Third fundamental form

In differential geometry, the third fundamental form is a surface metric denoted by $\mathrm {I\!I\!I}$ . Unlike the second fundamental form, it is independent of the surface normal.

Definition

Let $S$ be the shape operator and $M$ be a smooth surface. Also, let $u p$ and $v p$ be elements of the tangent space $T p (M)$ . The third fundamental form is then given by

\mathrm {I\!I\!I} (\mathbf {u} _{p},\mathbf {v} _{p})=S(\mathbf {u} _{p})\cdot S(\mathbf {v} _{p})\,.

Properties

The third fundamental form is expressible entirely in terms of the first fundamental form and second fundamental form. If we let $H$ be the mean curvature of the surface and $K$ be the Gaussian curvature of the surface, we have