Classification of flat slant surfaces in complex Euclidean plane

Abstract

It is well-known that the classification of flat surfaces in Euclidean 3space is one of the most basic results in differential geometry. For surfaces in the complex Euclidean plane $C^2$ endowed with almost complex structure $J$, flat surfaces are the simplest ones from intrinsic point of views. On the other hand, from $J$-action point of views, the most natural surfaces in $C^2$ are slant surfaces, i.e., surfaces with constant Wintinger angle. In this paper the author completely classifies flat slant surfaces in $C^2$. The main result states that, beside the totally geodesic ones, there are five large classes of flat slant surfaces in $C^2$. Conversely, every non-totally geodesic flat slant surfaces in $C^2$ is locally a surface given by these five classes.