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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:
 
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169101.png" /></td> </tr></table>
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$$Edu^2+2Fdudv+Gdv^2,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169102.png" /></td> </tr></table>
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$$Ldu^2+2Mdudv+Ndv^2,$$
  
 
the first one of which is positive definite, and let the coefficients of these forms satisfy the Gauss equations (cf. [[Gauss theorem|Gauss theorem]]) and the [[Peterson–Codazzi equations|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.
 
the first one of which is positive definite, and let the coefficients of these forms satisfy the Gauss equations (cf. [[Gauss theorem|Gauss theorem]]) and the [[Peterson–Codazzi equations|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169103.png" /> at all its points, then the external diameter of this surface is smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169104.png" />; this estimate cannot be improved. Stated by O. Bonnet in 1855.
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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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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) {{MR|0350630}} {{ZBL|0264.53001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) {{MR|}} {{ZBL|0326.53001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) {{MR|0350630}} {{ZBL|0264.53001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) {{MR|}} {{ZBL|0326.53001}} </TD></TR></table>
  
Bonnet's theorem on the mean value, second mean-value theorem [[#References|[2]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169105.png" /> be integrable functions on a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169106.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169107.png" /> be a positive decreasing function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169108.png" />; then there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b0169109.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b01691010.png" /> for which the equality
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Bonnet's theorem on the mean value, second mean-value theorem [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b01691011.png" /></td> </tr></table>
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$$\int\limits_a^bf(x)\phi(x)dx=\phi(a)\int\limits_a^\xi f(x)dx$$
  
is valid. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b01691012.png" /> is merely required to be monotone, Bonnet's theorem states that there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b01691013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b01691014.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016910/b01691015.png" /></td> </tr></table>
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$$\int\limits_a^bf(x)\phi(x)dx=\phi(a)\int\limits_a^\xi f(x)dx+\phi(b)\int\limits_\xi^bf(x)dx$$
  
 
is true.
 
is true.

Latest revision as of 14:47, 14 August 2014

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.


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 $f(x),\phi(x)$ 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

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
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