# Convergence of a sequence of points on a manifold

1. Jun 25, 2011

### 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?

2. Jun 25, 2011

### micromass

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.

3. Jun 26, 2011

### yifli

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 $k>N$?

4. Jun 26, 2011

### micromass

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 $\phi_2$.