Second fundamental form

The second fundamental form in mathematics is a function from differential geometry . The second fundamental form was initially defined in the theory of surfaces in three-dimensional space, a sub-area of ​​classical differential geometry. Today there is also a generalized definition in Riemannian geometry .

While the first fundamental form describes the inner geometry of a surface (i.e. properties that can be determined by length measurements within the surface), the second fundamental form depends on the position of the surface in the surrounding space. It is required for curvature calculations and occurs, for example, in the Mainardi-Codazzi equations . With their help and with the help of the first fundamental form, the main curvatures , the mean curvature and the Gaussian curvature of the surface are defined.

Classic differential geometry

definition

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. If the surface is regular , i.e. the first fundamental form of the surface is positive-definite , then a unit normal vector can be assigned to the surface . For the point of the area determined by the parameter values and , this is the vector product ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle \nu (u,v)}$ ${\displaystyle u}$ ${\displaystyle v}$

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

given. The coefficients of the second fundamental form in this point are defined as follows:

${\displaystyle L(u,v)=\nu (u,v)\cdot X_{uu}(u,v)}$

${\displaystyle M(u,v)=\nu (u,v)\cdot X_{uv}(u,v)}$

${\displaystyle N(u,v)=\nu (u,v)\cdot X_{vv}(u,v)}$

Are defined. Where , and are the second partial derivatives according to the parameters. The paint points express scalar products of vectors . To simplify the notation, the arguments are often left out and only , and . Some authors use the terms , and . ${\displaystyle X_{uu}(u,v)}$ ${\displaystyle X_{uv}(u,v)}$ ${\displaystyle X_{vv}(u,v)}$ ${\displaystyle L}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle e}$ ${\displaystyle f}$ ${\displaystyle g}$

The second fundamental form is then the quadratic form

${\displaystyle {\mathit {II}}\colon \mathbb {R} ^{2}\to \mathbb {R} ,\ (w_{1},w_{2})\mapsto L\,w_{1}^{2}+2M\,w_{1}w_{2}+N\,w_{2}^{2}}$

The notation with differentials is also occasionally used:

${\displaystyle d\sigma ^{2}=L\,du^{2}+2M\,du\,dv+N\,dv^{2}}$

Another (more modern) notation is:

${\displaystyle h_{11}=L;\quad h_{12}=h_{21}=M;\quad h_{22}=N}$,

the second fundamental form has the matrix representation

${\displaystyle (h_{ij})={\begin{pmatrix}L&M\\M&N\end{pmatrix}}.}$

The bilinear form represented by this matrix is ​​often referred to as the second fundamental form . ${\displaystyle h}$

properties

The discriminant (i.e. the determinant of the representation matrix) of the second fundamental form provides information about how the given surface is curved at the point under consideration. There are three different cases: ${\displaystyle LN-M^{2}}$

• For is elliptical curvature . (Example: surface of an ellipsoid or a sphere ) ${\displaystyle LN-M^{2}>0}$

• ${\displaystyle LN-M^{2}=0}$means parabolic curvature . (Example: surface of a straight circular cylinder )

• If true, it is called hyperbolic curvature . (Example: single-shell hyperboloid ) ${\displaystyle LN-M^{2}<0}$

Example spherical surface

Following the example from the article of the first fundamental form , the surface of a sphere of radius is again considered. 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 unit normal field can then go through

${\displaystyle \nu (u,v)={\frac {1}{r}}X(u,v)}$

to be discribed. The second partial derivatives of are ${\displaystyle X}$

${\displaystyle X_{uu}=-X\,}$as well as and . ${\displaystyle X_{uv}=\left({\begin{smallmatrix}-r\cos u\sin v\\r\cos u\cos v\\0\end{smallmatrix}}\right)\,}$ ${\displaystyle X_{vv}=\left({\begin{smallmatrix}-r\sin u\cos v\\-r\sin u\sin v\\0\end{smallmatrix}}\right)}$

Hence the coefficients , and . The representation of the second fundamental form of the spherical surface with the help of differentials then reads ${\displaystyle L=-r}$ ${\displaystyle M=0}$ ${\displaystyle N=-r\sin ^{2}(u)}$

${\displaystyle d\sigma ^{2}=-r\,du^{2}-r\sin ^{2}(u)\,dv^{2}.}$

Special case graph of a surface

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

${\displaystyle \nu ={\frac {1}{\sqrt {1+f_{u}^{2}+f_{v}^{2}}}}{\begin{pmatrix}-f_{u}\\-f_{v}\\1\end{pmatrix}}}$

and

${\displaystyle L={\frac {f_{uu}}{\sqrt {1+f_{u}^{2}+f_{v}^{2}}}},\quad M={\frac {f_{uv}}{\sqrt {1+f_{u}^{2}+f_{v}^{2}}}},\quad N={\frac {f_{vv}}{\sqrt {1+f_{u}^{2}+f_{v}^{2}}}}.}$

Here and denote the first and , and the second partial derivatives of . ${\displaystyle f_{u}}$ ${\displaystyle f_{v}}$ ${\displaystyle f_{uu}}$ ${\displaystyle f_{uv}}$ ${\displaystyle f_{vv}}$ ${\displaystyle f}$

Riemannian geometry

In contrast to the first fundamental form, which in Riemann's geometry was replaced by more descriptive constructions, the second fundamental form also has an important meaning and a generalized definition in Riemann's geometry.

definition

Be a submanifold of the Riemannian manifold starting point for the definition of the second fundamental form is the orthogonal decomposition of vector fields in in tangential and normal shares. If vector fields are open , these can be continued to vector fields . If the Levi-Civita connection is open , then the decomposition is obtained ${\displaystyle M}$ ${\displaystyle {\tilde {M}}.}$ ${\displaystyle T{\tilde {M}}|_{M}}$ ${\displaystyle X,Y\in \Gamma ^{\infty }(TM)}$ ${\displaystyle M}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle {\tilde {\nabla }}}$ ${\displaystyle {\tilde {M}}}$

${\displaystyle {\tilde {\nabla }}_{X}Y=({\tilde {\nabla }}_{X}Y)^{\top }+({\tilde {\nabla }}_{X}Y)^{\bot }.}$

The second fundamental form is a map

${\displaystyle {\mathit {II}}\colon \Gamma (TM)\times \Gamma (TM)\to \Gamma (NM),}$

which through

${\displaystyle {\mathit {II}}(X,Y):=({\tilde {\nabla }}_{X}Y)^{\bot }}$

is defined. In this case, referred to the normal bundle of which analogous to the tangent is defined and is the orthogonal projection onto the normal bundle. ${\displaystyle NM}$ ${\displaystyle M}$ ${\displaystyle ^{\bot }:T{\tilde {M}}\to NM}$

properties

The second fundamental form is

• regardless of the continuation of the vector fields and . ${\displaystyle X}$ ${\displaystyle Y}$
• bilinear over ${\displaystyle C^{\infty }(M).}$
• symmetrical in and ${\displaystyle X}$ ${\displaystyle Y.}$

Scalar second fundamental form

Let be a -dimensional Riemannian manifold with a Riemannian metric and be a -dimensional submanifold of . Such a submanifold of codimension 1 is also called a hypersurface . In this case, the normal space at each point of one-dimensional and there are exactly two unit normal vectors each span. These differ only in the sign . ${\displaystyle {\tilde {M}}}$ ${\displaystyle n}$ ${\displaystyle g}$ ${\displaystyle M}$ ${\displaystyle (n-1)}$ ${\displaystyle {\tilde {M}}}$ ${\displaystyle NM_{p}}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle NM_{p}}$

If a unit normal vector field is firmly selected, the associated scalar second fundamental form is defined by ${\displaystyle N\in \Gamma (NM)}$ ${\displaystyle h}$

${\displaystyle h(X,Y)=g({\mathit {II}}(X,Y),N)}$ for all ${\displaystyle X,Y\in \Gamma (TM).}$

Except for the sign, the scalar second fundamental form does not depend on the choice of the unit normal vector field: If one takes instead of the oppositely oriented second unit normal vector field, only the sign changes in the scalar second fundamental form. It follows that the scalar second fundamental form also symmetrical and from the properties of the second fundamental form is -linear in each case, ie a symmetrical (0,2) - tensor to . ${\displaystyle N}$ ${\displaystyle C^{\infty }(M)}$ ${\displaystyle M}$

Total geodetic submanifolds

→ Main article : Total geodetic submanifold

A submanifold is totally geodesic (ie geodesics in are also geodesics in ) if and only if its second fundamental form vanishes identically. ${\displaystyle N\subset M}$ ${\displaystyle N}$ ${\displaystyle M}$

See also

• Vineyard illustration

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., New Jersey, 1976, ISBN 0-13-212589-7
• Manfredo Perdigão do Carmo: Riemannian Geometry , Birkhäuser, Boston 1992, ISBN 0-8176-3490-8
• John M. Lee: Riemannian Manifolds. An Introduction to Curvature. Springer, New York 1997, ISBN 0387983228 .