## Complex Structure

The complex structure of a point in the Plane is defined by the linear Map

and corresponds to a clockwise rotation by . This map satisfies

where is the Identity 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.