Third fundamental form
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In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, the third fundamental form is a surface metric denoted by \mathrm. Unlike the second fundamental form, it is independent of the
surface normal In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
.


Definition

Let be the shape operator and be a smooth surface. Also, let and be elements of the tangent space . The third fundamental form is then given by : \mathrm(\mathbf_p,\mathbf_p)=S(\mathbf_p)\cdot S(\mathbf_p)\,.


Properties

The third fundamental form is expressible entirely in terms of the first fundamental form and second fundamental form. If we let be the mean curvature of the surface and be the Gaussian curvature of the surface, we have : \mathrm-2H\mathrm+K\mathrm=0\,. As the shape operator is self-adjoint, for , we find : \mathrm(u,v)=\langle Su,Sv\rangle=\langle u,S^2v\rangle=\langle S^2u,v\rangle\,.


See also

*
Metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
* First fundamental form * Second fundamental form *
Tautological one-form In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus p ...
{{differential-geometry-stub Differential geometry of surfaces Differential geometry Surfaces