First fundamental form

In mathematics, the first fundamental form or metric basic form is a function from the theory of surfaces in three-dimensional Euclidean space , a sub-area of ​​classical differential geometry . The first fundamental form enables, among other things, the following tasks to be dealt with:

• Calculation of the length of a curve on the given surface
• Calculation of the angle at which two curves intersect on the given surface
• Calculation of the area of ​​a patch of the given area

Furthermore, the Gaussian curvature (Brioschi's formula) and the Christoffel symbols of the second kind can be determined from the coefficients of the first fundamental form and their partial derivatives .

Those properties of a surface that can be investigated with the help of the first fundamental form are summarized under the term internal geometry .

Definition and characteristics

Let a surface be defined by a mapping on an open subset ${\displaystyle U\subset \mathbb {R} ^{2}}$

${\displaystyle X\colon U\to \mathbb {R} ^{3},\quad (u,v)\mapsto X(u,v)}$

given, i.e. through and parameterized. For the point of the surface determined by the parameter values and the coefficients of the first fundamental form are defined as follows: ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle v}$

${\displaystyle E(u,v)=X_{u}(u,v)\cdot X_{u}(u,v)=|X_{u}(u,v)|^{2}}$

${\displaystyle F(u,v)=X_{u}(u,v)\cdot X_{v}(u,v)}$

${\displaystyle G(u,v)=X_{v}(u,v)\cdot X_{v}(u,v)=|X_{v}(u,v)|^{2}}$

Here are the vectors

${\displaystyle X_{u}(u,v)={\frac {\partial X}{\partial u}}(u,v)\quad {\text{und}}\quad X_{v}(u,v)={\frac {\partial X}{\partial v}}(u,v)}$

the first partial derivatives according to the parameters or . The paint points denote the scalar product of the vectors . ${\displaystyle u}$ ${\displaystyle v}$

To simplify matters, one often leaves out the arguments and only writes , and for the coefficients. The first fundamental form is then the quadratic form ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$

${\displaystyle I\colon \mathbb {R} ^{2}\to \mathbb {R} ,\ (w_{1},w_{2})\mapsto E\,w_{1}^{2}+2F\,w_{1}w_{2}+G\,w_{2}^{2}}$,

The notation with differentials is also occasionally used:

${\displaystyle ds^{2}=E\,du^{2}+2F\,du\,dv+G\,dv^{2}}$

Another (more modern) notation is:

${\displaystyle g_{11}=E;\quad g_{12}=g_{21}=F;\quad g_{22}=G}$

If one sets and , then applies ${\displaystyle X_{1}=X_{u}}$ ${\displaystyle X_{2}=X_{v}}$

${\displaystyle g_{ij}=X_{i}\cdot X_{j}}$for . ${\displaystyle i,j=1,2}$

The numbers are the coefficients of the covariant metric tensor . So this one has the matrix representation ${\displaystyle g_{ij}}$

${\displaystyle (g_{ij})={\begin{pmatrix}E&F\\F&G\end{pmatrix}}}$.

This tensor , i.e. the bilinear form represented by this matrix , is often referred to as the first fundamental form ${\displaystyle g}$

The following applies to the coefficients of the first fundamental form:

${\displaystyle E\geq 0;\quad G\geq 0;\quad EG-F^{2}\geq 0}$.

It is the discriminant (i.e., the determinant of the matrix representation) of the first fundamental form. If it is also valid, it also follows and and the first fundamental form is positive definite . This is the case if and only if and are linearly independent. A surface with a positive definite first fundamental form is called differential geometry regular or differential geometry regular parameterized . ${\displaystyle EG-F^{2}}$ ${\displaystyle EG-F^{2}>0}$ ${\displaystyle E>0}$ ${\displaystyle G>0}$ ${\displaystyle X_{u}}$ ${\displaystyle X_{v}}$

Length of a surface curve

A curve on the given surface can be expressed by two real functions and : The point located on the surface is assigned to each possible value of the parameter . If all functions involved are continuously differentiable, the following applies to the length of the curve segment defined by : ${\displaystyle \varphi _{1}}$ ${\displaystyle \varphi _{2}}$ ${\displaystyle t}$ ${\displaystyle X(\varphi _{1}(t),\varphi _{2}(t))}$ ${\displaystyle t\in [a,b]}$

${\displaystyle l=\int \limits _{a}^{b}{\sqrt {I({\dot {\varphi }}_{1}(t),{\dot {\varphi }}_{2}(t))}}\,dt=\int \limits _{a}^{b}{\sqrt {E\cdot ({\dot {\varphi }}_{1}(t))^{2}+2F\cdot {\dot {\varphi }}_{1}(t){\dot {\varphi }}_{2}(t)+G\cdot ({\dot {\varphi }}_{2}(t))^{2}}}\,dt}$

Expressed with the help of the path element : ${\displaystyle ds={\sqrt {ds^{2}}}}$

${\displaystyle l=\int _{\varphi }ds}$

Content of a patch

The content of an area given by a parameter area can be calculated by ${\displaystyle B}$

${\displaystyle A=\int \limits _{B}{\sqrt {EG-F^{2}}}\,d(u,v)}$.

Example spherical surface

The surface of a sphere with a radius can be parameterized in spherical coordinates ${\displaystyle r}$

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

For the coefficients of the first fundamental form we get:

${\displaystyle E=X_{u}(u,v)\cdot X_{u}(u,v)={\begin{pmatrix}r\cos u\cos v\\r\cos u\sin v\\-r\sin u\end{pmatrix}}\cdot {\begin{pmatrix}r\cos u\cos v\\r\cos u\sin v\\-r\sin u\end{pmatrix}}=r^{2}}$

${\displaystyle F=X_{u}(u,v)\cdot X_{v}(u,v)={\begin{pmatrix}r\cos u\cos v\\r\cos u\sin v\\-r\sin u\end{pmatrix}}\cdot {\begin{pmatrix}-r\sin u\sin v\\r\sin u\cos v\\0\end{pmatrix}}=0}$

${\displaystyle G=X_{v}(u,v)\cdot X_{v}(u,v)={\begin{pmatrix}-r\sin u\sin v\\r\sin u\cos v\\0\end{pmatrix}}\cdot {\begin{pmatrix}-r\sin u\sin v\\r\sin u\cos v\\0\end{pmatrix}}=r^{2}\sin ^{2}u}$

The first fundamental form is therefore

${\displaystyle ds^{2}=r^{2}\,du^{2}+r^{2}\sin ^{2}(u)\,dv^{2}}$.

Special case graph of a function

If the area under consideration is the graph of a function over the parameter range , i.e. for all , then: ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle X(u,v)=(u,v,f(u,v))}$ ${\displaystyle (u,v)\in U}$

${\displaystyle X_{u}(u,v)=(1,0,f_{u}),\quad X_{v}(u,v)=(0,1,f_{v})}$

and thus

${\displaystyle E=1+f_{u}^{2},\quad F=f_{u}f_{v},\quad G=1+f_{v}^{2}}$

and

${\displaystyle EG-F^{2}=(1+f_{u}^{2})\,(1+f_{v}^{2})-(f_{u}f_{v})^{2}=1+f_{u}^{2}+f_{v}^{2}}$.

Here and denote the partial derivatives of to or . ${\displaystyle f_{u}}$ ${\displaystyle f_{v}}$ ${\displaystyle f}$ ${\displaystyle u}$ ${\displaystyle v}$

See also

• Second fundamental form

Individual evidence

1. ^ A. Hartmann: Surfaces, Gaussian curvature, first and second fundamental form, theorema egregium. (PDF) April 12, 2011, accessed September 29, 2016 .  Page 6, proof of Theorem 3.4.

literature

• Manfredo Perdigão do Carmo: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Upper Saddle River NJ 1976, ISBN 0-13-212589-7 .