Timelike surfaces in a Lorentz manifold with a canonical null direction

Abstract Let M be a three-dimensional Lorentz manifold and let Z be a non-parallel, closed conformal spacelike vector field. A timelike surface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> in M is said to have a canonical null direction with respect to Z if the projection $$Z^{\top }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mi>⊤</mml:mi> </mml:msup> </mml:math> on the tangent space of $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> 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 $$\mathbb {R}_1^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> </mml:math> and Z is a radial vector field, we proved that $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> 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 $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> lies in De Sitter space. Here, we found that $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> is neither of constant mean curvature nor flat. Later, we analyzed the case where $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> is in a warped product $$I\times _{\rho }N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msub> <mml:mo>×</mml:mo> <mml:mi>ρ</mml:mi> </mml:msub> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . We can assume that the surface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> is the graph of a function $$f:N \rightarrow \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . Then, it has a canonical null direction with respect to $$\rho \partial _t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> </mml:math> if and only if the gradient of f is a lightlike vector field on N . Another property is that the Gaussian curvature K of $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> and the Gaussian curvature $$K_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> of N satisfy that $$K=\frac{1}{\rho ^2}K_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:math> . Finally, we analyzed when $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> is the inverse image of a function.