The complex structure of a point
in the Plane is defined by the linear Map
More generally, if is a 2-D Vector Space, a linear map such that is called a complex structure on . If , this collapses to the previous definition.
References
Gray, A. Modern Differential Geometry of Curves and Surfaces.Boca Raton, FL: CRC Press,
pp. 3 and 229, 1993.