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).

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,v ∈ Tp(M), we find