# 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)
How to Cite This Entry:
Blaschke-Weyl formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Blaschke-Weyl_formula&oldid=46109
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article