HARMONIC MORPHISMS AND SUBMANIFOLDS WITH CONFORMAL SECOND FUNDAMENTAL FORMS

We show that surfaces such that the natural projections of the unit normal bundles are harmonic morphisms are composed of minimal points and totally umbilical points. As its application, we find a harmonic map from the torus to the complex quadric in CP^3 such that the projection map of the associated sphere bundle constructed by Gudmundsson is not a harmonic morphism. This contrasts sharply with the situation for holomorphic maps. We also establish sufficient conditions for reducing the codimension of an isometric immersion with conformal second fundamental form.