Mean curvature

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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)
How to Cite This Entry:
Mean curvature. L.A. Sidorov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098