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 $1/A^2$ 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.
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.
|[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 : 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
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
|[1a]||O. Bonnet, J. École Polytechnique , 24 (1865) pp. 204–230|
|[1b]||O. Bonnet, J. École Polytechnique , 25 (1867) pp. 1–151|
|||O. Bonnet, "Rémarques sur quelques intégrales définies" J. Math. Pures Appl. , 14 (1849) pp. 249–256|
Bonnet's original article is [a1].
|[a1]||O. Bonnet, C.R. Acad. Sci. Paris , 40 (1855) pp. 1311–1313 MR2017144 MR1888473 MR1860929 MR1248760 MR1226112 Zbl 1067.14068 Zbl 1047.14031 Zbl 1037.14018 Zbl 0801.57023 Zbl 0801.57022|
Bonnet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bonnet_theorem&oldid=32920