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Pull-back of vectors at a SINGLE point

  1. Oct 9, 2011 #1

    ivl

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    Dear all,

    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 defined if 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 defined if 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
     
  2. jcsd
  3. Oct 9, 2011 #2

    quasar987

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    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.
     
  4. Oct 9, 2011 #3

    ivl

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    Thanks Quasar897!

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

    xxxxxxx Inverse function theorem xxxxxxxxx
    If [itex]F_{*}[/itex] is invertible at the point [itex] x \in M[/itex], then [itex] F [/itex] is a local diffeomorphism around [itex]x \in M [/itex]. In other words, in some neighborhood of [itex]x \in M[/itex] the inverse [itex]F^{-1}[/itex] 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 [itex]F_{*}[/itex] invertible at the single point of interest.
    5. From the inverse function theorem, I am actually forced to assume that [itex]F[/itex] 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 [itex]M[/itex]. Hence, the pull-back of vector fields is defined if and only if [itex]F[/itex] is a local diffeomorphism from [itex]M[/itex] to [itex]N[/itex].

    Any help is very appreciated!
     
  5. Oct 9, 2011 #4

    quasar987

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    I would agree with all 7 points!
     
  6. Oct 9, 2011 #5
    Why don't you try some specific examples of pullbacks and pushforwards along maps F?
     
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