Complex Structure
The complex structure of a point x=x_1,x_2 in the plane is defined by the linear map J:R^2->R^2
| J(x_1,x_2)=(-x_2,x_1), |
(1)
|
and corresponds to a counterclockwise rotation by pi/2. This map satisfies
J^2 = -I
(2)
(Jx)·(Jy) = x·y
(3)
(Jx)·x = 0,
(4)
where I is the identity map.
More generally, if V is a two-dimensional vector space, a linear map J:V->V such that J^2=-I is called a complex structure on V. If V=R^2, this collapses to the previous definition.
See also
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References
Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 4 and 247, 1997.Referenced on Wolfram|Alpha
Complex StructureCite this as:
Weisstein, Eric W. "Complex Structure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexStructure.html