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Let $M$ be $n$-dimensional manifold, $p \in M$, $V$- open neighbourhood of $p$ and let $Y$ be a smooth vector field in $V$.

Do there exist an open neighbourhood $W \subset V$ of $p$ and a smooth vector field $X$ on the whole $M$ which extends $Y|_W$ ? How to do it. I know that similar fact, but for smooth real valued functions instead of smooth vector fields, holds.


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A related question, the answer of which you should be able to adapt to the present setting. – t.b. Apr 14 '12 at 12:30
up vote 3 down vote accepted

Let $p \in K \subset W \subset V$, where $K$ compact, $W$ open, let $\phi :W \rightarrow R^n$ be a chart, and let $Y(x)=\sum_{i=1}^n Y_i (x) \partial|_i(x) $ for $x \in W$. There exists $\chi \in C_c^\infty$ such that $\chi|_K=1$, $supp(\chi)\subset W$.

Put $X(x):=\chi(x) Y_i(x) \partial|_i(x)$ for $x \in W$ and $X(x)=0$ for $x \in M\setminus W$.

Then $X$ is smooth on $M$ and $X|_W=Y$.

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