Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms

Abstract

Let M be a Lorentz surface and F:M→N a time-like and conformal immersion of M into a 4-dimensional neutral space form N with zero mean curvature vector. We show that the curvature K of the induced metric on M by F is identically equal to the constant sectional curvature L0 of N if and only if the covariant derivatives of both of the time-like twistor lifts are zero or light-like. If K≡L0, then the normal connection ∇⟂ of F is flat, while the converse is not necessarily true. We also prove that a holomorphic paracomplex quartic differential Q on M defined by F is zero or null if and only if the covariant derivative of at least one of the time-like twistor lifts is zero or light-like. In addition, we get that K is identically equal to L0 if and only if not only ∇⟂ is flat but also Q is zero or null