Blaschke-Weyl formula

A variant of the Green formulas for the rotation field $ y $ of an infinitesimal deformation of a surface with position vector $ \mathbf x $:

$$ 2 {\int\limits \int\limits } _ { G } \left ( \mathbf x \frac{\partial \mathbf y }{\partial u } \frac{\partial \mathbf y }{\partial v } \right ) du dv = \ \oint _ {\partial G } ( \mathbf x\mathbf y d \mathbf y ) . $$

The proof and the idea of applying the Blaschke–Weyl formula to demonstrate the rigidity of ovaloids is due to W. Blaschke [1] and to H. Weyl [2]. For other applications see [3]. The Herglotz formula is an analogue of the Blaschke–Weyl formula. The formula has been generalized to the case of infinitesimal deformations of surfaces in spaces of constant curvature.

References

[1]	W. Blaschke, "Ein Beweis für die Unverbiegbarkeit geschlossener konvexer Flächen" Gött. Nachr. (1912) pp. 607–610
[2]	H. Weyl, "Ueber die Starrheit der Eiflächen und konvexer Polyeder" Sitzungsber. Akad. Wiss. Berlin (1917) pp. 250–266 (Also: Gesammelte Abh., Vol. 1, Springer, 1968, pp. 646–662)
[3]	N.V. Efimov, "Qualitative problems in the theory of deformations of surfaces" Uspekhi Mat. Nauk , 3 : 2 (24) (1948) pp. 47–158 (In Russian) (Translated into German as book)
[4]	W. Blaschke, "Einführung in die Differentialgeometrie" , Springer (1950)