Third fundamental form
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In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, the third fundamental form is a surface metric denoted by \mathrm. Unlike the
second fundamental form In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...
, it is independent of the
surface normal In geometry, a normal is an object (e.g. 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 straight line perpendicular to the tangent line to the ...
.


Definition

Let be the
shape operator In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...
and be a
smooth surface In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensiv ...
. 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 In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and ...
and
second fundamental form In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...
. 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

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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 In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and ...
*
Second fundamental form In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...
*
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 pro ...
{{differential-geometry-stub Differential geometry of surfaces Differential geometry Surfaces