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It is often advertised that affine connections are a way to differentiate a vector field along another (see e.g. here). For the little I know, Lie derivatives are advertised in the same way. On a more or less intuitive level, what distinguishes the two notions? Below is a possible answer that I couldn't formally verify.

Let $M$ be a smooth manifold and $X$, $Y$ be two smooth vector fields on $M$. Naively, the "derivative of $Y$ along $X$ at a point $p$" is $$ \lim_{\epsilon \rightarrow 0} \frac{Y_{p + \epsilon X_p} - Y_p}{\epsilon} . $$ But of course, $p + \epsilon X_p$ doesn't make sense. Furthermore, $Y_p$ and $Y_{p'}$ don't live in the same tangent space if $p \neq p'$. The idea with Lie derivative is to consider the flow $\theta$ of $X$ as a mediator: $$ (L_X Y)_p = \lim_{\epsilon \rightarrow 0} \frac{(d \theta_{-\epsilon}) Y_{\theta_\epsilon (p)} - Y_p}{\epsilon} . $$ Based on this, maybe $(\nabla_X Y)_p$ does the same thing, but instead of mediating with the derivative of the flow of $X$, it uses parallel transport along a (arbitrary) curve passing through $p$ with speed $X_p$?

Could you clarify the situation for me?

Thanks in advance.

cocompletehippopotamus
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  • The main point is that the covariant derivative $\nabla_X$ is $C^\infty$-linear in X, that is for any function f $\nabla_{fX}=f \nabla_X$, while the Lie derivative is not. – GFR Jan 22 '21 at 09:31
  • The Lie derivative is canonically defined thanks to the differential structure of $M$: it is a differential object. However, a connexion is an additive structure that can be derived from geometric structures (principal bundles, Riemannian metrics, etc.): it is a geometric object. There is no canonical connexion on a smooth manifold without any additional structure on it. They both are "derivations" for vector fields, but a connexion take cares of some geometric sense one would add to this notion, while the Lie derivative does not. – Didier Jan 22 '21 at 11:39
  • Thank you for your answers. I understand that a connection has different properties than Lie derivatives, and that they're added structure on a manifold. But I still don't see these two different notions represent the same idea of "the derivative of a vector field along another vector field"... – cocompletehippopotamus Jan 25 '21 at 00:19

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