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 $ k _ {1} $ and $ k _ {2} $ are the roots of the quadratic equation

$$ \tag{* } \left | \begin{array}{ll} L - kE &M - kF \\ M - kF &N - kG \\ \end{array} \right | = 0, $$

where $ E $, $ F $ and $ G $ are the coefficients of the first fundamental form, while $ L $, $ M $ and $ N $ are the coefficients of the second fundamental form of the surface, computed at the given point.

The half-sum of the principal curvatures $ k _ {1} $ and $ k _ {2} $ of the surface gives the mean curvature, while their product is equal to the Gaussian curvature of the surface. Equation (*) may be written as

$$ k ^ {2} - 2Hk + K = 0, $$

where $ H $ is the mean, and $ K $ is the Gaussian curvature of the surface at the given point.

The principal curvatures $ k _ {1} $ and $ k _ {2} $ are connected with the normal curvature $ \widetilde{k} $, taken in an arbitrary direction, by means of Euler's formula:

$$ \widetilde{k} = k _ {1} \cos ^ {2} \phi + k _ {2} \sin ^ {2} \phi , $$

where $ \phi $ is the angle formed by the selected direction with the principal direction for $ k _ {1} $.

Comments

In the case of an $ m $- dimensional submanifold $ M $ of Euclidean $ n $- space $ E ^ {n} $ principal curvatures and principal directions are defined as follows.

Let $ \xi $ be a unit normal to $ M $ at $ p \in M $. The Weingarten mapping (shape operator) $ A _ \xi $ of $ M $ at $ p $ in direction $ \xi $ is given by the tangential part of $ - \overline \nabla \; _ {\overline \xi \; } $, where $ \overline \nabla \; $ is the covariant differential in $ E ^ {n} $ and $ \overline \xi \; $ is a local extension of $ \xi $ to a unit normal vector field. $ A _ \xi $ does not depend on the chosen extension of $ \xi $. The principal curvatures of $ M $ at $ p $ in direction $ \xi $ are given by the eigen values of $ A _ \xi $, the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of $ A _ \xi $ define the higher mean curvatures of $ M $, which include as extremal cases the mean curvature as the trace of $ A _ \xi $ 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