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Difference between revisions of "Bonnet theorem"

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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall (1976)</TD></TR></table>
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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>
  
 
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
 
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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====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> O. Bonnet,   ''J. École Polytechnique'' , '''24''' (1865) pp. 204–230</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> O. Bonnet,   ''J. École Polytechnique'' , '''25''' (1867) pp. 1–151</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Bonnet,   "Rémarques sur quelques intégrales définies" ''J. Math. Pures Appl.'' , '''14''' (1849) pp. 249–256</TD></TR></table>
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<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> O. Bonnet, ''J. École Polytechnique'' , '''24''' (1865) pp. 204–230</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> O. Bonnet, ''J. École Polytechnique'' , '''25''' (1867) pp. 1–151</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Bonnet, "Rémarques sur quelques intégrales définies" ''J. Math. Pures Appl.'' , '''14''' (1849) pp. 249–256</TD></TR></table>
  
 
''T.Yu. Popova''
 
''T.Yu. Popova''
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Bonnet,   ''C.R. Acad. Sci. Paris'' , '''40''' (1855) pp. 1311–1313</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Bonnet, ''C.R. Acad. Sci. Paris'' , '''40''' (1855) pp. 1311–1313 {{MR|2017144}} {{MR|1888473}} {{MR|1860929}} {{MR|1248760}} {{MR|1226112}} {{ZBL|1067.14068}} {{ZBL|1047.14031}} {{ZBL|1037.14018}} {{ZBL|0801.57023}} {{ZBL|0801.57022}} </TD></TR></table>

Revision as of 17:31, 31 March 2012

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
How to Cite This Entry:
Bonnet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bonnet_theorem&oldid=18094
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article