Complex curvilinear coordinates(?)

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SUMMARY

A system of curvilinear coordinates can indeed be expressed using complex-valued functions, leading to the concept of "Complex manifolds." This transformation involves real scalar functions represented as x_i = x_i(u_1, ..., u_n), where both x_i and u_i can take complex values. This approach effectively doubles the dimensionality of the coordinate system, resulting in 2n coordinates. Important examples of complex manifolds in physics include Calabi-Yau manifolds and Riemann surfaces, which are crucial in string theory and complex geometry.

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mnb96
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Hello,
a system of curvilinear coordinates is usually expressed by an admissible transformation represented by a set of real scalar functions x_i=x_i(u_1,\ldots,u_n).

Does it make sense to form a system of curvilinear coordinates where the x_i[/tex] and u_i[/tex] functions are <i>complex</i> valued?<br /> If that exists, I cannot find any source in the net.
 
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Wouldn't that be equivalent to a 2n coordinates? But I'm not quite sure
 

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