Third fundamental form

In differential geometry, the third fundamental form is a surface metric denoted by $\mathbf{III}$. 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 $\mathbf{u}_p$ and $\mathbf{v}_p$ be elements of the tangent space $T_pM$. The third fundamental form is then given by

$\mathbf{III}(\mathbf{u}_p,\mathbf{v}_p)=S(\mathbf{u}_p)\cdot S(\mathbf{v}_p)\,.$

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

$\mathbf{III}-2H\mathbf{II}+K\mathbf{I}=0\,.$

See also

• Metric tensor
• First fundamental form
• Second fundamental form