Spacelike surfaces in Lorentzian manifolds

Abstract

In this work we first study spacelike surfaces in the Minkowski space which are minimal or have parallel mean curvature vector and which are isotropic. Minimal isotropic surfaces in Euclidean space are essentially holomorphic curves. In Minkowski space there is no complex structure, but even so we are able to characterize minimal isotropic surfaces explicitly. We study the Gauss map of isotropic surfaces into the Grassmannian of spacelike 2-planes in the Minkowski space, and give a generalization of a theorem of Chern. For the study of spacelike surfaces in general Lorentzian manifolds, we consider the Grassmann bundle of oriented spacelike tangent planes. We extend a general construction in the Riemannian case to the construction of a pseudo-Riemannian metric on this Grassman bundle. We give necessary and sufficient conditions for the Gauss map to be harmonic. We also investigate isotropic Codazzi surfaces with parallel mean curvature.