  We study the problem posed by F. Burstall of developing a theory of
isothermic Euclidean submanifolds of dimension greater than or equal to three.
As a natural extension of the definition in the surface case, we call a
Euclidean submanifold {\it isothermic} if it is locally the image of a
conformal immersion of a Riemannian product of Riemannian manifolds whose
second fundamental form is adapted to the product net of the manifold. Our main
result is a complete classification of all such conformal immersions of
Riemannian products of dimension greater than or equal to three. We derive
several consequences of this result. We also study whether the classical
characterizations of isothermic surfaces as solutions of Christoffel's problem
and as envelopes of nontrivial conformal sphere congruences extend to higher
dimensions.
