Why has to be M,N be orientable?

  • Thread starter Canavar
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In summary, the conversation discusses the problem of proving that if MxN is orientable, then M and N must also be orientable. One person has solved the direction "<=" with no problem but is struggling with the other direction. They mention using different criteria for orientability such as atlases and differential forms. The other person suggests relating the volume form on MxN to those on M and N, possibly using the product structure of MxN. The first person clarifies that they want to prove that if M is not orientable, then MxN is not orientable, but they are unsure how to do so. The second person suggests using the relationship between the volume forms.
  • #1
Canavar
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Hello,

I want to solve this Problem:

If M,N are manifolds then MxN is orientable iff M,N is orientable.



I have solved the direction "<=" This was no problem.

But i have a lot of problems to solve the other direction!

Let us assume that MxN are orientable.
Why has to be M,N be orientable?

We have defined orientable in different ways. Once per atlases and also with a differential non-vanishing form...
But i could't see a connection.

Can you please help me?

Regards
 
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  • #2


If MxN is orientable, it has a nonvanishing volume form. Can you relate this form to forms on M and N? Presumably the way you proved the other part of theorem would shed some light on this.
 
  • #3


Hello,

There are different äquivalent criteria for a manifold to be orianted.
A manifold is oriented if it has an atlas, s.d. the differential of the coordinate changes have positiv determinant.
Therefore the product MxN has a canonically atlas, which also satisfy this criterion.

I hope you understand me.
We can assume that MxN has an atlas A, s.t. forall maps x,y in A we have
det d(x\circ y^{-1})>0. That is MxN is orientable.

But why has M to be orientable, if we assume that MxN has to be?

Regards
 
  • #4


You have to use the product structure of MxN in your proof. Is there any way to write an atlas on MxN in terms of an atlas on M and an atlas on N?
 
  • #5


Yes of course there is! But not any atlas has to be in that form! This is a real problem.

For example, if (U_i,f_i) is a atlas for M and (V_i,g_i) is a atlas for N => (U_i x V_i, f_i x g_i) is a atlas for MxN.
But it has not to be in this form. That is f,g can depend on both, the element in M and N.

I mean in general the maps have the form (U_i x V_i, h) with h(x,y).

What can i do?
 
  • #6


A single atlas satisfying the correct conditions should be enough to show orientabliltiy.

If it's too hard to construct, you might try using differential forms, which seems simpler.
 
  • #7


Oh i think it is a misunderstanding. You are right, we inly need a single atlas, which is oriented, to show that a manifold is oriented.
But I want to show something else. i want to show tha MxN is not orientable, if M is not. That is i have to show that any atlas is not orientable of MxN!

I hope the problem is now clear.
Once again:
I want to show:

If M is not orientable =>MxN is not orientable

I have no idea how i can show this. Neither with atlases nor with differential forms.

Regards
 
  • #8


There is a relationship between the volume form on MxN and those on M and N. You can use that.
 

1. Why is orientability important in mathematics and science?

Orientability is important because it allows us to define a consistent direction or orientation for a surface or object. This is crucial in many mathematical and scientific concepts, such as vector fields, integration, and differential equations.

2. What does it mean for a surface to be orientable?

A surface is orientable if it can be assigned a consistent direction or orientation at every point on the surface. This means that if you were to walk along the surface, you would always know which direction you were facing.

3. How can you determine if a surface is orientable?

One way to determine if a surface is orientable is to check for the existence of a continuous, non-vanishing tangent vector field on the surface. If such a vector field exists, then the surface is orientable. Another way is to look for a consistent orientation of the normal vectors at each point on the surface.

4. What happens if a surface is non-orientable?

If a surface is non-orientable, it means that it cannot be consistently assigned a direction or orientation at every point. This can lead to mathematical and physical inconsistencies, and certain concepts and equations may not be applicable to non-orientable surfaces.

5. Can any surface be made orientable?

No, not all surfaces can be made orientable. For example, the Möbius strip is a non-orientable surface that cannot be continuously deformed into an orientable surface. However, some non-orientable surfaces can be made orientable by adding additional dimensions or "twists" to the surface.

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