Charts on submersions (manifolds).

[center o bass]

As far as I have understood it submersions play the analogous role in manifold theory to quotient spaces in topology. Now is that suppose that we have submersion $\pi: M \to N$ with $M$ having a certain differential structure, then how is the differential structure of $N$ related to that of $M$ if we want $M$ to be a manifold? Or more specifically how are the charts on $N$ related to those on $M$?

 

[jgens]

The standard relationship: Suppose p:Mⁿ→N^m is a submersion and (x¹,...,x^m) is a coordinate system around some point p(x). Then there exist maps y^k:M→R (where 1≤ k ≤ n-m) such that (x¹p,...x^mp,y¹,...,y^n-m) is a coordinate system around x. So this gives a relationship between charts on M and N and just about every book on manifolds will prove this fact.

 

[center o bass]

Alright. Those charts define the differential structure on the submersion. Can one say something about what charts are not compatible with this structure?

If $(U,\psi)$ s.t. $\psi: N \to \mathbb{R}^n$ is a chart of the type you mentioned above. If we wonder if a chart $(V,\phi)$ is compatible with the differential structue, can one say something about what the transition function $\phi \circ \psi^{-1}(x)$ must satisfy?

 

[center o bass]

Does it for example make sense to you that the only allowable coordinate transition functions on $\mathbb{R} \times S^1$ constructed from a submersion $\pi: \mathbb{R}^2 \to \mathbb{R} \times S^1$ is of the form

$$x' = f(x) \ \ y' = cy + g(x) \ \ $$

where $y$ is a coordinate on the circle and c is a constant?

 

[jgens]

Can one say something about what charts are not compatible with this structure?

I am unsure. Someone more knowledgeable than me might know definitively. My hunch, however, is that even if the submersion does give conditions on compatibility these will likely be no easier to check than simply verifying (non)differentiability of the transition functions. Is this question motivated by a concrete problem or just a random thought?

Does it for example make sense to you that the only allowable coordinate transition functions on $\mathbb{R} \times S^1$ constructed from a submersion $\pi: \mathbb{R}^2 \to \mathbb{R} \times S^1$ is of the form

$$x' = f(x) \ \ y' = cy + g(x) \ \ $$

where $y$ is a coordinate on the circle and c is a constant?

What does "contructed from a submersion" mean in this context? In the context above a submersion p:M→N essentially starts with a charts on N and then pulls this back to charts on M. It sounds like you want to turn this procedure around, which is possibly doable (I have not given it much thought), but in that case you can probably generate every chart on N this way and classifying these guys is hopeless. Maybe I am missing something here.

 

[center o bass]

I am unsure. Someone more knowledgeable than me might know definitively. My hunch, however, is that even if the submersion does give conditions on compatibility these will likely be no easier to check than simply verifying (non)differentiability of the transition functions. Is this question motivated by a concrete problem or just a random thought?

It is motivated by a concrete problem in Kaluza-Klein theory where one assumes a five-dimensional spacetime manifold $M^5$ with a five-dimensional metric tensor $g$. Then one assumes that the space is compactified to a topology $\mathbb{M}^4\times S^1$. The analogue coordinate transformations as in the case with cylinder ($x'^\mu = f^\mu(x^\nu)$ and $y' = cy + g(x^\mu)$ for $\mu = 0,1,2,3$) then shows that the five-dimensional metric tensor reduces to a four-dimensional metric tensor and a vector - the vector can be interpreted as the electromagnetic vector potential.

Thus I've wondered if this is something that is just assumed or rather if it is something that is forced by the mathematics in requiring that $\mathbb{M}^4\times S^1$ have a certain differential structure.

 

[jgens]

I see. I have absolutely zero familiarity with Kaluza-Klein theory so I am of no help here. Hopefully someone with knowledge about this can chime in.

 

[center o bass]

I am unsure. Someone more knowledgeable than me might know definitively. My hunch, however, is that even if the submersion does give conditions on compatibility these will likely be no easier to check than simply verifying (non)differentiability of the transition functions. Is this question motivated by a concrete problem or just a random thought?



What does "contructed from a submersion" mean in this context? In the context above a submersion p:M→N essentially starts with a charts on N and then pulls this back to charts on M. It sounds like you want to turn this procedure around, which is possibly doable (I have not given it much thought), but in that case you can probably generate every chart on N this way and classifying these guys is hopeless. Maybe I am missing something here.

I have tried to read up on the subject and I think a much more precise formulation would be the following:

Consider a space M with a group G acting transitively on M . This could well be the isometry group, but it does not necessary need to be so.

1. Find a discrete subgroup Γ ⊂ G which acts properly discontinuous on M.
2. Construct the quotient M/Γ, given by the identification of points in M under the action of Γ. Hence, define an equivalence relation ∼: p ∼ q if there exists a γ ∈ Γ such that γ(p) = q. The quotient M/Γ is then the quotient M/ ∼.
3. If the action of Γ is free, then M/Γ is a smooth manifold.

In Kaluza-Klein theory which I mentioned above a Killing field, or equivalently an one-parameter isometry group is assumed so G would then be this group. The y-coordinate on the 'circle' would then (i think) be a coordinate on $\Gamma$.

 

[center o bass]

I see. I have absolutely zero familiarity with Kaluza-Klein theory so I am of no help here. Hopefully someone with knowledge about this can chime in.

I've tried the physics forum. No help there either. It's really just mathematical question about if the stated coordinate transformations are forced by the math or assumed.

 

[center o bass]

I see. I have absolutely zero familiarity with Kaluza-Klein theory so I am of no help here. Hopefully someone with knowledge about this can chime in.

I think a simpler model to relate to would be to consider $\mathbb{R}^2$ with the metric tensor $g = dx \otimes dx + dy\otimes dy$. We wish to construct a quotient manifold $\mathbb{R} \times S^1 = \mathbb{R}^2 /\Gamma$ with it's canonical differential structure (which is the unique structure such that$\pi: \mathbb{R}^2 \to \mathbb{R}^2/\Gamma$ smooth) from this Riemannian space. The vector $\partial_x$ would then certainly be a Killing vector field, so let's use it's one-parameter isometry group as G and construct a quotient manifold according to the above procedure.

Can we then say something like that the coordinate transformation

$$x' = f(x), \ \ y' = c y + g(x)$$

are forced if we demand that all charts be compatible with the differential structure on $\mathbb{R}^2/\Gamma$, where $y$ is a coordinate on $\Gamma$?