Orientability of 1-Dimensional Manifolds: A Closer Look

In summary, it is proven that any 1-dimensional topological manifold is either the real line or the circle. Furthermore, every 1-dimensional topological manifold is orientable in the sense of orientation on simplices. To show orientability, a nowhere vanishing 1-form is needed. For the real line, the differential dx works as the 1-form, but for the circle, dx is not everywhere non-zero. Instead, two angular coordinate charts can be used to define a well-defined nowhere vanishing 1-form. It should be noted that "manifold" refers to a special case of "manifold with boundary", and that connectedness and second-countability are important considerations in the definition of a manifold.
  • #1
jem05
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I have the result that any 1 dim topological manifold is either R or S1. And I have the fact that every 1-dim topological manifold is orientable in the sense of orientation on simplices.

i want to get that any 1-dim manifold (smooth) is orientable, where orientability is given by the existence of a nowhere vanishing 1-form. Since i know my manifold is either the real line or the circle, does the section s:M --> T∗M that takes each point p to the differential dx at p work as the 1-form i need?
 
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  • #2
For the real line, this argument works, because R is covered by a single coordinate chart. But the circle cannot be covered by a single coordinate chart (why?), so dx only defines a 1-form locally on some open subset of the circle.

So what you try to do for the circle is to find a collection of charts U_1,...,U_n such that dx^i coincides with dx^j on U_i n U_j. This way, the 1-form a defined by "a=dx^i (on U_i)", is a well defined nowhere vanishing 1-form.

Hint: use two angular coordinate charts.
 
  • #3
jem05 said:
I have the result that any 1 dim topological manifold is either R or S1. And I have the fact that every 1-dim topological manifold is orientable in the sense of orientation on simplices.

i want to get that any 1-dim manifold (smooth) is orientable, where orientability is given by the existence of a nowhere vanishing 1-form. Since i know my manifold is either the real line or the circle, does the section s:M --> T∗M that takes each point p to the differential dx at p work as the 1-form i need?

dx works on the line

dtheta works on the circle.

dx is not everywhere non-zero on the circle.
 
  • #4
Minor nitpick: every connected one-dimensional topological manifold without boundary is R or S1.
 
  • #5
"Manifold" is a special kind of "manifold with boundary", not the other way around. A "manifold with boundary" that has a non-empty boundary is, in fact, not a manifold at all. (Of course, if you are considering "manifolds with boundary", you might decide to rename "manifold" to "manifold without boundary" to avoid this trap of language)

The connected bit is relevant though. (Also, we need second-countable, although some people opt to include that in their definition of manifold)
 

1. What is a 1-dimensional manifold?

A 1-dimensional manifold is a mathematical concept that describes a smooth, curved shape that can be embedded in a higher-dimensional space. It is characterized by the fact that locally, it resembles a straight line.

2. What is the orientation of a 1-dimensional manifold?

The orientation of a 1-dimensional manifold refers to the direction in which the manifold is traversed. It can be clockwise or counterclockwise, or have no defined orientation at all.

3. How is the orientation of a 1-dimensional manifold determined?

The orientation of a 1-dimensional manifold is determined by a choice of basis vectors at each point on the manifold. These basis vectors can be used to define a consistent direction for traversing the manifold.

4. Can the orientation of a 1-dimensional manifold change?

No, the orientation of a 1-dimensional manifold is fixed and cannot be changed. However, the choice of basis vectors used to determine the orientation can be changed.

5. Why is the orientation of a 1-dimensional manifold important?

The orientation of a 1-dimensional manifold is important in mathematical and scientific applications, as it can affect the results of calculations and experiments. It also plays a role in understanding the global structure of the manifold.

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