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 , and are the coefficients of the first fundamental form, while , and are the coefficients of the second fundamental form of the surface, computed at the given point.
The half-sum of the principal curvatures and of the surface gives the mean curvature, while their product is equal to the Gaussian curvature of the surface. Equation (*) may be written as
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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_curvature&oldid=12102