Third fundamental form

In differential geometry, the third fundamental form is a surface metric denoted by . Unlike the second fundamental form, it is independent of the surface normal.

Definition

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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

 

Properties

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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

 

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

 

See also

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