The Gauss Map & Second Fundamental Form

The Gauss map, the shape operator dN, and the form e, f, g.

The idea

The Gauss map and the second fundamental form record how a surface bends inside the surrounding space. The first fundamental form cannot: it records only measurements made within the surface, and a plane and a rolled-up cylinder share the same one. Bending is an extrinsic matter, and we read it off the unit normal.

Orient the surface by choosing a unit normal $N(p)$ that varies smoothly. Each $N(p)$ is a unit vector, hence a point of the unit sphere, so the assignment is a map $N \colon S \to \mathbb{S}^{2},$ the Gauss map. A plane has a constant normal, so its Gauss map is a single point; a sharply curved surface has a normal that turns quickly, sending nearby points $p$ and $q$ to visibly separated images $N(p)$ and $N(q)$. The bending is therefore recorded in the derivative of $N$: the differential $dN_p$ is the shape operator. Since $\langle N, N \rangle = 1$, differentiating shows that $dN_p(v)$ is perpendicular to $N$, so $dN_p$ maps the tangent plane to itself.

Pairing the operator against the direction of motion gives the second fundamental form, $II_p(v) = -\langle dN_p(v),\, v \rangle,$ with coefficients $e = \langle N, \mathbf{x}_{uu} \rangle$, $f = \langle N, \mathbf{x}_{uv} \rangle$, $g = \langle N, \mathbf{x}_{vv} \rangle$ in a parametrization $\mathbf{x}(u,v)$. Rigid motions of space leave $II$ unchanged, but reversing the normal reverses its sign: $II$ is defined only once an orientation is chosen.

Ways to work on it

• Walkthrough. The Gauss map, shape operator, and second fundamental form on the sphere.
• Practice. Compute a coefficient of the second fundamental form.
• Hardest. Build the full second fundamental form and classify the point.

Not sure where to start? Take the ten-question placement test.