Could a Non-Euclidean Complex Plane Be Defined Using Differential Geometry?

[Klaus_Hoffmann]

do not know if such generalization exist, my question is...

we have Euler identity e^{ix}=cos(x)+isin(x)

considering that complex plane defined by real and complex part, is Euclidean (a,b) but could we define using the axioms and tools of Differential Geommetry a Non-Euclidean complex plane ?? with different Euler identities.. for example if space is Elliptic instead of an exponential we would have an elliptic function.

and the same with Quaternions, since Space-time is a four dimensional manifold perhaps there would be an isomorphism between the quaternions and the usual curved space-time.

 

[fresh_42]

The Euler identity is about numbers. This has nothing to do with any representation in the complex plane. Of course you can draw it there, and deform the plane anyway you like, so that your drawing will be deformed as well. But you cannot re-attach this to a numerical equation. Addition and multiplication are Euclidean.