Convergence of a sequence of points on a manifold

[yifli]

I have a question regarding the following definition of convergence on manifold:
Let M be a manifold with atlas A. A sequence of points \{x_i \in M\} converges to x\in M if

1. there exists a chart (U_i,\phi_i) with an integer N such that x\in U_i and for all k>N,x_i\in U_i
2. \phi_i(x_k)_{k>N} \rightarrow \phi_i(x)

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 U_1 (resp., upper semi-sphere U_2) from the south pole (resp., north pole) to the x-y plane, i.e.,
\phi_1(x_1,x_2,x_3)=\langle \frac{x_1}{1+x_3}, \frac{x_2}{1+x_3} \rangle
\phi_2(x_1,x_2,x_3)=\langle \frac{x_1}{1-x_3}, \frac{x_2}{1-x_3} \rangle

Since the sequence of points converge to the north pole, we can find an N such that x_k \in U_2, k > N; however, \phi_2(x_k) \rightarrow \infty, which means this sequence is not convergent. How come?

 

[micromass]

Hi yifli!

I have a question regarding the following definition of convergence on manifold:
Let M be a manifold with atlas A. A sequence of points \{x_i \in M\} converges to x\in M if

1. there exists a chart (U_i,\phi_i) with an integer N such that x\in U_i and for all k>N,x_i\in U_i
2. \phi_i(x_k)_{k>N} \rightarrow \phi_i(x)

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 U_1 (resp., upper semi-sphere U_2) from the south pole (resp., north pole) to the x-y plane, i.e.,
\phi_1(x_1,x_2,x_3)=\langle \frac{x_1}{1+x_3}, \frac{x_2}{1+x_3} \rangle
\phi_2(x_1,x_2,x_3)=\langle \frac{x_1}{1-x_3}, \frac{x_2}{1-x_3} \rangle

Since the sequence of points converge to the north pole, we can find an N such that x_k \in U_2, k > N; however, \phi_2(x_k) \rightarrow \infty, which means this sequence is not convergent. How come?

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.

 

[yifli]

Hi yifli!



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.

If I choose the chart (U_1,\phi_1), where U_1 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 x_k \in U_1 for k>N?

 

[micromass]

If I choose the chart (U_1,\phi_1), where U_1 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 x_k \in U_1 for k>N?

Oh, but I think there is a mistake. Your map \phi_1 isn't a map for the lower semi-sphere. Indeed, the value \phi_1(0,0,-1) isn't well-defined.
I think you'll need to swap \phi_1 and \phi_2.