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 $\phi_2$.

 

[blue_raver22]

I would like to first clarify that the definition of convergence on a manifold is a mathematical concept and may not always have a physical interpretation. In this case, the sequence of points converging to the north pole of a sphere may not have a physical significance and may simply be used as an example for understanding the concept of convergence on a manifold.

Now, to address the question at hand, the reason why the sequence of points is not convergent is because the charts used in the definition of convergence are not compatible with each other. In other words, the transition map between the charts, which is necessary for the compatibility of charts on a manifold, is not defined for this particular case.

In the given example, the transition map between the charts \phi_1 and \phi_2 is given by \phi_2 \circ \phi_1^{-1}, which results in a singularity at the point (0,0) in the x-y plane. This singularity causes the sequence of points to diverge and not converge to the north pole.

In summary, the lack of compatibility between charts in this case results in the sequence of points not being convergent, highlighting the importance of compatible charts in the definition of convergence on a manifold.