Mean curvature

In the theory of surfaces in three-dimensional Euclidean space, a field of differential geometry, the mean curvature in addition to the Gaussian curvature is an important curvature term.

Definition

Given a regular surface and a point in this area. The mean curvature of the surface at this point is the arithmetic mean of the two principal curvatures and. That is, the mean curvature is defined as

Of particular interest are so-called minimal surfaces for which respectively applies.

Generally, one can mean curvature for n-dimensional hypersurfaces of define by. Here is the Weingarten map and called the trace of a matrix.

Examples

• In the case of a sphere (surface ) with the mean curvature radius is given by.

• At an arbitrary point on the curved surface of a straight circular cylinder having a radius of curvature equal to the average.

• Be a graph over the plain. Then the mean curvature by the formula:

Properties

• Are, respectively, the coefficients of the first and second fundamental form, the formula is

• If the first fundamental form is parameterized isothermally, ie it applies and is then written as

• For a surface, the equation is with the unit normal, the first fundamental form and the covariant derivative.

• If a surface is parameterized isothermal, so it satisfies the Rellichschen H- surface system

• If the surface is given as a level surface of an immersion, it shall The divergence and the unit normal field is This formula is called Bonnet formula and applies generally to n-dimensional hypersurfaces.