Preprint A103/2001
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.