[1rudeboy]

Ok, let's throw Euclidean space out the window. [there it goes]

Wouldn't your theory depend on whether space is positively or negatively curved?

[LeGrande]

“Wouldn’t your theory depend on whether space is positively or negatively curved?”

We are dealing with space-time which is curved. Space can be flat or positively or negatively curved by itself, but that is only half of the equation.

This illustration may help you to visualize it : ) http://curious.astro.cornell.edu/question.php?number=649