Linear transition maps on manifolds

[vanesch]

Hi,

I have a question. Consider a differentiable manifold. This structure is imposed by requiring differentiability of the transition functions between charts of the atlas. Does requiring on top of that, linearity or affinity of the transition functions, result in any specific extra structure on the manifold ? (I would be tempted to call it an "affine manifold", but I think the name is already taken). Clearly a flat affine space would be such kind of manifold, and also things like S1. I'm not sure about S^2 though.

 

[Tom1992]

For S^n (n>=2), you the transition functions would require square root functions if it has the 2n+2 hemispherical charts, so linearity is ruled out. Also, the stereographic projection charts for S^n have the inverse function in the transition function, so linearity is ruled out there as well.

The projective space has inverse functions in the transition maps as well.

The circle and cylinder satisfy the linearity you seek. That might give some insight.

 

[vanesch]

For S^n (n>=2), you the transition functions would require square root functions if it has the 2n+2 hemispherical charts, so linearity is ruled out. Also, the stereographic projection charts for S^n have the inverse function in the transition function, so linearity is ruled out there as well.

The projective space has inverse functions in the transition maps as well.

The circle and cylinder satisfy the linearity you seek. That might give some insight.

Yes, I understand that many charts are eliminated by this linearity requirement - that's pretty obvious - but the question is: are there still sufficient charts left to form an atlas over, say, S^n ? In more general terms, by imposing this extra requirement, do we:
1) eliminate certain manifolds ?
2) introduce some extra structure on those that remain ?

 

[Doodle Bob]

It seems to me that at least even-dimensional spheres would *not* have such an atlas. Recall that on such spheres there are no nowhere-zero vector fields yet it seems to me that, if one had one of these affine atlases on a manifold, one could define a vector field on the entire manifold that is nowhere-zero.

The construction would run along these lines: the transition maps will preserve linearity, i.e. we have a well-defined family of "submanifolds" corresponding to the linear subspaces of R^n (i put the quotes in there because some of them might actually end up doing weird things on the manifold such as being dense). On one coordinate patch (x1,...xN), just look at the flow generated by, say, the vector field d/dx1. Since the transition maps are all affine and nonsingular, this should generate a nowhere-stationary flow on the surrounding coordinate patches, and so on. From here, you would need to show that this construction creates a well-defined *global* flow.

If you do that, then we have a global nowhere-zero vector field generated by the flow: which eliminates the possibility that the manifold is an even-dimensional sphere.

Toruses of course have these affine atlases (the transition maps are all translations). And to illustrate my remark about weird linear "submanifolds" If you look at the square torus R^2/Z^2, any line in R^2 with an irrational slope will give you a dense curve in the torus.

 

[vanesch]

It seems to me that at least even-dimensional spheres would *not* have such an atlas. Recall that on such spheres there are no nowhere-zero vector fields yet it seems to me that, if one had one of these affine atlases on a manifold, one could define a vector field on the entire manifold that is nowhere-zero.

The construction would run along these lines: the transition maps will preserve linearity, i.e. we have a well-defined family of "submanifolds" corresponding to the linear subspaces of R^n (i put the quotes in there because some of them might actually end up doing weird things on the manifold such as being dense). On one coordinate patch (x1,...xN), just look at the flow generated by, say, the vector field d/dx1. Since the transition maps are all affine and nonsingular, this should generate a nowhere-stationary flow on the surrounding coordinate patches, and so on. From here, you would need to show that this construction creates a well-defined *global* flow.

If you do that, then we have a global nowhere-zero vector field generated by the flow: which eliminates the possibility that the manifold is an even-dimensional sphere.

Toruses of course have these affine atlases (the transition maps are all translations). And to illustrate my remark about weird linear "submanifolds" If you look at the square torus R^2/Z^2, any line in R^2 with an irrational slope will give you a dense curve in the torus.

Right ! That seems indeed to be a convincing argument that not every differentiable manifold can get this structure. I was wondering what kind of beast is such a "manifold with affine atlas". I thought it might be a known thing, given the simplicity of the requirement (linear or affine transition maps), but apparently not... Intuitively there's some kind of "flatness" to such a structure, but I wonder if there is a relationship.

 

[Doodle Bob]

Right ! That seems indeed to be a convincing argument that not every differentiable manifold can get this structure. I was wondering what kind of beast is such a "manifold with affine atlas". I thought it might be a known thing, given the simplicity of the requirement (linear or affine transition maps), but apparently not... Intuitively there's some kind of "flatness" to such a structure, but I wonder if there is a relationship.

I do think that these beasts have been studied (and possibly even classified), but for the life of me I can't remember where I've seen it. You might want to look at "Structures on Manifolds" by Yano and Kon. Also, maybe even Boothby's intro to Riemannian manifolds and/or do Carmo's Riemannian Manifolds.

 

[Doodle Bob]

Although it doesn't give much more info than what we've talked about here, another place that you might want to look is chapter 3 of Thurston's very fine text "Three-Dimensional Geometry and Topology". He actually defines such beasts as "affine manifolds" and talks briefly about a result by Bieberbach which says that if the transition maps are all isometries of E^n then the manifold is covered by a torus.

 

[vanesch]

Thanks a lot for this information, Bob !