The Geometry of Non-Euclidean Complex Planes

[maline]

If the Parallel Axiom is just one of several possible assumptions, why is it that so many mathematical relationships seem to only be expressible in the Euclidean plane? Do planes with positive or negative curvature give analogues to the Agrand plane for complex algebra, or the Cartesian plane for, say, the representation of differentials as slopes and integrals as area?

 

[micromass]

The complex plane actually has a lot of relations to hyperbolic and spherical geometry. The great book Visual Complex Analysis goes a bit into that.

Something you should know is that most spaces studied in mathematics locally are Euclidean. Even the hyperbolic and spherical spaces are locally Euclidean, which means that they locally satisfy the parallel axiom. Even the spaces studied in physics and general relativity are locally Euclidean. Curvature is something that really shows up more in global situations (it shows up locally too but it's very small, so everything is approximately Euclidean).

Doing integral and differential calculus on spaces with curvature is definitely possible and is studied in differential geometry. Trigonometry on such spaces is possible as well.

 

[maline]

The complex plane actually has a lot of relations to hyperbolic and spherical geometry. The great book Visual Complex Analysis goes a bit into that.

Something you should know is that most spaces studied in mathematics locally are Euclidean. Even the hyperbolic and spherical spaces are locally Euclidean, which means that they locally satisfy the parallel axiom. Even the spaces studied in physics and general relativity are locally Euclidean. Curvature is something that really shows up more in global situations (it shows up locally too but it's very small, so everything is approximately Euclidean).

Doing integral and differential calculus on spaces with curvature is definitely possible and is studied in differential geometry. Trigonometry on such spaces is possible as well.

Thanks so much! Can you you give me a few more details? For instance, can multiplication of complex numbers be interpreted geometrically in a curved plain?