Spacelike surfaces in Lorentzian manifolds (Q1924636)

The author studies spacelike surfaces in Lorentz-Minkowski 4-space \(L^4\). First he shows that minimal isotropic surfaces are locally graphs of harmonic functions on a domain in the plane. To do that he mainly uses the moving frame method. For oriented surfaces he considers its Gauss map into the Grassmannian \(G^s_2(3,1)\) of oriented spacelike planes in \(L^4\). Then the following result is shown: Let \(f\colon M \to L^4\) be an oriented spacelike surface and let \(n_f\colon M \to G^s_2(3,1)\) be its Gauss map. Then \(n_f\) is harmonic if and only if \(f\) has parallel mean curvature vector field. This result is extended for spacelike surfaces in any 4-dimensional Lorentz manifold. Namely, conditions to achieve the harmonicity of the Gauss lift of a spacelike surface are given. Finally, for Codazzi surfaces with parallel mean curvature vector field the author finds some interesting formulas relating umbilicity, curvature and Euler characteristic of the surface.