Pushforward/Pullback of Vector Field

[knowwhatyoudontknow]

TL;DR

Pushforward of vector field.

I am looking at the following document. In section 2.3 they have the formula for the pushforward:

f_*(X) := Tf o X o f^-1

I am having trouble trying to reconcile this with the more familiar equation:

f_*(X)(g ) = X(g o f)

Any help would be appreciated.

 

[fresh_42]

I read this as an unfortunate way to reference the point of evaluation: Given $f\, : \,M\longrightarrow N$ we need a point $p\in M$ to evaluate at: $p=f^{-1}(q)$ for $q\in N.$

Have a look at eq. 38 at the end in https://www.physicsforums.com/insights/pantheon-derivatives-part-iii/
I also collected a few of the many ways to write it.

 

[knowwhatyoudontknow]

OK. Thanks. If I have γ(t) = exp(tv) so that γ(0) = p and γ'(0) = v and L_g is a left action then is the following correct?

L_g*γ'(0) = (L_g o γ(t))'(0)

= d/dt|_t=0L_g(γ(t))

= dL_g/dt|_t=0.v

= gv

In other words dL_g/dt|_t=0 ≡ g

 

[fresh_42]

OK. Thanks. If I have γ(t) = exp(tv) so that γ(0) = p and γ'(0) = v and L_g is a left action then is the following correct?

L_g*γ'(0) = (L_g o γ(t))'(0)

= d/dt|_t=0L_g(γ(t))

= dL_g/dt|_t=0.v

= gv

In other words dL_g/dt|_t=0 ≡ g

Looks ok to me.

 

[knowwhatyoudontknow]

Great. Thanks.