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An integral relation between two closed isometric oriented regular surfaces. Let local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h0469601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h0469602.png" /> be introduced on the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h0469603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h0469604.png" /> so that equality of the coordinates realizes an isometric mapping. Let
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An integral relation between two closed isometric oriented regular surfaces. Let local coordinates $u$ and $v$ be introduced on the surfaces $S_1$ and $S_2$ so that equality of the coordinates realizes an isometric mapping. Let
  
<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/h/h046/h046960/h0469605.png" /></td> </tr></table>
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$$ds^2=Edu^2+2Fdudv+Gdv^2$$
  
be the first fundamental form, having the same coefficients for both surfaces in the given coordinates, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h0469606.png" /> be the Gaussian curvature, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h0469607.png" /> be the mean curvatures, and let
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be the first fundamental form, having the same coefficients for both surfaces in the given coordinates, let $K$ be the Gaussian curvature, let $H_\alpha$ be the mean curvatures, and let
  
<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/h/h046/h046960/h0469608.png" /></td> </tr></table>
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$$\sqrt{EG-F^2}(\lambda_\alpha du^2+2\mu_\alpha dudv+\nu_\alpha dv^2)$$
  
be the second fundamental forms of the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h0469609.png" />. Herglotz' formula then takes the following form:
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be the second fundamental forms of the surfaces $S_\alpha$. Herglotz' formula then takes the following form:
  
<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/h/h046/h046960/h04696010.png" /></td> </tr></table>
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$$\int\limits_{S_1}\begin{vmatrix}\lambda_2-\lambda_1&\mu_2-\mu_1\\\mu_2-\mu_1&\nu_2-\nu_1\end{vmatrix}(\mathbf n,\mathbf x)d\tau=\int\limits_{S_2}H_2d\tau-\int\limits_{S_1}H_1d\tau,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h04696011.png" /> is the position vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h04696012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h04696013.png" /> is the unit vector of the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h04696014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046960/h04696015.png" /> is the surface element. It was obtained by G. Herglotz [[#References|[1]]].
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where $\mathbf x=\mathbf x(u,v)$ is the position vector of $S_1$, $\mathbf n$ is the unit vector of the normal to $S_1$ and $d\tau$ is the surface element. It was obtained by G. Herglotz [[#References|[1]]].
  
 
====References====
 
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Latest revision as of 14:03, 12 August 2014

An integral relation between two closed isometric oriented regular surfaces. Let local coordinates $u$ and $v$ be introduced on the surfaces $S_1$ and $S_2$ so that equality of the coordinates realizes an isometric mapping. Let

$$ds^2=Edu^2+2Fdudv+Gdv^2$$

be the first fundamental form, having the same coefficients for both surfaces in the given coordinates, let $K$ be the Gaussian curvature, let $H_\alpha$ be the mean curvatures, and let

$$\sqrt{EG-F^2}(\lambda_\alpha du^2+2\mu_\alpha dudv+\nu_\alpha dv^2)$$

be the second fundamental forms of the surfaces $S_\alpha$. Herglotz' formula then takes the following form:

$$\int\limits_{S_1}\begin{vmatrix}\lambda_2-\lambda_1&\mu_2-\mu_1\\\mu_2-\mu_1&\nu_2-\nu_1\end{vmatrix}(\mathbf n,\mathbf x)d\tau=\int\limits_{S_2}H_2d\tau-\int\limits_{S_1}H_1d\tau,$$

where $\mathbf x=\mathbf x(u,v)$ is the position vector of $S_1$, $\mathbf n$ is the unit vector of the normal to $S_1$ and $d\tau$ is the surface element. It was obtained by G. Herglotz [1].

References

[1] G. Herglotz, "Ueber die Starrheit von Eiflächen" Abh. Math. Sem. Univ. Hamburg , 15 (1943) pp. 127–129
[2] N.V. Efimov, "Qualitative questions of the theory of deformations of surfaces" Uspekhi Mat. Nauk , 3 : 2 (1948) pp. 47–158 (In Russian)


Comments

This formula can be used to prove rigidity or congruence theorems for surfaces. For related formulas and results see [a1].

References

[a1] H. Huck, R. Roitzsch, U. Simon, W. Vortisch, R. Walden, B. Wegner, W. Wendland, "Beweismethoden der Differentialgeometrie im Grossen" , Lect. notes in math. , 335 , Springer (1973)
[a2] W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German)
How to Cite This Entry:
Herglotz formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Herglotz_formula&oldid=32870
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article