I don't understand how the author calculated the pullback of the 1 form (differential geometry)

[DiffMani]

Homework Statement

I am doing self study on differential geometry

Relevant Equations

I understand everything until author calculated the pullback of the one form
How did he get the result for L*g wg?

I am doing self study on differential geometry using Analysis and Algebra on Differentiable Manifolds by Gadea. I don't understand the step where he calculates the pullback of the left translation of the one form. Why did all the variables become x? What is the formula for the pullback on w?
I have tried many hours on that step. Is Lg upper star equal to Ls lower star? Please help, I really appreciate it.

 

[fresh_42]

The formula is $L^*_g\omega_g=\omega_g \circ L_g.$ That explains the second half of the equation. I think
$$
L^*_g\omega_g=\left\{\begin{pmatrix}x&0\\0&x\end{pmatrix}\begin{pmatrix}1/x\\0\end{pmatrix}\, , \,\begin{pmatrix}x&0\\0&x\end{pmatrix}\begin{pmatrix}0\\1/x\end{pmatrix}\right\}
$$
is simply the coordinate notation of vectors in the tangent space. The tangent space is spanned by $dx=\begin{pmatrix}1\\0\end{pmatrix}$ and $dy=\begin{pmatrix}0\\1\end{pmatrix}.$ $\omega_g$ contributes the factor $1/x$ and $L_g$ the left-multiplication. I wouldn't have switched to coordinates and had simply written
$$
L^*_g\omega_g=\omega_g \circ L_g=\dfrac{1}{x}\begin{pmatrix}dx&dy\\0&0\end{pmatrix}\cdot \begin{pmatrix}x&0\\0&x\end{pmatrix}=\begin{pmatrix}dx&dy\\0&0\end{pmatrix}=\omega_e
$$
$x$ comes as coordinates of $g.$

 

[WWGD]

Pullbacks of Differential Forms have a standard definition. They're contravariant. EDIT: And IIRC, pullbacks are functorial too. Maybe @fresh_42 can elaborate on that.

 

[fresh_42]

Pullbacks of Differential Forms have a standard definition. They're contravariant. EDIT: And IIRC, pullbacks are functorial too. Maybe @fresh_42 can elaborate on that.

I am so glad that I got rid of that ...
https://www.physicsforums.com/insights/pantheon-derivatives-part-iii/
and that ...
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/

I have even lost the right to correct typos.

 

[WWGD]

I didn't know what you were referring to by "It" in "Getting rid of it", until I read more carefully. Your roomate's body?

 

[fresh_42]

I didn't know what you were referring to by "It" in "Getting rid of it", until I read more carefully. Your roomate's body?

That was actually a bit funny. Some guys here literally talked me into writing a guide to differentiation and this undertaking resulted in five articles. And all I wanted to say is that I like Weierstraß's formula
$$
f(x+v)=f(x)+J(v)+r(v)
$$