Surfaces in the 3-dimensional Lorentz-Minkowski space satisfying Δx=Ax+B

Hypersurfaces in the Lorentz-Minkowski space satisfying L k ψ = Aψ + b

“…. , E n } in which S adopts its canonical form, and we need to distinguish four cases, according to the canonical form of the shape operator, see (1).…”

“…In this case, the shape operator of the hypersurface needs not be diagonalizable, condition which plays a chief role in the Riemannian case. The first attempt in this line was made by the first author, jointly with Alías and Ferrández, in [1], where the authors classify the surfaces x : M 2 s → L 3 in the 3-dimensional Lorentz-Minkowski space L 3 satisfying the condition x = Ax + B. Later on [2] (see also [3]), they also studied pseudo-Riemannian submanifolds M n s in pseudo-Euclidean spaces R n+m t satisfying the condition x = Ax + B, where A is a constant endomorphism of R n+m t and B is a constant vector in R n+m t , and gave a characterization theorem.…”

Submanifolds in pseudo-Euclidean spaces satisfying the condition Δx=Ax+B

“…It seems reasonable to hope for a richer classification when working on pseudo-Riemannian submanifolds in an indefinite Euclidean space. The first attempt in this line has been made by the authors in [1], where they classify the surfaces in the 3-dimensional Lorentz-Minkowski space ~_a satisfying the quoted condition. Now we are going to generalize that work not only by considering hypersurfaces but also taking them in any pseudo-Euclidean space.…”

Surfaces of revolution in the three dimensional pseudo-Galilean space

“…In other words, each coordinate function is of 1-type in the sense of Chen ([6]). For the Lorentzian version of surfaces satisfying (1.2), Alías, Ferrández and Lucas ( [1]) proved that the only such surfaces are minimal surfaces and open pieces of Lorentz circular cylinders, hyperbolic cylinders, Lorentz hyperbolic cylinders, hyperbolic spaces or pseudo-spheres.…”