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.


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


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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