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?