Differential geometry of surfaces#Shape operator

The Weingarten map ( after the German mathematician Julius Weingarten ), also called shape operator is a function from the theory of surfaces in three-dimensional Euclidean space ( ), a branch of classical differential geometry.

Preparation

A regular surface either by the parametric representation

Given. It should be at least twice continuously differentiable at each point have the derivative, a linear mapping from to, full rank. The image of this linear map is then a two-dimensional subspace of, the tangent space of the surface at the point. The image vectors one thinks tacked on point. The tangent is of the two vectors

Spanned. ( Here and denote the unit vectors of the standard basis. )

The unit normal at the point of the surface can be calculated with the aid of the vector product:

Thus, a differentiable map from the parameter range in the vector space. The Vector thinks you clung to the point. The derivation in point is a linear mapping from to. From the condition that a unit vector, it follows that for each pair of parameters the image of the figure in the tangent space of the surface is at point and thus the image of the figure. Since is injective, the inverse map exists as a map on the tangent space at the point.

Definition

One can now define the Weingarten map as a linear map in the parameter range ( classical view ) or on the tangent space ( modern view ).

In the parameter range

The figure is the tangent space onto the surface at the point. The picture makes this tangent again to the. The resulting therefrom by concatenation and sign change linear map

By means of Weingarten map at the location.

On the surface

The figure represents a vector of the tangent space of the surface at the point in the. The figure is the image vector in the tangent space again. The resulting therefrom by concatenation and sign change linear map

Is the tangent space at the point onto itself and is called Weingarten map at the point. It is therefore

Coordinate representation

The two versions of the Weingarten map defined on completely different vector spaces. However one chooses the parameter range the standard basis in the tangent space and the base, so tune the associated imaging matrices

Match. They are defined by the equations

Connection with the second fundamental form

For each parameter pair, the first fundamental form is an inner product in and the second fundamental form a symmetric bilinear form. These are connected by the Weingarten map as follows: For vectors

For the corresponding matrix representations applicable in shear Einstein summation convention

And

Properties

  • The Weingarten map is self-adjoint with respect to the first fundamental form, ie, valid for all In each point of the surface, therefore, there exists a basis of eigenvectors of which is orthonormal with respect.
  • The directions of the eigenvectors are called principal directions.
  • The eigenvalues ​​of the Weingarten map indicate the principal curvatures of the surface.
  • For a vector describing the change of the surface normal to this direction at this point.
  • The Weingarten map is the derivative of the Gauss map.

Example

Following the example of the articles first fundamental form and second fundamental form following, the surface of a sphere of radius is considered again. This area is again

The matrix representation of the first fundamental form consists of the components, as well.

The matrix representation of the second fundamental form consists of the components, as well.

Both of which are connected to one another by the equation. This provides through contests of the Einstein summation convention the following four equations:

Substituting the components of the matrix representations we obtain the components of the Weingarten map:

Alternatively, the explicit formula could be used. This, however, the matrix of the first fundamental form would need to be inverted to obtain the.

Pushforward (differential)
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