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I have the following problem:

Let $U,V\subset\mathbb{R}^n$ be open subsets, $f:U \rightarrow V$ a diffeomorphism and $\omega \in \Omega^nV$. Find a formula for $f^*\omega$.

My problem is that I don't know the meaning of $f^*\omega$.

$\Omega^nV$ is a vector space of k-forms, which are continuously differentiable once.

md2perpe
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Tobi92sr
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2 Answers2

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Well, $\omega$ is something like a sum of terms of the form
$$g_1(y_1,y_2, \dots y_n)\wedge g_2(y_1,y_2, \dots y_n)\wedge \dots \wedge g_k(y_1,y_2, \dots y_n)$$

and given $$ f = (f_1(x_1,x_2, \dots x_n), f_2(x_1,x_2, \dots x_n), \dots f_n(x_1,x_2, \dots x_n)) $$ you just have $f^* \omega$ as a sum of terms of the form $$g_1(f_1(x_1,x_2, \dots x_n), f_2(x_1,x_2, \dots x_n), \dots f_n(x_1,x_2, \dots x_n))\wedge g_2(f_1(x_1,x_2, \dots x_n), f_2(x_1,x_2, \dots x_n), \dots f_n(x_1,x_2, \dots x_n))\wedge \dots \wedge g_k(f_1(x_1,x_2, \dots x_n), f_2(x_1,x_2, \dots x_n), \dots f_n(x_1,x_2, \dots x_n)).$$

That is, just substitute $\bf y$ in the image with $\bf f (\bf x)$

Luca
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  • Does that mean, your answer is already the solution to my problem? Because I'm not sure, if I understand it completely. Is it possible for you to elaborate? For example what eactly is the meaning of ∧? Because we`ve never written it down like that. – Tobi92sr May 25 '17 at 17:21
  • @Tobi92sr What notation have you used for forms? – md2perpe May 25 '17 at 21:23
  • @Tobi92sr sorry for being late. $\wedge$ is just the antisymmetric product of forms, that is $dx \wedge dy = - dy \wedge dx$. So if $x = wz- z^2$ and $y= w^2$ then $dx \wedge dy $ becomes $(zdw +(w -2z)dz) \wedge 2wdw = (4wz - 2w^2)dw \wedge dz $ – Luca Jun 09 '17 at 18:51
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$f^*\omega$ is the pull-back of $\omega$. If $f: U \to V$ and $\omega$ is a differential form on $V$ then $f^*\omega$ is a differential form on $U$. How do we define this differential form on $U$? Given tangent vectors $u_1,\ldots,u_k$ we define $(f^*\omega)(u_1,\ldots,u_k)=\omega(f_*u_1,\ldots,f_*u_k)$, where $f_* : TU \to TV$ is the differential of $f$. The differential is a linear map whose matrix representation is the Jacobian matrix.

Calculate the Jacobian matrix. Apply the Jacobian matrix to each of the $u_i$ to give $f_*u_i$. Then apply the differential form $\omega$ on the $f_*u_i$.

Fly by Night
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  • Do you have any generic definition of pull-back? It's easy to find formulas for different types of object, but not a generic definition. – md2perpe May 26 '17 at 12:32
  • @md2perpe What do you mean by "generic"? My post works for manifolds $M$ and $N$ and differential $k$-forms. Wikipedia says you can generalise to certain types of tensor field: https://en.wikipedia.org/wiki/Pullback_(differential_geometry)#Pullback_of_.28covariant.29_tensor_fields but I don't have any experience of anything other than differential forms. – Fly by Night May 26 '17 at 18:26
  • The Wikipedia page defines the pullback for several types of objects: smooth functions, bundles and sections, multilinear forms, cotangent vectors, tensor fields, differential forms. I'm looking for a unifying definition of pullbacks, or at least a description of what a pullback does, no matter what type of object it's working on. – md2perpe May 26 '17 at 19:45
  • @md2perpe Perhaps you should post a question of your own. That would get more attention. – Fly by Night May 26 '17 at 22:07