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:

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

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

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.

Contents

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) MR0350630 Zbl 0264.53001 [a2] M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) Zbl 0326.53001

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

$$\int\limits_a^bf(x)\phi(x)dx=\phi(a)\int\limits_a^\xi f(x)dx$$

is valid. If $\phi(x)$ is merely required to be monotone, Bonnet's theorem states that there exists a point $\xi$ in $[a,b]$ such that

$$\int\limits_a^bf(x)\phi(x)dx=\phi(a)\int\limits_a^\xi f(x)dx+\phi(b)\int\limits_\xi^bf(x)dx$$

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