On the infinitesimal affine rigidity of ellipsoids in the \(n\)-space
Ellipsoids are rigid in the sense of the Blaschke/Berwald hypersurface theory in the equiaffine space \( A^{n} \). As a counterpart to this theorem by \textit{R. Schneider} [Math. Z. 101, 375-406 (1967; Zbl 0156.20101)], the author proves the following: Every ellipsoid in the \( n \)-dimensional space \( A^{n} \) is infinitesimally \( S \)-rigid. This has not been known so far, except for \( n = 3 \). A hypersurface in \( A^{n} \) is called infinitesimally \( S \)-rigid if each infinitesimal deformation of it satisfying \( \delta S = 0 \), where \( S \) is the Einstein scalar curvature of the Blaschke-Berwald metric, is trivial. The method is that of partial differential equations, leading finally to eigenfunctions of the Laplace-Beltrami operator for eigenvalues \( n-1 \) and \( 2n \). Then the theory of spherical harmonics can be applied.
scientific article; zbMATH DE number 1309649