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

TL;DR

This paper characterizes time-like surfaces with zero mean curvature in 4D neutral space forms, linking their curvature, twistor lifts, and differential invariants, revealing conditions for flat normal connection and differential nullity.

Contribution

It provides new criteria relating curvature, twistor lifts, and differential forms for time-like surfaces in neutral space forms with zero mean curvature.

Findings

Curvature equals the ambient space curvature iff twistor derivatives are zero or light-like.

Normal connection is flat when curvature equals ambient space curvature.

Holomorphic quartic differential is zero or null iff twistor derivatives are zero or light-like.

Abstract

Let $M$ be a Lorentz surface and $F:M\rightarrow N$ a time-like and conformal immersion of $M$ into a 4-dimensional neutral space form $N$ with zero mean curvature vector. We see that the curvature $K$ of the induced metric on $M$ by $F$ is identically equal to the constant sectional curvature $L_0$ of $N$ if and only if the covariant derivatives of both of the time-like twistor lifts are zero or light-like. If $K\equiv L_0$, then the normal connection $\nabla^{\perp}$ of $F$ is flat, while the converse is not necessarily true. We see 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 see that $K$ is identically equal to $L_0$ if and only if not only $\nabla^{\perp}$ is flat but also $Q$ is zero or null.