# 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 |

**How to Cite This Entry:**

Principal curvature.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Principal_curvature&oldid=12102