I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." In p. 69, it gives the definition of the Lie derivative as follows:

2.24 Definition (summerized) Fix a smooth vector field $X$ on a differentiable manifold $M$. Let $Y$ be another smooth vector field on $M$. The Lie derivative of $Y$ with respect to $X$ at $m$, which is denoted by $(L_XY)_m$, is $$(L_XY)_m = \lim_{t\to 0} \frac{dX_{-t}(Y_{X_t(m)}) - Y_m} {t}$$.

Here, $X_t(m) = \gamma_m(t)$, where $\gamma_m$ is an integral curve of $X$ such that $0$ is in the domain of $\gamma_m$ and $\gamma_m(0) = m$. Also, $Y_m = Y(m)$.

I see that the function $t\mapsto\frac{dX_{-t}(Y_{X_t(m)}) - Y_m} {t}$ is a function from the subset of $\mathbb R$ into $M_m$, the tangent space to $M$ at $m$. But to define a limit of such function, there should be a topology on $M_m$, right? But I haven't learned any inner product, norm, metric, or topology given on $M_m$. So what is the topology on $M_m$?

  • $\begingroup$ Is $M_m$ the tangent space to $M$ at the point $m$? So a finite-dimensional vector space? $\endgroup$ – Lord Shark the Unknown Feb 17 at 10:50
  • $\begingroup$ @Lord Yes, $M_m$ is the tangent space to $M$ at $m$, so $\operatorname{dim}(M_m)$ = $\operatorname{dim}(M)$ so that it is finite. $\endgroup$ – zxcv Feb 17 at 10:56
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    $\begingroup$ You know that on a finite-dimensional real vector space, there is only one sensible topology.... $\endgroup$ – Lord Shark the Unknown Feb 17 at 11:03

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