# Third fundamental form

In differential geometry, the third fundamental form is a surface metric denoted by ${\displaystyle \mathrm {I\!I\!I} }$. 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 up and vp be elements of the tangent space Tp(M). The third fundamental form is then given by

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

## 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

${\displaystyle \mathrm {I\!I\!I} -2H\mathrm {I\!I} +K\mathrm {I} =0\,.}$

As the shape operator is self-adjoint, for u,vTp(M), we find

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