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

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

See also

• Metric tensor
• First fundamental form
• Second fundamental form
• Tautological one-form