Classification of flat slant surfaces in complex Euclidean plane

In memory of Professor Seiichi Yamaguchi

抄録

It is well-known that the classification of flat surfaces in Euclidean 3-space is one of the most basic results in differential geometry. For surfaces in the complex Euclidean plane \bm{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 \bm{C}² are slant surfaces, i.e., surfaces with constant Wintinger angle. In this paper the author completely classifies flat slant surfaces in \bm{C}². The main result states that, beside the totally geodesic ones, there are five large classes of flat slant surfaces in \bm{C}². Conversely, every non-totally geodesic flat slant surfaces in \bm{C}² is locally a surface given by these five classes.