Resumen de Tension field, iterated Laplacian, type number and Gauss maps

Chen Bang-Yen

• Let M be a Riemannian manifold. By applying the finite type theory we study maps from M into a Euclidean space whose tension field is an eigenmap of a p-iterated Laplacian for some natural number p. First, we prove that such maps are either of 1-type, of null 2-type, or of infinite type. Several examples are then given to illustrate that this result is sharp. Some applications of this result are also presented. The simplest examples of maps whose tension field is an eigenmap of an iterated Laplacian are those which have constant tension field. Next, we study hypersurfaces whose (classical or spherical) Gauss map has constant tension field. Finally, we prove that every spherical hypersurface with 2-type spherical Gauss map must have non-constant mean curvature.