Pull-back of vectors at a SINGLE point

[ivl]

Dear all,

xxxxxxPreliminaryxxxxxxx
the push-forward of vectors is FIRST defined at a single point, as

(F_{*}v)_{F(x)}(f)=v_{x}(f\circ F)

where

1. F_{*} is the push-forward associated with the smooth map F:M\rightarrow N
2. v_{x} is a vector at the point x\in M (a member of the tangent space of M at x)
3. f is a smooth function f:N\rightarrow \mathbb{R}

xxxxxxPreliminaryxxxxxxx
the pull-back of a covector is FIRST defined at a single point, as

(F^{*}\alpha)_{x}(v)=\alpha_{F(x)}(F_{*}v)

where

\alpha_{y} is a member of the cotangent vector of N at y.

xxxxxQuestionxxxxxx
At this point, I would like to pull-back vectors and push-forward covectors, but STILL at a SINGLE point. In other words, I do NOT want to consider FIELDS yet (I plan to do that later, as doing things step by step is more instructive).

I propose that the pull-back of vectors at a single point is defined if and only if F_{*} is invertible at that point. Then, I can write (F^{*}u)_{y}:=((F_{*})^{-1}u)_{y} where u belongs to the tangent space of N at y.

Similarly, I propose that the push-forward of covectors at a single point is defined if and only if F^{*} is invertible at that point. Then, I can write (F_{*}\beta)_{x}=((F^{*})^{-1}\beta)_{x} where \beta belongs to the cotangent space of M at x

Does anyone know if this is right? is there a book that deals with things completely point-wise (until using fields becomes unavoidable)?

Thanks a lot

 

[quasar987]

It looks ok. Except the notation ((F_∗))^-1u)_y is weird.

Better is ((F^-1)_∗)u)_y where F^-1 stands for a local inverse of F around y.

 

[ivl]

Thanks Quasar897!

the reason I was using the notation ((F_{∗})^{-1}u)_{y} is that I was not aware of the inverse function theorem:

xxxxxxx Inverse function theorem xxxxxxxxx
If F_{*} is invertible at the point x \in M, then F is a local diffeomorphism around x \in M. In other words, in some neighborhood of x \in M the inverse F^{-1} is defined.
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Would you mind giving me your opinion on the following chain of reasoning?

1. I want to introduce the pull-back of vector fields.
2. I begin by trying and pulling-back vectors at a single point.
3. The only sensible definition for a single-point pull-back of a vector is the inverse of the push-forward at that point.
4. I have no choice but to assume F_{*} invertible at the single point of interest.
5. From the inverse function theorem, I am actually forced to assume that F is a local diffeomorphism.
6. Hence, I automatically get a pull-back that works for vector FIELDS, albeit locally around a point.
7. Then, I just need to repeat the procedure for every point on M. Hence, the pull-back of vector fields is defined if and only if F is a local diffeomorphism from M to N.

Any help is very appreciated!

 

[quasar987]

I would agree with all 7 points!

 

[Bacle]

Why don't you try some specific examples of pullbacks and pushforwards along maps F?