Timelike surfaces in [formula omitted] with a canonical null direction

Abstract

Let N be a Lorentz surface and let ∂t be a unit vector field on the Lorentz manifold N×R tangent to R. A timelike surface Σ in N×R is said to have a canonical null direction with respect to ∂t if the projection ∂t⊤ on the tangent space of Σ gives a lightlike vector field. We prove that these surfaces are minimal and ruled, whose rulings are lightlike geodesics and lines of curvature. Other results include the fact that these surfaces are flat if they are totally geodesic. Furthermore, let us assume that the surface Σ is the graph of a function f:N→R. Then it has a canonical null direction if and only if the gradient of f is a lightlike vector field on N. Another property is that the Gaussian curvature of Σ coincides with the Gaussian curvature of N and the sectional curvature of N×R, along tangent planes to Σ. In particular such surfaces in S12×R have constant curvature. We prove, using isothermal coordinates, that functions on N whose gradient is lightlike locally always exist. We give several examples of such surfaces.