# 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

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

## Properties

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*,*v* ∈ *T _{p}*(

*M*), we find

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