The twistor lifts of surfaces in 4-spaces

Abstract

This is a survey of the twistor lifts of surfaces in $4$-dimensional spaces. In most part of this survey, the space is Euclidean $4$-space $E^4$. The definitions of the Gauss maps and the twistor lifts of surfaces in $E^4$ are given by orthogonal complex structures of $E^4$. Based on these definitions, we can understand holomorphicity of the Gauss maps of minimal surfaces in $E^4$ and isotropicity of such surfaces.