Transforming to curved manifolds

[masudr]

I know that the Riemann tensor vanishes in a flat space. And no amount of co-ordinate transformations can go from a flat space to a curved space.

Does that mean there is no transformation that will go from, say Cartesian 2D, to (\theta,\phi), the co-ordinates usually used for the unit 2-sphere? I say this since the first space is flat, and the second is curved.

[George Jones]

Topoloically, the 2-dimensional plane is non-compact, while S^2, the 2-dimensional (surface of a) sphere is compact, so, globally, these two are not "the same", i.e., not topologically isomorphic.

[masudr]

Thanks for your reply. I did know that, but is there no co-ordinate transformation (however exotic) that can make me think I'm on the plane instead of the sphere?

[HallsofIvy]

No, there is no such coordinate transformation. There are, of course, transformations for the surface of a sphere that are 'approximately' flat and will look like R2 on a sufficiently small region.

[masudr]

Many thanks, to the both of you.