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