ee_i = \nn\cdot d\ee_i = \omega_{3i} \end{equation} so (some of the) connection 1-forms appear to be telling us something about curvature!

Recall now the structure equations, one of which says that \begin{equation} d\sigma^3 + \omega^3{}_i\wedge\sigma^i = 0 \end{equation} But on the surface we can assume that $\sigma^3=0$ everywhere, so that $d\sigma^3=0$, and we have \begin{equation} \omega^3{}_1\wedge\sigma^1 + \omega^3{}_2\wedge\sigma^2 = 0 \end{equation} If we write \begin{equation} \omega^3{}_i = \Gamma^3{}_{ij}\,\sigma^j \end{equation} then the structure equation forces $\Gamma^3{}_{12}=\Gamma^3{}_{21}$ on the surface. The $2\times2$ matrix \begin{equation} S = -\begin{pmatrix}\Gamma^3{}_{ij}\end{pmatrix} \end{equation} (evaluated on the surface, and with $i,j=1,2$) is therefore symmetric, and is called the shape operator.

The shape operator $S$ is, of course, dependent on the basis used to compute it. However, its trace and determinant are not. The eigenvalues of $S$, usually denoted $\kappa_1$ and $\kappa_2$, are called the principal curvatures of the surface, and give the maximum and minimum curvatures for any curve through the given point. Since $S$ is symmetric, these eigenvalues correspond to orthogonal eigenvectors, and the directions of maximum and minimum curvature are perpendicular. The trace $\kappa_1+\kappa_2$ of $S$ is called the mean curvature, and the determinant $K=\kappa_1\kappa_2$ of $S$ is called the Gaussian curvature. As we will see, only the Gaussian curvature is an intrinsic property of the surface.