Complex Structure (original) (raw)
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The complex structure of a point 
in the plane is defined by the linear map 
 |
(1) |
|---|
and corresponds to a counterclockwise rotation by 
. This map satisfies
 |
 |
 |
(2) |
|---|---|---|---|
 |
 |
 |
(3) |
 |
 |
 |
(4) |
where 
is the identity map.
More generally, if 
is a two-dimensional vector space, a linear map 
such that 
is called a complex structure on 
. If 
, 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.
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Cite this as:
Weisstein, Eric W. "Complex Structure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexStructure.html