Medium curvature

In the theory of surfaces in three-dimensional Euclidean space , a field of differential geometry , the mean curvature is an important concept of curvature in addition to the Gaussian curvature . ${\displaystyle \mathbb {R} ^{3}}$

definition

Let a regular area in and a point of this area be given. The mean curvature of the surface at this point is the arithmetic mean of the two main curvatures and . That is, the mean curvature is defined as ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle H}$ ${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$

${\displaystyle H:={\frac {1}{2}}(k_{1}+k_{2}).}$

So-called minimum areas are of particular interest , for which or applies. ${\displaystyle H=0}$ ${\displaystyle k_{1}=-k_{2}}$

General can be the mean curvature for n-dimensional hyper-surfaces of the through define. Here is the Weingarten figure and denotes the trace of a matrix . ${\displaystyle \mathbb {R} ^{n+1}}$ ${\displaystyle H:={\tfrac {1}{n}}\operatorname {Spur} (S)}$ ${\displaystyle S}$ ${\displaystyle \operatorname {Spur} }$

calculation

• Are , , and , , the coefficients of the first and second fundamental form of the surface, so applies the formula ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle L}$ ${\displaystyle M}$ ${\displaystyle N}$

${\displaystyle H={\frac {LG-2MF+NE}{2(EG-F^{2})}}}$

If the area is parameterized isothermally, that is, if the coefficients of the first fundamental form and hold, then this formula is simplified to ${\displaystyle E=G}$ ${\displaystyle F=0}$

${\displaystyle H={\frac {L+N}{2E}}.}$

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

${\displaystyle H={\frac {(1+f_{v}^{2})f_{uu}-2f_{u}f_{v}f_{uv}+(1+f_{u}^{2})f_{vv}}{2{\sqrt {1+f_{u}^{2}+f_{v}^{2}}}^{3}}}}$.

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

Examples

• The surface of a sphere with a radius has the mean curvature . ${\displaystyle r}$ ${\displaystyle H={\tfrac {1}{r}}}$

• At any point on the curved surface of a straight circular cylinder with a radius , the mean curvature is the same ${\displaystyle r}$ ${\displaystyle H={\tfrac {1}{2r}}}$

Other properties

• The equation applies to an area ${\displaystyle X=X(u,v)}$

${\displaystyle H{\vec {n}}=g^{ij}\nabla _{i}\nabla _{j}X,}$

with the unit normal , the first fundamental form and the covariant derivative. ${\displaystyle {\vec {n}}}$ ${\displaystyle g_{ij}}$ ${\displaystyle \nabla _{i}}$

• If a surface is parameterized isothermally, it satisfies Rellich's H-surface system ${\displaystyle X=X(u,v)}$

${\displaystyle \Delta X=2HX_{u}\times X_{v}.}$

• If the area is given as the level area of a function , then applies ${\displaystyle F}$

${\displaystyle 2H=-\operatorname {div} {\vec {n}}=-\operatorname {div} {\frac {\nabla F}{|\nabla F|}}.}$

This is the divergence and the unit normal field. This formula is called the Bonnet formula and applies generally to n-dimensional hypersurfaces . ${\displaystyle \operatorname {div} }$ ${\displaystyle {\vec {n}}}$ ${\displaystyle {\tfrac {\nabla F}{|\nabla F|}}.}$

literature

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