  We give a complete solution of a problem in submanifold theory posed and
partially solved by the eminent algebraic geometer Pierre Samuel in 1947.
Namely, to determine all pairs of immersions of a given manifold into Euclidean
space that have the same Gauss map and induce conformal metrics on the
manifold. The case of isometric induced metrics was solved in 1985 by the first
author and D. Gromoll.
