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:
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 .
|[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)|
Mean curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mean_curvature&oldid=12526