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xxxxxxPreliminaryxxxxxxx

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

[itex](F_{*}v)_{F(x)}(f)=v_{x}(f\circ F)[/itex]

where

1. [itex]F_{*}[/itex] is the push-forward associated with the smooth map [itex]F:M\rightarrow N[/itex]

2. [itex]v_{x}[/itex] is a vector at the point [itex]x\in M[/itex] (a member of the tangent space of [itex]M[/itex] at [itex]x[/itex])

3. [itex]f[/itex] is a smooth function [itex]f:N\rightarrow \mathbb{R}[/itex]

xxxxxxPreliminaryxxxxxxx

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

[itex](F^{*}\alpha)_{x}(v)=\alpha_{F(x)}(F_{*}v)[/itex]

where

[itex]\alpha_{y}[/itex] is a member of the cotangent vector of [itex]N[/itex] at [itex]y[/itex].

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 definedif and only if[itex]F_{*}[/itex] is invertible at that point. Then, I can write [itex](F^{*}u)_{y}:=((F_{*})^{-1}u)_{y}[/itex] where [itex]u[/itex] belongs to the tangent space of [itex]N[/itex] at [itex]y[/itex].

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

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

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# Pull-back of vectors at a SINGLE point

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