Solving linear systems in non-Euclidean geometries?

Question

I am interested in knowing the method(s) used to solve systems of linear equations in non-Euclidean geometries, i.e., geometries where Euclid's fifth postulate is false.

Also, how exactly can these solutions be interpreted and used? If we have a system of linear equations and we solve it on a spherical surface, for example, what additional information do we then have about the system's properties?

Answer 1

A "linear system" of equations, which can be solved by matrix means, does depend on linearity. A linear system of equations has the property that solutions are always explicitly available, which is not true of non-linear systems of equations. The idea of forming a "linear system of equations" based upon non-Euclidiean geometries is an intriquing one, but the chief problem lies in finding those quantities in such non-Euclidean geometries that can form the basis of LINEAR equations! For example, it's not hard to devise a "system of equations" in terms of spherical coordinates, but explicit algebraic solutions cannot be expected in the general case, because it's not a linear system of equations.

So, let's see now....?

Maybe one thing we can do is to consider a mapping from cartesian coordinates onto a spherical map, like for example an inverse Mercator projection. Then linear equations which are represented by straight lines in the cartesian plane would be represented by loxodromes on the sphere? So then, yes, we could consider a "linear system" of loxodromes on a sphere, and solutions would be found by ordinary matrix means. This is an example of a morphism, and many others should be possible. Let's see now...

In fact, a rich field of investigation might be found in conformal mappings of functions of a complex variable, because they have the property perserving orthogonality, and the mapping is entire, meaning that the entire x-y complex plane is remapped onto itself with no pieces missing. Yet, a "linear system of equations" which is possible in the x-y complex plane is mapped onto itself, so that we can have a complex system of curves in place of striaght lines. Now, that would be another morphism.

And then we can have "local geometries", as with Lie Geometries, which do exhibit linearities, and are excellent tools for dealing with non-linear geometries, but that's now getting beyond the scope of this forum.

Answer 2

The system's properties would then not be directly precise due to the fact that the 5th postulate is false. Stick with the Euclidean linear way of doing equations. All that other stuff is rubbish.