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Anyhow, given a Riemann manifold ##(M,g)## I would like to take the underlying set of ##M## and give it another metric ##h##, hence create a new manifold ##(N,h)##. Furthermore assume I have a family of equations defining function and vector fields on ##M## that I would like to carry over to ##N##. Assuming I can find a smooth map ##\phi: M \rightarrow N## shouldn't I be able to do that? Those equations are supposed to represent the laws of physics i.e. differential equations defining the time evolution of fields and whatnot. So the general question is, what are the restricting factors between switching the geometry of my physics space at will?

A simple example would be starting with ##(M,g)## being the Euclidean space in ## \mathbb {R}^3## and defining ##(N,h)## via the spherical coordinates mapping and taking canonical metric from its codomain to define ##h## thus in the spherical coordinates ##h## will be represented by an identity matrix. Well, for sake of dodging problems I take out the entire polar region.

As an example for family of equations I would just pick only Newtons equation of motion, perhaps in Earth's gravity field centered at ##(0,0,0)## (which should be the geodesic equation with a term added accounting for the gravity field). Yeah, for the sake of the joke I'd like to know if I can correctly/equivalently describe physics in a "flat earth" geometry :D. The underlying thing I want to know is whether the geometry of the physical space is just a convention where the laws of physics take the simplest form or whether there is more to it.