For now I would be content with understanding why the eigenvalues of the shape operator of a surface are the principle curvatures, let's call them $k_1,k_2$.
Let $f: M \rightarrow S^2$ be the Gauss map of an oriented surface $M$ into the sphere. This map simply sends the unit normal vector at any point of our surface to it's point on the sphere, I like to think of this map sort of like a trippy compass.
The differential of this map is called the Shape Operator.
Given a point $x \in M$, the tangent plane at $x$ is denoted $T_xM$ is an inner product space. The shape operator can be defined as a linear operator on $T_xM$ by the equation:
$$ (S_x(v),w)=(df_x(v),w) \quad \text{for any $v,w \in T_x M.$} $$
Could help me understand why the eigen-vectors of the shape operator are the principal directions of curvature? Thanks!!