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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 $(h_{ij})$?

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    I don't think Kuhnel is talking about arbitrary parametrizations -- he says specifically in your quote that he's restricting to the situation as in your example $f(u,v)=(u,v,h(u,v))$ since up to a rotation of space, that's exactly what's going on when you express the surface as the graph of a function over the surface's tangent space (at a point).2011-02-28

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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 $(0,0)$.