Convergence of a sequence of points on a manifold

In summary, the definition of convergence on a manifold states that a sequence of points converges to a point on the manifold if there exists a chart in the atlas of the manifold such that the points are contained in the chart after a certain integer N, and the corresponding values of the points in the chart converge to the value of the point on the manifold. However, in the case of a sphere with two charts, the sequence of points converging to the north pole does not converge because the north pole is not an element of the chart that should contain it. This can be resolved by swapping the charts in the atlas.
  • #1
yifli
70
0
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?
 
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  • #2
Hi yifli! :smile:

yifli said:
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?

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
micromass said:
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.

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]?
 
  • #4
yifli said:
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]?

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].
 
  • #5


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.
 

Related to Convergence of a sequence of points on a manifold

What is the definition of convergence of a sequence of points on a manifold?

The convergence of a sequence of points on a manifold is a mathematical concept that describes the behavior of a sequence of points on a manifold as the number of points in the sequence increases.

What is the significance of convergence of a sequence of points on a manifold?

The concept of convergence is important in understanding the behavior of a sequence of points on a manifold and can help us determine whether a sequence will approach a certain point or diverge in different directions.

How is convergence of a sequence of points on a manifold different from convergence in Euclidean space?

Convergence of a sequence of points on a manifold is different from convergence in Euclidean space because on a manifold, the points are not necessarily moving closer to each other as the sequence progresses, but rather approaching a specific point on the manifold.

What are the conditions for convergence of a sequence of points on a manifold?

The conditions for convergence of a sequence of points on a manifold include the existence of a limit point and the points in the sequence approaching the limit point as the number of points in the sequence increases.

How is convergence of a sequence of points on a manifold related to topology?

Convergence of a sequence of points on a manifold is closely related to topology, as it can help us determine the topology of a manifold by examining the behavior of the sequence of points. It also plays a crucial role in studying continuity and differentiability on manifolds.

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