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) 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 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 MR2017144 MR1888473 MR1860929 MR1248760 MR1226112 Zbl 1067.14068 Zbl 1047.14031 Zbl 1037.14018 Zbl 0801.57023 Zbl 0801.57022