Third fundamental form

In differential geometry, the third fundamental form is a surface metric denoted by [math]\displaystyle{ \mathrm{I\!I\!I} }[/math]. 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

[math]\displaystyle{ \mathrm{I\!I\!I}(\mathbf{u}_p,\mathbf{v}_p)=S(\mathbf{u}_p)\cdot S(\mathbf{v}_p)\,. }[/math]

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

[math]\displaystyle{ \mathrm{I\!I\!I}-2H\mathrm{I\!I}+K\mathrm{I}=0\,. }[/math]

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

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

See also

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