Second fundamental form (shape operator)

The second fundamental form is similar to the first fundamental form (sec:FirstFundamentalForm), except that it relates a vector $v$ in the tangent plane to the change in normal in the direction of $v$ . Specifically, the second fundamental form or shape operator at a point $p$ is defined as

$$II_p(v) = -\ensuremath{\langle dN_p(v), v \rangle}.$$ (2.30)

Recall that $dN_p(v)$ is the change in surface normal in the direction of $v$ at a point $p$ . Similarly to the case of curves (sec:CurvatureCurves), the change in surface normals encodes information about the curvature of the surface.

To see how this might be true, consider a regular arc length-parameterized curve $\alpha(s)$ on the surface. The field of normals restricted to the curve is also a function of $s$ : $N(s) = N \circ \alpha(s)$ . Since the normal is always perpendicular to the tangent plane, we have $\ensuremath{\langle N(s), \alpha'(s) \rangle} = 0$ , which we can differentiate to obtain $\ensuremath{\langle N'(s), \alpha'(s) \rangle} + \ensuremath{\langle N(s), \alpha''(s) \rangle} = 0$ . Therefore

$$II_p(\alpha'(0)) = -\ensuremath{\langle dN_p(\alpha'(0)), \alpha'(0) \rangle}$$ (2.31)

$$= -\ensuremath{\langle N'(0), \alpha'(0) \rangle}$$ (2.32)

$$= \ensuremath{\langle N(0), \alpha''(0) \rangle}$$ (2.33)

$$= \ensuremath{\langle N, kn \rangle}(p)$$ (2.34)

$$= k_n(p)$$ (2.35)

So along a curve on the surface, the second fundamental form at a point gives the normal curvature to an arc length-parameterized curve.