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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

Moduli Space

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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 Structure

Cite this as:

Weisstein, Eric W. "Complex Structure." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexStructure.html

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