Pages 120, 121 of W. Boothby's An Introduction to Differentiable Manifolds contain an explanation of infinitesimal generator, with the quotient:

$$\frac{1}{\Delta{t}}[f(\theta_{\Delta{t}}(p)) - {f}(p)]$$

Then the author assumes that $\hat{f} (x^{1}, \ldots, x^{n})$ is the local expression for $f \in C^{\infty}(p)$, and then he equals to the above quotient with

$$\frac{1}{\Delta{t}}[\hat{f}(h(\Delta{t}, x)) - \hat{f}(x)].$$

I am confused as to how these quotients may be equal.

I'd appreciate any useful explanation to help me out.

(I argue that $\theta : \mathbb{R} \times M \to M$ and $f : M \to \mathbb{R}$, so that $f \circ \theta : \mathbb{R} \times M \to \mathbb{R}$. On the other hand, $h : \mathbb{R} \times \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $\hat{f} : \mathbb{R}^{n} \to \mathbb{R}$, which gives $ \hat{f} \circ h : \mathbb{R} \times \mathbb{R}^{n} \to \mathbb{R}$.)


1 Answer 1


You put a chart $\phi:M\to\Bbb R^n$ in between, $$f∘θ=f∘\phi^{-1}∘\phi∘θ$$ and split in the middle, to get $$\hat f=f∘\phi^{-1},\\ h=\phi∘θ∘(id_{\Bbb R}, \phi^{-1})$$ with the described properties.


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