What is the meaning of constant on each others fibres in differential geometry?

[Kreizhn]

This should hopefully be a quick and easy answer.

I'm running through Lee's Introduction to Smooth Manifolds to brush up my differential geometry. I love this book, but I've come to something I'm not sure about. He states a result for which the proof is an exercise:

Proposition: Suppose [itex] \pi_1: M\to N_1, \pi_2: M \to N_2 [/itex] are surjective submersions that are constant on each others fibres. Then there exists a unique diffeomorphism [itex] F: N_1 \to N_2 [/itex] such that [itex] F \circ \pi_1 = \pi_2 [/itex].

I'm not quite clear on what he means by "constant on each others fibres." It is okay to assume that [itex] M, N_1, N_2 [/itex] are smooth manifolds, but there is no mention of them being fibre bundles. Does he just mean the preimage of points in [itex] N_1, N_2[/itex]? That is, the fibre of [itex]q \in N_1 [/itex] in M would be [itex] M_q = \pi_1^{-1}(q) [/itex]? This seems reasonable since we assumed that [itex] \pi_1,\pi_2 [/itex] are surjective and hence this is well defined, but I've never heard "fibres" used in this manner before.

If this is the case, what does it mean to be constant on each others fibres? Does this mean that if [itex] q_1 \in N_1[/itex] then [itex] \pi_2(\pi_1^{-1}(q_1)) = c(q_1) [/itex], where [itex] c(q) [/itex] is some constant function that varies only with q?

 

[Hurkyl]

What you've said looks like how I would interpret those words. (but disclaimer: I am not an expert in this field!)

 

[Kreizhn]

Thanks for the support Hurkyl. I can't think of another way of possibly interpreting it, and it is unlike Lee to be ambiguous in such a statement. I guess I should try proving the result with these assumptions and see if it works.

 

[Bacle]

You may want to visit Lee's errata page to save yourself some time:

http://www.math.washington.edu/~lee/Books/Smooth/errata.pdf

I don't have time now to check whether he addressed your point.

 

[arkajad]

That means [tex]\pi_2[/tex] is constant on the fibers of [tex]\pi_1[/tex] and [tex]\pi_1[/tex] is constant on the fibers of [tex]\pi_2[/tex].

 

[quasar987]

Does this mean that if [itex] q_1 \in N_1[/itex] then [itex] \pi_2(\pi_1^{-1}(q_1)) = c(q_1) [/itex], where [itex] c(q) [/itex] is some constant function that varies only with q?

Yes, and also that [itex] q_2 \in N_2[/itex] then [itex] \pi_1(\pi_2^{-1}(q_2)) = c'(q_2) [/itex], where [itex] c'(q) [/itex] is some constant function that varies only with q (as arkajad just said).

Given a function f:A-->B and b in B, the fiber of f over b is by definition f^-1({b}).

For instance, the (rough) idea of a fiber bundle over a space B is that of a space E such that E is the union of all the fibers of some map p:E-->B.

So, yes, the meaning of the word "fiber" here is the same as when used in the term "fiber bundles".

 

[Kreizhn]

Thanks everyone for your contributions.

Quasar, is that really just the definition of a fibre? The preimage of a point in any map? I just want to make sure - without ever having seen one formally defined, I just inferred from the definition of a fibre bundle that each fibre would need to have the same dimension, and there would need to be a local trivialization.

Now maybe it's possible that surjective submersions are indeed bundle projection maps? I've been thinking about this. Surjective submersions are open maps, and hence are quotient maps. We can then view [itex] N_1 [/itex] as a quotient space with [itex] \pi_1 [/itex] identifying elements of the same equivalence class. I think we can get the local trivialization using charts from [itex] M [/itex], so it comes down to this:

We know that equivalence classes by [itex] \pi_1 [/itex] will partition [itex] M [/itex]. Must it necessarily form equal partitions, each of the same dimension?

 

[arkajad]

See "http://en.wikipedia.org/wiki/Foliation" " and in particular the note on Submersions there.

 

[Kreizhn]

Thanks for the link. Does the map have to be a submersion for pre-images to be fibers though? Unfortunately, the wiki article again uses fibres without defining them, and I really want to make sure that I know the rigorous definition.

Also, what if the map is a submersion but not surjective? Then do we only define fibres on elements in the image of the map?

 

[Kreizhn]

On an interesting note, I just found the wiki-article on fibres. The reason I couldn't find it before is that I search for "Fibres (mathematics)" and wiki couldn't figure out my CanE spelling of Fibre and connect that to "Fibers (mathematics)". It took until the 28th entry on the search page to get there.

Anyway, the wiki page suggests that it holds for any map. Sorry to waste your time on this.