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I'm reading John M. Lee's Introduction to smooth manifolds, First Edition, Proposition 18.16 and some question arises :

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Why the underlined statements are true? ; i.e., why the map $T$ is smooth? Under what smooth structure on $T^{k}(T_pM)$ ?

And why $T(t) = T(0) = \tau _p$ for all $t \in \mathcal{D}^{(p)}$? A priori it seems possible but I can't make formal proof until now.

Can anyone help?

Plantation
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    Look at the problem in coordinates: This is the usual existence and uniqueness result for first order differential equation in $\Bbb R^n$. The functions involved in the differential equation are smooth, so is the solution. (And you are looking at the maximal solution whose interval of definition contains $0$.) – Didier Dec 10 '22 at 13:11
  • Thanks. And it seems a bit vague to understand what you are saying :) Can I ask a queation? 1) What equation is $T$ a solution to? And you wrote, "The functions involved ~" . What functions you indicated? 2) Does situation "we are looking at the maximal solution whose interval of definition contains $0$" have anything to do with $T(t)=T(0)$ ? Why $T(t)=T(0)$? – Plantation Dec 11 '22 at 03:55
  • I think that I am begginer of differential geometry. Perhaps can you explain more in detail? If so, I am appreciated ~~ – Plantation Dec 11 '22 at 03:56
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    $T$ is solution, in coordinates, to $\frac{d}{dt}T_{i_1,\ldots,i_k}^{j_1,\ldots,j_l}=0$, on an interval containing zero. Hence, these components are constant functions, equal to $T_{i_1,\ldots,i_k}^{j_1,\ldots,j_l}(t)=T_{i_1,\ldots,i_k}^{j_1,\ldots,j_l}(0)$. Regarding your question $1$, it is $\theta$ which is solution to a first order ODE in coordinates with smooth coefficients. You should go back to the chapter where the flow of a vector field is studied – Didier Dec 11 '22 at 11:28
  • @Didier Thanks! , I will try it.~ – Plantation Dec 11 '22 at 12:03

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