The normal curvature of a surface in a principal direction, i.e. in a direction in which it assumes an extremal value. The principal curvatures and are the roots of the quadratic equation
where is the mean, and is the Gaussian curvature of the surface at the given point.
The principal curvatures and are connected with the normal curvature , taken in an arbitrary direction, by means of Euler's formula:
where is the angle formed by the selected direction with the principal direction for .
In the case of an -dimensional submanifold of Euclidean -space principal curvatures and principal directions are defined as follows.
Let be a unit normal to at . The Weingarten mapping (shape operator) of at in direction is given by the tangential part of , where is the covariant differential in and is a local extension of to a unit normal vector field. does not depend on the chosen extension of . The principal curvatures of at in direction are given by the eigen values of , the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of define the higher mean curvatures of , which include as extremal cases the mean curvature as the trace of and the Lipschitz–Killing curvature as its determinant.
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Principal curvature. E.V. Shikin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Principal_curvature&oldid=12102