Imersões isométricas k-umbílicas em formas espaciais

Abstract

Let x be an isometric immersion in its second fundamental form. Newton's polynomials are defined inductively by Po = I, Pk = Sk I-A Pk-1, where Sk is the immersion k-curvature. This work defines k-umbilicity by the proviso that the product of A by the Newton polynomial of k-1 is a multiple of identity. As a general consequence of the k-umbilicity concept, it is shown that if a k-umbilical isometric immersion has a zero main curvature, then it has n-k + 1 null principal curvatures and also shows that in every k-umbilical immersion the perk Lk is elliptical whenever Sk is nonzero, considering Lk (f) = dash (Pk Hess (f)) a second-order differential operator. For the case k = 2, it shows that all 2-umbilical immersion in spatial forms has S2 constant. It also partially classifies the closed 2-umbilical hypersurfaces on the unit sphere, exhibiting an enumerable family of such immersions.