Third fundamental form
From HandWiki
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 up and vp be elements of the tangent space Tp(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 ∈ Tp(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
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