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Cone as surface

  1. Nov 14, 2011 #1
    cone as surface....

    In Barrett O'Neill's Elementary Differential Geometry book...he says that the cone
    M:[itex]x^{2}[/itex] + [itex]y^{2}[/itex]=[itex]z^{2}[/itex] is not a surface in that
    there exists a point p in M such that there exists no proper patch in M which can cover a neighbourhood of p in M
    Intuitively I realise that this is point is the apex...here [itex]\left(0,0,0\right)[/itex]
    but how would the patches for the rest of points be??...and what goes wrong with the patch at the apex of the cone??
    Note: A PATCH IS A 1-1 REGULAR FUNCTION X:Dā†’[itex]ā„^{3}[/itex]..D open in [itex]ā„^{2}[/itex]
    This is an exercise problem.
  2. jcsd
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