F must be a Diffeomorphism: Why & What if?

[fk378]

In the definition of F-related vector fields, F must be a diffeomorphism. Why must it be a diffeomorphism? What if F is smooth and bijective, but not a diffeo?

 

[Tinyboss]

If you post the definition I might be able to help. I left my manifolds book at school.

 

[quasar987]

The definition does not even require F-->N to be bijective. If F is not bijective, then a vector field on M might to push-foward to a vector field on N, and a vector field on N might not have an F-related vector field on M. But if F is a diffeo, then we are in the nice situation where to every vector field on N, then exists a unique F-related vector field on M given, of course, by the pushfoward by F^-1.

 

[fk378]

If you post the definition I might be able to help. I left my manifolds book at school.

Suppose F: M-->N is a diffeomorphism. For every Y in TM (tangent bundle to M), there is a unique smooth vector field on N that is F-related to Y.

 

[fk378]

The definition does not even require F-->N to be bijective. If F is not bijective, then a vector field on M might to push-foward to a vector field on N, and a vector field on N might not have an F-related vector field on M. But if F is a diffeo, then we are in the nice situation where to every vector field on N, then exists a unique F-related vector field on M given, of course, by the pushfoward by F^-1.

Yes, I understand the smooth and bijective part, but what about the non-diffeo part?

 

[quasar987]

I said that the notion of "F-relatedness" can be defined if F is merely smooth. In particular, it makes sense to speak about F-relatedness if F is smooth and bijective.

Then I hinted to the fact that your book defines the notion when F is diffeo probably because in that case, we are in the nice situation where to every vector field on N there exists a unique F-related vector field on M... which is probably the property that the authors needed.