Differential Geometry of Surfaces - Local Metric Structure - Shape Operator

Further information: Peterson operator

The differential df of the Gauss map f can be used to define a type of extrinsic curvature, known as the shape operator or Weingarten map. This operator first appeared implicitly in the work of Wilhelm Blaschke and later explicitly in a treatise by Burali-Forti and Burgati. Since at each point x of the surface, the tangent space is an inner product space, the shape operator Sx can be defined as a linear operator on this space by the formula

for tangent vectors v, w (the inner product makes sense because df(v) and w both lie in E3). The right hand side is symmetric in v and w, so the shape operator is self-adjoint on the tangent space. The eigenvalues of Sx are just the principal curvatures k1 and k2 at x. In particular the determinant of the shape operator at a point is the Gaussian curvature, but it also contains other information, since the mean curvature is half the trace of the shape operator. The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically. Its mean curvature is not zero, though; hence extrinsically it is different from a plane.

In general, the eigenvectors and eigenvalues of the shape operator at each point determine the directions in which the surface bends at each point. The eigenvalues correspond to the principal curvatures of the surface and the eigenvectors are the corresponding principal directions. The principal directions specify the directions that a curve embedded in the surface must travel to have maximum and minimum curvature, these being given by the principal curvatures.

The shape operator is given in terms of the components of the first and second fundamental forms by the Weingarten equations:

S= (EG-F^2)^{-1}\begin{pmatrix}
eG-fF& fG-gF \\
fE-eF & gE- fF\end{pmatrix}.

Read more about this topic:  Differential Geometry Of Surfaces, Local Metric Structure

Famous quotes containing the word shape:

    There are moments when very little truth would be enough to shape opinion. One might be hated at extremely low cost.
    Jean Rostand (1894–1977)