TITLE: Why can we think of the second fundamental form as a Hessian matrix?
QUESTION [5 upvotes]: Let $f: U \rightarrow \mathbb{R}^3$ be an immersion that parametrizes a piece of a surface, and let $(h_{ij})$ be the matrix for the second fundamental form of that surface.
According to pg. 70 of the text Differential Geometry by Wolfgang Kuhnel, we can think of the $(h_{ij})$ as "the Hessian of matrix of a function $h$, which represents the surface as a graph over its tangent plane".
I have a "heuristic" understanding of what's going on, but I'd like to be a bit more careful about this.  What exactly is the function $h$?  Can we write it down explicitly (perhaps in terms of the parametrization $f$, the unit normal $\nu$, and their derivatives), so that we can directly check that its Hessian is indeed the second fundamental form $(h_{ij})$?

REPLY [2 votes]: Thanks Ryan!  So in conclusion, if we choose coordinates such that the surface is locally described by $f(u,v) = (u,v,h(u,v))$, then the second fundamental form at the point $f(0,0)$ is precisely the Hessian of $h$ at $(0,0)$.