Vineyard illustration

The Weingarten mapping (after the German mathematician Julius Weingarten ), also known as the shape operator , is a function from the theory of surfaces in three-dimensional Euclidean space ( ), a sub-area of ​​classical differential geometry . ${\displaystyle \mathbb {R} ^{3}}$

preparation

Let the parametric representation be a regular surface

${\displaystyle {\begin{aligned}X\colon \mathbb {R} ^{2}\supset A&\to \mathbb {R} ^{3}\\(u,v)&\mapsto X(u,v)\end{aligned}}}$

given. Let it be at least twice continuously differentiable and at every point the derivative , a linear mapping from to , has full rank. The image of this linear mapping is then a two-dimensional subspace of the , the tangent space of the surface in the point . The image vectors are thought to be attached to the point . The tangent space is defined by the two vectors ${\displaystyle X}$ ${\displaystyle (u,v)}$ ${\displaystyle DX_{(u,v)}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle p=X(u,v)}$ ${\displaystyle p=X(u,v)}$

${\displaystyle X_{1}(u,v)=X_{u}(u,v)={\tfrac {\partial X}{\partial u}}(u,v)=DX_{(u,v)}(e_{1})}$ and

${\displaystyle X_{2}(u,v)=X_{v}(u,v)={\tfrac {\partial X}{\partial v}}(u,v)=DX_{(u,v)}(e_{2})}$

stretched. (Here and denote the unit vectors of the standard basis des .) ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle \mathbb {R} ^{2}}$

The unit normal at the point of the surface can be calculated using the vector product : ${\displaystyle N(u,v)}$ ${\displaystyle p=X(u,v)}$

${\displaystyle N(u,v)={\frac {X_{u}(u,v)\times X_{v}(u,v)}{|X_{u}(u,v)\times X_{v}(u,v)|}}}$

Thus there is a differentiable mapping from the parameter area into the vector space . You think of the image vector attached to the point . The derivative in the point is a linear mapping from to . From the condition that there is a unit vector, it follows that for each pair of parameters the image of the image lies in the tangent space of the surface in the point and thus in the image of the image . Since is injective, the inverse mapping exists as mapping on the tangent space in the point . ${\displaystyle N}$ ${\displaystyle A\subset \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle N(u,v)}$ ${\displaystyle p=X(u,v)}$ ${\displaystyle DN_{(u,v)}}$ ${\displaystyle (u,v)}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle N(u,v)}$ ${\displaystyle (u,v)}$ ${\displaystyle DN_{(u,v)}}$ ${\displaystyle p=X(u,v)}$ ${\displaystyle DX_{(u,v)}}$ ${\displaystyle DX_{(u,v)}}$ ${\displaystyle (DX_{(u,v)})^{-1}}$ ${\displaystyle X(u,v)}$

definition

The vineyard mapping can now be defined as a linear mapping in the parameter area (classic view) or on the tangential space (modern view).

In the parameter area

The figure is the on the tangent of the surface at the point from. The image maps this tangential space back onto the . The resulting linear mapping through concatenation and sign change ${\displaystyle DN_{(u,v)}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle X(u,v)}$ ${\displaystyle (DX_{(u,v)})^{-1}}$ ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle L_{(u,v)}=-(DX_{(u,v)})^{-1}\circ DN_{(u,v)}}$

from to is called vineyard illustration at the point . ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle (u,v)}$

On the surface

The figure shows a vector of the tangent space of the surface at the point in the . The image maps the image vector back into the tangential space. The resulting linear mapping through concatenation and sign change ${\displaystyle (DX_{(u,v)})^{-1}}$ ${\displaystyle p=X(u,v)}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle DN_{(u,v)}}$

${\displaystyle L_{X(u,v)}=-DN_{(u,v)}\circ (DX_{(u,v)})^{-1}}$

maps the tangential space on itself at the point and is called vineyard mapping at the point . So it applies ${\displaystyle p=X(u,v)}$ ${\displaystyle p=X(u,v)}$

${\displaystyle L_{X(u,v)}X_{i}(u,v)=-N_{i}(u,v)}$for . ${\displaystyle i=1,2}$

Coordinate representation

The two versions of the vineyard map are defined on completely different vector spaces. However one chooses the parameter area the standard basis and in the tangent the base , so tune the associated mapping matrices ${\displaystyle X_{u}(u,v)}$ ${\displaystyle X_{v}(u,v)}$

${\displaystyle {\begin{pmatrix}h^{1}{_{1}}(u,v)&h^{1}{_{2}}(u,v)\\h^{2}{_{1}}(u,v)&h^{2}{_{2}}(u,v)\end{pmatrix}}}$

match. You are through the equations

${\displaystyle L_{X(u,v)}(X_{u}(u,v))=-N_{u}(u,v)=h^{1}{_{1}}(u,v)X_{u}(u,v)+h^{2}{_{1}}(u,v)X_{v}(u,v)}$

${\displaystyle L_{X(u,v)}(X_{v}(u,v))=-N_{v}(u,v)=h^{1}{_{2}}(u,v)X_{u}(u,v)+h^{2}{_{2}}(u,v)X_{v}(u,v)}$

characterized. In Einstein's summation convention shear , with , , , and the omission of the argument: ${\displaystyle X_{1}=X_{u}}$ ${\displaystyle X_{2}=X_{v}}$ ${\displaystyle N_{1}=N_{u}=DN_{(u,v)}(e_{1})}$ ${\displaystyle N_{2}=N_{v}=DN_{(u,v)}(e_{2})}$

${\displaystyle L(X_{j})=-N_{j}=h^{i}{}_{j}X_{i}}$

Connection with the second fundamental form

For each parameter pair , the first fundamental form is a scalar product im and the second fundamental form is a symmetrical bilinear form. These are connected by the vineyard map as follows: The following applies to vectors ${\displaystyle (u,v)}$ ${\displaystyle g_{(u,v)}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle h_{(u,v)}}$ ${\displaystyle w_{1},w_{2}\in \mathbb {R} ^{2}}$

${\displaystyle h(w_{1},w_{2})=g(w_{1},Lw_{2})}$.

Einstein's summation convention applies to the associated matrix representations

${\displaystyle h_{ik}=g_{ij}h^{j}{_{k}}}$

and

${\displaystyle h^{i}{}_{k}=g^{ij}h_{jk}.}$

properties

The main curvatures are eigenvalues ​​of the vineyard map

• The Weingarten mapping is self-adjoint with respect to the first fundamental form , that is, for all of them , there is therefore a basis of eigenvectors of which is orthonormal with respect to each point of the surface . ${\displaystyle L}$ ${\displaystyle g}$ ${\displaystyle w_{1},w_{2}\in \mathbb {R} ^{2}}$
  ${\displaystyle g(w_{1},Lw_{2})=g(Lw_{1},w_{2})\,.}$
  ${\displaystyle L}$ ${\displaystyle g}$
• The directions of the eigenvectors are called main directions of curvature .
• The eigenvalues ​​of the vineyard map indicate the main curvatures of the surface.
• For a vector describes the change of the surface normal in this direction at this point. ${\displaystyle w\in T_{(u,v)}\mathbb {R} ^{2}}$ ${\displaystyle Lw}$
• The vineyard map is the derivation of the Gauss map .

example

Following the example from the articles first fundamental form and second fundamental form , the surface of a sphere is again considered from the radius . This area is back through ${\displaystyle r>0}$

${\displaystyle X(u,v)={\begin{pmatrix}r\sin u\cos v\\r\sin u\sin v\\r\cos u\end{pmatrix}}}$ parameterized.

The matrix representation of the first fundamental form consists of the components , , and . ${\displaystyle g_{uu}=r^{2}}$ ${\displaystyle g_{uv}=g_{vu}=0}$ ${\displaystyle g_{vv}=r^{2}\sin ^{2}u}$

The matrix representation of the second fundamental form consists of the components , , and . ${\displaystyle h_{uu}=-r}$ ${\displaystyle h_{uv}=h_{vu}=0}$ ${\displaystyle h_{vv}=-r\sin ^{2}u}$

Both are linked by the equation . By writing out Einstein's sum convention, this yields the following four equations: ${\displaystyle h_{ik}=g_{ij}h^{j}{_{k}}}$

${\displaystyle h_{uu}=g_{uu}h^{u}{_{u}}+g_{uv}h^{v}{_{u}}}$

${\displaystyle h_{uv}=g_{uu}h^{u}{_{v}}+g_{uv}h^{v}{_{v}}}$

${\displaystyle h_{vu}=g_{vu}h^{u}{_{u}}+g_{vv}h^{v}{_{u}}}$

${\displaystyle h_{vv}=g_{vu}h^{u}{_{v}}+g_{vv}h^{v}{_{v}}}$

By inserting the components of the matrix representations, the components of the vineyard map are obtained:

${\displaystyle h^{u}{_{u}}=h^{v}{_{v}}=-{\frac {1}{r}}}$

${\displaystyle h^{u}{_{v}}=h^{v}{_{u}}=0}$

Alternatively, the explicit formula could have been used. To do this, however, the matrix of the first fundamental form would have to be inverted in order to obtain the. ${\displaystyle h^{i}{}_{k}=g^{ij}h_{jk}}$ ${\displaystyle g^{ij}}$

literature

• Wolfgang Kühnel : Differential Geometry. Curves - surfaces - manifolds . 4th revised edition. Vieweg, Wiesbaden 2007, ISBN 978-3-8348-0411-2 .