# Bonnet theorem

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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:

$$Edu^2+2Fdudv+Gdv^2,$$

$$Ldu^2+2Mdudv+Ndv^2,$$

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 $1/A^2$ at all its points, then the external diameter of this surface is smaller than $\pi A$; this estimate cannot be improved. Stated by O. Bonnet in 1855.

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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.

How to Cite This Entry:
Bonnet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bonnet_theorem&oldid=32920
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article