Rigidity of pseudo-isotropic immersions

Abstract

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 pseudoisotropy and then to study the rigidity of this class of immersion into the pseudoEuclidean 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.

Isotropic submanifolds of pseudo-Riemannian spaces

“…A pseudo-Riemannian submanifold is isotropic if, roughly speaking, the geometry of the submanifold is the same regardless of direction. This notion remind us of the cosmological principle that no matter where we look in the Universe, we still see the same distribution of objetcs (see [8] p. 835 and Prop. 5.1).…”

“…(d) The composition of isotropic immersions is an isotropic immersion, and the isotropy function of the composition is the sum of the corresponding isotropy functions. Now we provide a characterization of isotropic immersions [8].…”

Null pseudo-isotropic Lagrangian surfaces

“…This leads in a natural way to the notions of (i) Timelike pseudo-isotropic if, for any point p and any timelike tangent vector v at a point p, equation (1.1) is satisfied, (ii) Spacelike pseudo-isotropic if, for any point p and any spacelike tangent vector v at a point p, equation (1.1) is satisfied, (iii) Lightlike isotropic if, for every lightlike vector v at the point p, we have that h (v, v) is again a lightlike vector. It was shown in [2] that the notions of pseudo-isotropic, timelike pseudo-isotropic and spacelike pseudo-isotropic are equivalent. In the same paper they also included an example of an immersion which is lightlike pseudo-isotropic but not pseudo-isotropic.…”

Lorentzian isotropic Lagrangian immersions

“…The notion of isotropic submanifold was first introduced in [8] by O'Neill for immersions of Riemannian manifolds and recently extended by Cabrerizo, Fernandez and Gomez in [2] to the pseudo-Riemannian case. A submanifold is called isotropic if and only if for any point p and any tangent vector X at a point p, we have that h(X, X), h(X, X) = λ(p) X, X 2 ,…”