Third fundamental form explained

In differential geometry, the third fundamental form is a surface metric denoted by

III . Unlike the second fundamental form, it is independent of the surface normal.

Definition

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

III ( u _{p,v} _{p)=S(u} _{p) ⋅} 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 be the mean curvature of the surface and be the Gaussian curvature of the surface, we have

III -2 H II + K I =0 . As the shape operator is self-adjoint, for, we findIII (u,v) = \langle Su,Sv\rangle = \langle u,S ^{2v\rangle=\langle} S ^{2u,v\rangle.}

See also

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