Transnormality and the infinitesimal rectangle property (Q1116460)

\textit{L. W. Danzer} [Proc. Symp. Pure Math. 7, 99-100 (1963; Zbl 0135.225)] and \textit{T. Zamfirescu} [Geom. Dedic. 27, No.2, 209-212 (1988; Zbl 0654.52001)] studied the global and the infinitesimal rectangle property for curves in the plane. Here the investigations are extended to smooth closed curves in 3-space and differentially immersed hypersurfaces in \(E^ n\) (n\(\geq 3).\) The infinitesimal rectangle property is proved to characterize transnormal space curves, the global version additionally implies that these curves are spherical and symmetric with respect to the center of the sphere. Examples show that the infinitesimal version does not imply the global rectangle property and that the latter does not characterize circles among space curves. Furthermore all transnormal spherical curves which are centrally symmetric must satisfy the infinitesimal rectangle property. In contrast to these results the infinitesimal version is shown to characterize hyperspheres among differentiably immersed closed hypersurfaces. The case of higher codimension is open.