Reducibility of Dupin submanifolds

Ruy Tojeiro | Dajczer, Marcos | Florit, Luis A.

Keywords: Dupin submanifolds | Ribaucour Transform

In this paper we locally characterize those proper Dupin hypersurfaces that are reducible with respect to one of its principal curvatures, where reducibility should now be understood in the following broader sense. We say that a hypersurface is reducible with respect to a principal curvature r if the conullity of r, that is, Im(A-rI) where A stands for the second fundamental form of the hypersurface, is integrable, a property that is also invariant under Lie transformations. The starting goal of this paper was to give an answer to the last problem for arbitrary codimension. In the process, we were naturally led to deal with a much more general situation. Namely, to describe the structure of a submanifold of arbitrary codimension that carries a 'Dupin principal normal' with integrable conullity. This problem is important on its own right in the theory of submanifolds in space forms, and it is completely solved in the paper. A key observation in the characterization of submanifolds carrying a Dupin principal normal with integrable conullity is that the leaves of the conullity distribution of such a submanifold are always Ribaucour transforms one of each other. This is in the sense of the extended notion of Ribaucour transformation for submanifolds of arbitrary dimension and codimension developed from the classical notion for surfaces of Euclidean space. This observation can be seen as a generalization of the classical fact that the orthogonal surfaces of a cyclic system are Ribaucour transforms one of each other. In order to turn the above observation into an explicit description of all such submanifolds, it was convenient to introduce the notion of N-Ribaucour transform of a submanifold h carrying a flat parallel normal subbundle N. This is an explicitly parametrized submanifold foliated by Ribaucour transforms of h, each one corresponding to a parallel section of N. The orthogonal distribution to this foliation is precisely the nullity distribution of a Dupin principal normal. One of the main results of this paper is that any submanifold that carries a Dupin principal normal with integrable conullity arises locally this way.

Anexos:

• dupin.dvi