Year: 2025
Type: article
Abstract: Let M be a three-dimensional Lorentz manifold and let Z be a non-parallel, closed conformal spacelike vector field. A timelike surface $$\Sigma $$ Σ in M is said to have a canonical null direction with respect to Z if the projection $$Z^{\top }$$ Z ⊤ on the tangent space of $$\Sigma $$ Σ gives a lightlike vector field. We prove that these surfaces are not minimal. We also find that they are ruled, whose rulings are lines of curvature and lightlike geodesics in M. When M is the Minkowski space $$\mathbb {R}_1^3$$ R 1 3 and Z is a radial vector field, we proved that $$\Sigma $$ Σ is a quasi-umbilic surface and it is not flat. Subsequently, we found a condition for quasi-umbilic surfaces to be surfaces with canonical null direction. We explored the case where $$\Sigma $$ Σ lies in De Sitter space. Here, we found that $$\Sigma $$ Σ is neither of constant mean curvature nor flat. Later, we analyzed the case where $$\Sigma $$ Σ is in a warped product $$I\times _{\rho }N$$ I × ρ N . We can assume that the surface $$\Sigma $$ Σ is the graph of a function $$f:N \rightarrow \mathbb {R}$$ f : N → R . Then, it has a canonical null direction with respect to $$\rho \partial _t$$ ρ ∂ t if and only if the gradient of f is a lightlike vector field on N. Another property is that the Gaussian curvature K of $$\Sigma $$ Σ and the Gaussian curvature $$K_N$$ K N of N satisfy that $$K=\frac{1}{\rho ^2}K_N$$ K = 1 ρ 2 K N . Finally, we analyzed when $$\Sigma $$ Σ is the inverse image of a function.
Institution: Universidad Nacional Autónoma de México
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4410923170