Local extension of a vector field along a submanifold

For an embedded submanifold $i:M\hookrightarrow N$, a smooth section of a vector bundle $X\in \Gamma (i^{\ast }TN)$ 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 $(\tilde{X}F){|}_M=X(F{|}_M)$ for every local smooth function $F$ on $N$. Testing coordinate functions suffices. For tangent fields $X,Y$, this criterion also proves $[\tilde{X},\tilde{Y}]{|}_M=[X,Y]$. General fields with normal components have no extension-independent ambient Lie bracket.

// Target: riemannian-geometry.bigb