The quadratic slice theorem and the equiaffine tube theorem for equiaffine Dupin hypersurfaces
This is an interesting paper. It presents a theory of Dupin hypersurfaces of equiaffine immersions, generalizing definitions of Niebergall and Ryan for the case of so called Blaschke hypersurfaces. In the Riemannian hypersurface theory, a Dupin hypersurfaces, is characterized as a hypersurface whose focal set consists of some connected submanifolds. In the affine (or equiaffine) hypersurface theory, the author considers the corresponding hypersurfaces that are not necessarily Blaschke hypersurfaces as in the paper of \textit{R. Niebergall} and \textit{P. J. Ryan} [Trans. Am. Math. Soc. 348, No. 3, 1093--1115 (1996; Zbl 0860.53008)]. There are five major theorems for the class considered under the additional assumption that the affine shape operator is diagonizable, but there are no examples or applications. There are still many open questions left, however it is worthwhile to have the author's results in print as they will provide a useful framework for further work.
scientific article; zbMATH DE number 2174563