Isothermic submanifolds of Euclidean space

Abstract

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.