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Transforming to curved manifolds

  1. Sep 6, 2006 #1
    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 [itex](\theta,\phi)[/itex], the co-ordinates usually used for the unit 2-sphere? I say this since the first space is flat, and the second is curved.
     
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  3. Sep 6, 2006 #2

    George Jones

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    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.
     
  4. Sep 6, 2006 #3
    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?
     
  5. Sep 6, 2006 #4

    HallsofIvy

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    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.
     
  6. Sep 6, 2006 #5
    Many thanks, to the both of you.
     
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