Airysche Spannungsfunktion und isotrope Differentialgeometrie
In the classical plane elasticity theory the Airy function \(z(x, y)\) plays an important role satisfying the equation \(\Delta\Delta z(x, y) = 0\), The author observes that this equation is invariant with respect to the group \(G_6\) of transformations \[x' = a + x \cos \varphi - y \sin \varphi,\quad y' =b + x \sin \varphi + y \cos \varphi,\quad z' = c + c_1x + c_2y +z \] which defines the isotropic geometry already developed by the same author [Math. Z. 47, 743--777 (1942; Zbl 0026.26301); 48, 369--427 (1942; Zbl 0027.25301); 50, 1--94 (1944; Zbl 0063.07216)], with the metric defined by the invariant form \(ds^2 = dx^2 + dy^2\). lt is therefore to be expected that the isotropic geometry of the Airy surface \(z = z(x, y)\) yields the elasticity properties. That this is so is actually proved by showing that all invariant notions concerning a surface in isotropic geometry (like isotropic normal curvature, geodesic torsion, principal curvatures, mean curvature, curvature lines) do have a definite meaning in elasticity theory. The interesting link so put in evidence between this theory and isotropic geometry is apt to give further results.
differential geometry in Euclidean spaces
scientific article; zbMATH DE number 3174087