Principal curvature

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 .

Comments

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.

References

[a1]	N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a2]	B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)
[a3]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a4]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a5]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145
[a6]	H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60
[a7]	D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)
[a8]	B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
[a9]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5