What is a Differential Structure on a Manifold?

[LearningDG]

Hi,
I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help!

Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for x>-2. The transition map is C-infinity in (-2,-1).
Next I consider another altas with r(x,y)=x for x<1 and s(x,y)=y for x>-1. The transition is not differentiable at 0 inside (-1,1).

Can I say I've defined two differential structures of the same manifold?
But what does it mean by a one-deminsional manifold having only one differential structure up to diffeomorphism?

Thanks!

 

[micromass]

Hi,
I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help!

Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for x>-2. The transition map is C-infinity in (-2,-1).
Next I consider another altas with r(x,y)=x for x<1 and s(x,y)=y for x>-1. The transition is not differentiable at 0 inside (-1,1).

Can I say I've defined two differential structures of the same manifold?

No, since the transition map for the second atlas is not differentiable. So it will not be a smooth atlas.

But what does it mean by a one-deminsional manifold having only one differential structure up to diffeomorphism?

It involves a notion of maximal atlas. It's certainly true that atlases are not unique. For example, if we define p(x,y)=x for x<-2 in the first atlas, then we would have obtained a different atlas. But the differential structure doesn't change: we will have the same differentiable maps.
To solve this, we see that every smooth atlas generates a maximal smooth atlas. We say that two manifolds are the same if their maximal smooth atlas is the same. And it turns out that one-dimensional manifolds have only one maximal smooth atlas.

 

[LearningDG]

Thanks Micromass! But I still got some questions.

No, since the transition map for the second atlas is not differentiable. So it will not be a smooth atlas.

That's exactly what I mean by two structures, viz., not smooth in one altas but smooth in the other. Or, should the transition map at least be C1? Then may be put this way, change my representation to a function which is smooth everywhere but only C1 at x = 0. Using the same two altas. The first altas A1 is C-inifinity. According to my understand, the maximal atlas consists of all other altas compatible A1. I can keep it C-infinity provided I don't add charts overlapping at x = 0. For my second altas A2, I can also extend it to a maximal C-1 atlas.

Are these two called different differential structures of the same manifold?

 

[micromass]

Thanks Micromass! But I still got some questions.



That's exactly what I mean by two structures, viz., not smooth in one altas but smooth in the other. Or, should the transition map at least be C1? Then may be put this way, change my representation to a function which is smooth everywhere but only C1 at x = 0. Using the same two altas. The first altas A1 is C-inifinity. According to my understand, the maximal atlas consists of all other altas compatible A1. I can keep it C-infinity provided I don't add charts overlapping at x = 0. For my second altas A2, I can also extend it to a maximal C-1 atlas.

Are these two called different differential structures of the same manifold?

Well, in order to have a differentiable manifold, your transition maps should at least be C1 (some authors even demand smoothness).
What you're talking about are topological manifolds. There the transition maps don't need to be C1. And I'm afraid that your two atlasses determine thesame topological manifold.

 

[LearningDG]

Thanks micromass!
Yes, they determine the same topological manifold. What I am not sure is if I have constructed two maximal altas of it which are not compatible (say, one is C1 and the other is smooth), can I say these are two differential structures? If yes, I guess mathematicians have proved that all the C-p altas (p>0) are diffeomorphic (meaning C-infinity diffeomorphic?). When we say two manifolds are diffeomorphic, we don't require each altas to be C-infinity themselves, right?

 

[micromass]

Thanks micromass!
Yes, they determine the same topological manifold. What I am not sure is if I have constructed two maximal altas of it which are not compatible (say, one is C1 and the other is smooth), can I say these are two differential structures? If yes, I guess mathematicians have proved that all the C-p altas (p>0) are diffeomorphic (meaning C-infinity diffeomorphic?). When we say two manifolds are diffeomorphic, we don't require each altas to be C-infinity themselves, right?

This depends on the definitions we use. Some authors do require each atlas to be C-infinity. For others, C1 is good enough.
So for some, diffeomorphic is between C-infinity atlases, for others, it will be between C1-atlases. Check the definitions in the beginning of each book to see what the author is working with...

 

[lavinia]

Hi,
I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help!

Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for x>-2. The transition map is C-infinity in (-2,-1).
Next I consider another altas with r(x,y)=x for x<1 and s(x,y)=y for x>-1. The transition is not differentiable at 0 inside (-1,1).

Can I say I've defined two differential structures of the same manifold?
But what does it mean by a one-deminsional manifold having only one differential structure up to diffeomorphism?

Thanks!

Two atlases may not be compatible yet the two differentiable manifolds may be diffeomorphic.

The classic example is the atlases with single charts, x, and x^3,on the real line. These atlases are not compatible yet the two manifolds are diffeomorphic. The map x -> x^1/3 form the x atlas manifold to the x^3 atlas manifold is a diffeomorphism - I think.

The definition of differentiable manifold only requires that the transition functions be differentiable - maybe continuously - but there are more special differentiable manifolds whose charts are required to be C^r for r>1 and also smooth manifolds where the transition functions are infinitely differentiable. There are also analytic manifolds where the transition functions are analytic.

There are theorems on when C^r manifolds have diffeomorphic smooth structures but I am not educated in them. Further there are topological manifolds that have no compatible differentiable sub-atlas. Also on some topological manifolds there are smooth structures that are not diffeomorphic.

there is also the idea of a combinatorial structure on a topological manifold. on a smooth manifold a combinatorial structure is called smooth if it derives from a smooth triangulation. There are examples of triangulations of topological manifolds that do not come from any smooth triangulation.

 

[LearningDG]

Thanks a lot Lavinia.
So correct me if I am wrong:
For the same manifold, if I have two different atlas, say one is Cn and another is Cm for general m,n, which are not compatible, then I can say these are two different differential structures.
Then the fact that for a manifold with dimension<4, there is only one differential structure means one can always construct a diffeomorphism between the two atlas so that the map from one the other is infinitely differentiable.

 

[lavinia]

Thanks a lot Lavinia.
So correct me if I am wrong:
For the same manifold, if I have two different atlas, say one is Cn and another is Cm for general m,n, which are not compatible, then I can say these are two different differential structures.
Then the fact that for a manifold with dimension<4, there is only one differential structure means one can always construct a diffeomorphism between the two atlas so that the map from one the other is infinitely differentiable.

I believe that two atlases may be incompatible yet the manifolds may be diffeomorphic.

The example I gave on the real line illustrates this. If a manifold has a C^r structure then sometimes it will have a C^r+K substructure. i think if r is large enough then the manifold will have a smooth substructure but don't quote me on this.