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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

I I I ( u p , v p ) = S ( u p ) ⋅ S ( v p ) . {\displaystyle \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

I I I − 2 H I I + K I = 0 . {\displaystyle \mathrm {I\!I\!I} -2H\mathrm {I\!I} +K\mathrm {I} =0\,.}

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

I I I ( u , v ) = ⟨ S u , S v ⟩ = ⟨ u , S 2 v ⟩ = ⟨ S 2 u , v ⟩ . {\displaystyle \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