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

\mathrm{I\!I\!I}-2H\mathrm{I\!I}+K\mathrm{I}=0\,. $$ As the shape operator is self-adjoint, for $u,v ∈ T_{p}(M)$, we find

\mathrm{I\!I\!I}(u,v)=\langle Su,Sv\rangle=\langle u,S^2v\rangle=\langle S^2u,v\rangle\,. $$