In differential geometry, the third fundamental form is a surface metric denoted by I I I . 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 u p and v p be elements of the tangent space T p M . The third fundamental form is then given by
I I I ( u p , v p ) = S ( u p ) ⋅ S ( v p ) . 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 . 