# Bonnet theorem

Bonnet's theorem on the existence and the uniqueness of a surface with given first and second fundamental forms . Let the following two quadratic forms be given:

the first one of which is positive definite, and let the coefficients of these forms satisfy the Gauss equations (cf. Gauss theorem) and the Peterson–Codazzi equations. Then there exists a surface, which is unique up to motions in space, for which these forms are, respectively, the first and the second fundamental forms.

Bonnet's theorem on the diameter of an oval surface: If the curvature of an oval surface is larger than or equal to at all its points, then the external diameter of this surface is smaller than ; this estimate cannot be improved. Stated by O. Bonnet in 1855.

#### Comments

A proof of this theorem of Bonnet may be found in [a1] or [a2]. The Peterson–Codazzi equations are usually called the Mainardi–Codazzi equations, cf. [a1], after G. Mainardi (1857) and D. Codazzi (1868), who established them.

#### References

[a1] | W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) |

[a2] | M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) |

Bonnet's theorem on the mean value, second mean-value theorem [2]: Let be integrable functions on a segment and let be a positive decreasing function of ; then there exists a number in for which the equality

is valid. If is merely required to be monotone, Bonnet's theorem states that there exists a point in such that

is true.

#### References

[1a] | O. Bonnet, J. École Polytechnique , 24 (1865) pp. 204–230 |

[1b] | O. Bonnet, J. École Polytechnique , 25 (1867) pp. 1–151 |

[2] | O. Bonnet, "Rémarques sur quelques intégrales définies" J. Math. Pures Appl. , 14 (1849) pp. 249–256 |

*T.Yu. Popova*

#### Comments

Bonnet's original article is [a1].

#### References

[a1] | O. Bonnet, C.R. Acad. Sci. Paris , 40 (1855) pp. 1311–1313 |

**How to Cite This Entry:**

Bonnet theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Bonnet_theorem&oldid=18094