Complex Structure

The complex structure of a point ${\bf x}={x_1, x_2}$ in the Plane is defined by the linear Map $J: \Bbb{R}^2\to\Bbb{R}^2$

$$J(x_1, x_2)=(-x_2, x_1),$$

and corresponds to a clockwise rotation by $\pi/2$. This map satisfies

$$J^2 = -I$$

$$(J{\bf x})\cdot(J{\bf y}) = {\bf x}\cdot{\bf y}$$

$$(J{\bf x})\cdot{\bf x} = 0,$$

where $I$ is the Identity Map.

More generally, if $V$ is a 2-D Vector Space, a linear map $J:V\to V$ such that $J^2=-I$ is called a complex structure on $V$. If $V=\Bbb{R}^2$, 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.