Which is the best way define Lie derivative?

Question

I read 2 main definitions of the Lie derivative. One is done using the concept of flow and the pullback-pushfoward while the other one uses the algebraic with axioms. Both of these can be found on wikipedia. I know that these ways are equivalent but I sort of need to find a principal one. Since I am studying this alone and I don't have a unique definition given by a professor I started wondering which one is the best/most correct way to define the Lie derivative for vector and tensor fields so here I am in trouble also in understanding it. So, in which way should I define this in differential geometry ?

Answer 1

I can write my opinion. Consider that I am just a student. The ideal variant is to define something axiomatically, then to give an example of such object, a realization of it. You ask why? I give an explonation. I think you know that real numbers can be defined with at least three ways. The first is a set of classes of equivalence of fundamental sequences of rational numbers. The second is a set of Dedecind's sections of rational numbers. The third is a set of sequences of numbers from zero to nine. All these constructions can be used as a definition. But from set theory point of view these sets are not the same, they are different as sets. "Geometrically" they are about one thing, but formally they are different. So where are "real" real numbers? To avoid this problem it is better as I think to define real numbers with a list of axioms. I hope you heard about these axioms. Then you prove two things. The first is the existence, for it you may give one of constructions I mentioned. Then you prove uniqness up to an "isomorphism". I hope you get my idea. This way is ideal for me and as I think it is good to try to correspond with defining something.

Answer 2

My opinion is that the definition using flows is the clearest, because it shows in what sense this is a derivative. Indeed, for $\theta$ the flow of $X\in \mathfrak{X}(M)$, and $W$ another vector field we get $$ (\mathcal{L}_XW)_p=\lim_{t\to 0}\frac{d(\theta_{-t})_{\theta_t(p)}W_{\theta_t(p)}-W_p}{t} $$ which shows that the Lie derivative is measuring how $W$ changes as we flow along $X$. Ideally, you would then prove the algebraic characterization.