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I've finished a first course on differential geometry and I still find it confusing on how to compose/multiply two vector fields.

Let's assume that $X$ and $Y$ are two vector fields on a smooth manifold. Then my understanding of $XY$ is that: $$XY f (p)=X_p (Y f).$$ Therefore if $X$ and $Y$ belong to the Lie algebra of a group $G$, then $$XY f(g)=X_g(Yf)=\frac{d}{dt} (Yf)(g\exp tX_e)|_{t=0}= \frac{d}{dt}\frac{d}{ds} f(g\exp tX_e\exp sY_e)|_{s=0}|_{t=0}.$$

Does this sound OK so far? Now I want to set $X=Y$ to get $$XX f(g)= \frac{d}{dt}\frac{d}{ds} f(g\exp (t+s)X_e)|_{s=0}|_{t=0}.$$ However, the expression that I have in my notes for $XX$ is $$XX f(g)= \frac{d^2}{dt^2}f(g\exp (t)X_e)|_{t=0}.$$

Are these two expressions the same?!

student
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    Yes. Those are two different ways of taking the second derivative of something. Can you prove the corresponding fact for just ordinary functions of one variable? – Qiaochu Yuan May 02 '15 at 06:15
  • What leads you to believe they aren't? – Robin Goodfellow May 02 '15 at 12:02
  • @Qiaochu Ah, stupid me! This is just $\frac{d^2}{dt^2}h(t)|{t=0}=\frac{d}{dt}\frac{d}{ds}h(t+s)|{s=0}|_{t=0}$ (chain rule). Thanks! Could I do my interpretation of $XX$ in a way to avoid the extra variable $s$ from the beginning or my thought process is OK? – student May 02 '15 at 14:54

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