Gluing vector fields

Question

Let $M$ be a smooth manifold and $\mathcal{A} = \{(U_i, \varphi_i)~|~i \in I\}$ the smooth associated atlas. I have smooth vector fields $X_i$ on each $U_i$ such that for $i, j \in I$, $X_i$ and $X_j$ are equal on $U_i \cap U_j$. Is there a theorem that allows me to "glue" these vector fields, $i.e.$ is there a (unique) vector field $X$ on $M$ such that $X\big|_{U_i} = X_i$ ? Thank you very much!

Answer 1

Yes, in fact we have a slightly more general result, appear in Lee's Introduction to Smooth Manifolds 2nd ed, as follows:

Corollary 2.8 (Gluing Lemma for Smooth Maps) Let $M$ and $N$ be smooth manifolds with or without boundary, and let $\{U_{\alpha}\}_{\alpha \in A}$ be an open cover of $M$. Suppose that for each $\alpha \in A$, we are given a smooth map $F_{\alpha} : U_{\alpha} \to N$ such that the maps agree on overlaps : $F_{\alpha}|_{U_{\alpha}\cap U_{\beta}} =F_{\beta}|_{U_{\alpha}\cap U_{\beta}}$ for all $\alpha$ and $\beta$. Then there exists a unique smooth map $F : M \to N$ such that $F|_{U_{\alpha}} = F_{\alpha}$ for each $\alpha \in A$.

This is just a consequence of the fact that if a map $F: M \to N$ is locally smooth, then the map $F$ is smooth.