Herglotz formula

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)