Rigidity of pseudo-isotropic immersions

Several notions of isotropy of a (pseudo-)Riemannian manifold have been introduced in the literature, in particular, the concept of pseudo-isotropic immersion. The aim of this paper is to look more closely at this notion of pseudo-isotropy and then to study the rigidity of this class of immersion into the pseudo-Euclidean space. It is worth pointing out that we first obtain a characterization of the pseudo-isotropy condition by using tangent vectors of any causal character. Then, rigidity theorems for pseudo-isotropic immersions are proved, and in particular, some well known results for the Riemannian case arise. Later, we bring together the notions of pseudo-isotropy, intrinsically and extrinsically isotropic manifolds, and prove interesting relations among them. Finally, we pay special attention to the case of codimension two Lorentz surfaces.