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Convergence of a sequence of points on a manifold

  1. Jun 25, 2011 #1
    I have a question regarding the following definition of convergence on manifold:
    Let M be a manifold with atlas A. A sequence of points [itex]\{x_i \in M\}[/itex] converges to [itex]x\in M[/itex] if
    1. there exists a chart [itex](U_i,\phi_i)[/itex] with an integer [itex]N[/itex] such that [itex]x\in U_i[/itex] and for all [itex]k>N,x_i\in U_i[/itex]
    2. [itex]\phi_i(x_k)_{k>N} \rightarrow \phi_i(x)[/itex]

    Given a sphere (2-manifold) centered at origin and a sequence of points converging to the north pole. The atlas of this sphere contains two charts, which projects all the points on the lower semi-sphere [itex]U_1[/itex] (resp., upper semi-sphere [itex]U_2[/itex]) from the south pole (resp., north pole) to the x-y plane, i.e.,
    [tex]\phi_1(x_1,x_2,x_3)=\langle \frac{x_1}{1+x_3}, \frac{x_2}{1+x_3} \rangle[/tex]
    [tex]\phi_2(x_1,x_2,x_3)=\langle \frac{x_1}{1-x_3}, \frac{x_2}{1-x_3} \rangle[/tex]

    Since the sequence of points converge to the north pole, we can find an N such that [itex]x_k \in U_2, k > N[/itex]; however, [itex]\phi_2(x_k) \rightarrow \infty[/itex], which means this sequence is not convergent. How come?
  2. jcsd
  3. Jun 25, 2011 #2
    Hi yifli! :smile:

    The problem is that the north pole is not an element of U2, so it isn't a good chart. You'll need the chart U1 for a sequence converging to the north pole, and the sequence does converge in this chart.
  4. Jun 26, 2011 #3
    If I choose the chart [itex](U_1,\phi_1)[/itex], where [itex]U_1[/itex] is the lower semi-sphere, then the first condition is not met because the north pole is in the upper semi-sphere; also, since the points converge to the north pole, how do you find an integer N such that [itex]x_k \in U_1[/itex] for [itex]k>N[/itex]?
  5. Jun 26, 2011 #4
    Oh, but I think there is a mistake. Your map [itex]\phi_1[/itex] isn't a map for the lower semi-sphere. Indeed, the value [itex]\phi_1(0,0,-1)[/itex] isn't well-defined.
    I think you'll need to swap [itex]\phi_1[/itex] and [itex]\phi_2[/itex].
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