Vtyjkyuk

This is actually a detail in Lee’s smooth manifold 2nd ed p.234, 2nd paragraph in the Theorem 9.46. What i found is possibly a correction for the book. Forgive me if it’s not even close.

Let $M$ be a smooth manifold of dimension $n$ and $p in M$ be an arbitrary point and $U$ is a domain of a smooth chart centered at $p$. Suppose that we have $k$-tuple of smooth vector fields $(V_1, dots, V_k)$ defined on $U$. Let $theta_i$ denote the flow of $V_i$.

Before i state the problem that has bugged me, i want to note that actually; first, the chart $U$ above is a slice chart. Second the vector fields $(V_i)$ above is linearly independent and mutually commuting. But for the following problem, i think we will not use those assumptions. Here’s the problem

Lee claim that there exists $epsilon>0$ and a neighbourhood $Y$ of $p$ in $U$ such that the composition $(theta_1)_t_1 circ (theta_2)_t_2 circ cdots circ (theta_k)_t_k$ defined on $Y$ and maps $Y$ into $U$ whenever $|t_1|,dots,|t_k|$ are all less than $epsilon$.

After this claim, he also said that

To see this, just choose $epsilon_k>0$ and $U_k subseteq U$ such that $theta_k$ maps $(-epsilon_k,epsilon_k)times U_k$ into $U$, and then inductively choose $epsilon_i$ and $U_i$ such that $theta_i$ maps $(-epsilon_i,epsilon_i)times U_i$ into $U_i+1$. And then taking $epsilon = textmin epsilon_i$ and $Y=U_1$.

I did the suggested construction but i think the suggestion should be

To see this, just choose $epsilon_1>0$ and $U_1 subseteq U$ such that $theta_1$ maps $(-epsilon_1,epsilon_1)times U_1$ into $U$, and then inductively choose $epsilon_i$ and $U_i$ such that $theta_i$ maps $(-epsilon_i,epsilon_i)times U_i$ into $U_i-1$. And then taking $epsilon = textmin epsilon_i$ and $Y=U_k$.

I may get this wrong. I hope somebody could clarify this for me. Is the suggestion is correct or it should be other way around as i did. Any help will be appreciated. Thank you