# Mean curvature

*of a surface in -dimensional Euclidean space *

Half of the sum of the principal curvatures (cf. Principal curvature) and , calculated at a point of this surface:

For a hypersurface in the Euclidean space , this formula is generalized in the following way:

where , , are the principal curvatures of the hypersurface, calculated at a point .

The mean curvature of a surface in can be expressed by means of the coefficients of the first and second fundamental forms of this surface:

where are the coefficients of the first fundamental form, and are the coefficients of the second fundamental form, calculated at a point . In the particular case where the surface is defined by an equation , the mean curvature is calculated using the formula:

which is generalized for a hypersurface in , defined by the equation , as follows:

where

#### Comments

For an -dimensional submanifold of an -dimensional Euclidean space of codimension , the mean curvature generalizes to the notion of the mean curvature normal

where is an orthonormal frame of the normal space (cf. Normal space (to a surface)) of at and ( denotes the tangent space to at ) is the shape operator of at in the direction , which is related to the second fundamental tensor of at by .

#### References

[a1] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) |

[a2] | M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) |

[a3] | W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) |

[a4] | B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973) |

**How to Cite This Entry:**

Mean curvature.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mean_curvature&oldid=12526