= Local extension of a vector field along a submanifold
For an embedded <submanifold> $i:M\hookrightarrow N$, a smooth <section of a vector bundle> $X\in\Gamma(i^*TN)$ locally extends to an ambient <vector field>. In an adapted chart $M=\{y=0\}$, extend each component $X^a(x)$ independently of the transverse coordinates $y$. If $X$ takes values in $TM$, the extension criterion is $(\widetilde XF)|_M=X(F|_M)$ for every local smooth function $F$ on $N$. Testing coordinate functions suffices. For tangent fields $X,Y$, this criterion also proves $[\widetilde X,\widetilde Y]|_M=[X,Y]$. General fields with normal components have no extension-independent ambient <Lie bracket>.
// Target: riemannian-geometry.bigb
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