Third fundamental form

In differential geometry, the third fundamental form is a surface metric denoted by $\mathrm{I}\mathrm{I}\mathrm{I}$. 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

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

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

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

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

$\mathrm{I}\mathrm{I}\mathrm{I}(u,v)=⟨Su,Sv⟩=⟨u,S^{2}v⟩=⟨S^{2}u,v⟩.$

See also

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