Concerning charts on a manifold

In summary: By the previous exercise, the family {V_k} is finite, and has a maximum, call it V_0. In summary, there can be charts of a smooth manifold that do not overlap with any other charts, such as in the case of R^n as a manifold. However, in a maximal smooth atlas, charts can be restricted to proper subsets and still overlap with other charts. Additionally, open sets in a manifold can have pairwise-disjoint closures, which would result in a disconnected manifold.
  • #1
hansenscane
6
0
I'm new to manifolds, so please forgive me if this sounds ignorant. I was just wondering whether the charts of a smooth manifold (within some atlas) always "overlap". If I'm not mistaken they map to open subsets of R^n, and being homeomorphisms should have the inverse image as open. But I'm not entirely sure, so I'd appreciate it if somebody could either confirm my understanding or explain what I'm missing.
 
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  • #2
There can be charts that do not overlap any other. Take for instance R^n itself as a manifold. There is an atlas for this manifold consisting of the single chart id:R^n-->R^n. So trivially, this chart intersects no other since there are no other. More generally, if a manifold has a connected component C homeomorphic to an open subset U of R^n, then there is an atlas for this manifold with a chart mapping C to U and intersecting no other chart.
 
  • #3
Of course, you almost always deal with a maximal (possibly oriented) smooth atlas, which is defined to be an atlas which contains every possible chart that would be compatible with it (that's a loose paraphrase, but it conveys the idea). Then you can restrict a chart to a proper subset of its domain, and that will be a different chart also in the atlas, and of course they overlap.
 
  • #4
If your manifold is the union of open sets with pairwise-disjoint closure, then it is disconnected.
 
  • #5
Thanks you guys, that cleared a lot up. :smile:
 
  • #6
I'm worried you guys are painting a misleading picture -- you're answering "can they overlap" but not "must they overlap".

For any open interval contained in the real line, there is a coordinate chart that covers only that interval and nothing else.

So, for example, a chart covering (0,1) and a chart covering (2,3) have empty overlap.


(And for the record, the empty set is an open subset)
 
  • #7
Actually, using the fact that n is the Lebesgue Covering Dimension, an n-manifold
has, in each component, a refinement of each cover in which each element is covered
by a maximum of n+1 charts.

To give a proof to the previous claim that a manifold M covered by a union
of pairwise-disjoint open sets must be disconnected ( I think that disjoint closure
is not necessary):

Let U_(i in I) {V_i} cover M, with V_j/\V_k ={ }

Then any V_k is open in M, but also closed in it, since its complement in M is the
union of a collection of open sets V_k- \/{V_i}-V_k.
 

Related to Concerning charts on a manifold

1. What is a manifold?

A manifold is a mathematical concept that describes a space that is locally similar to Euclidean space. In other words, a manifold is a space that looks flat when zoomed in on a small enough scale, but may have a more complex structure when viewed on a larger scale.

2. How are charts used to describe a manifold?

Charts are used to map points on a manifold to points in a Euclidean space, typically with the use of coordinates. These charts can then be used to define the geometry and topology of the manifold, allowing for mathematical analysis and visualization.

3. What is the purpose of using charts on a manifold?

Using charts on a manifold allows for the application of standard tools and techniques from Euclidean geometry and calculus. This makes it easier to study the properties of the manifold and make predictions or calculations about it.

4. Can charts be used to describe any type of manifold?

Yes, charts can be used to describe any type of manifold, including smooth manifolds, topological manifolds, and more abstract mathematical objects. However, the specific type of charts and their properties may vary depending on the type of manifold being described.

5. How do charts relate to the concept of a tangent space on a manifold?

Charts are used to define a coordinate system on a manifold, which in turn allows for the definition of tangent vectors at each point. These tangent vectors make up the tangent space at that point, providing a way to measure changes and rates of change on the manifold.

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