Third fundamental form

Template:Unreferenced 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 Template:Mvar be the shape operator and Template:Mvar be a smooth surface. Also, let Template:Math and Template:Math be elements of the tangent space Template:Math. The third fundamental form is then given by

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

Properties

The third fundamental form is expressible entirely in terms of the first fundamental form and second fundamental form. If we let Template:Mvar be the mean curvature of the surface and Template:Mvar 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 Template:Math, 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

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