Connection laws in directional form

Fibrewise linearity of $(\mathrm{\nabla }s{)}_p$ gives the first identity by evaluating at $f(p)X(p)+g(p)Y(p)$. Real linearity of $\mathrm{\nabla }$ gives the second. Evaluating $\mathrm{\nabla }(fs)=df\otimes s+f\mathrm{\nabla }s$ on $X$ gives the third because $df(X)=X(f)$.

We will use a cutoff $\chi$ supported in a prescribed neighbourhood and equal to one on a smaller neighbourhood of a chosen point. On an interior chart this is precisely the construction in the proof of [F3]: apply [F4] to two nested balls about the coordinate point and extend by zero. For a boundary chart with image relatively open in a closed half-space $H$, choose radii $0<r<R$ such that ${\bar{B}}_R(a)\cap H$ lies in the chart image. Restrict a Euclidean bump equal to one on ${\bar{B}}_r(a)$ and supported in $B_R(a)$ to $H$, then pull back and extend by zero. Its support lies in the compact chart preimage of ${\bar{B}}_R(a)\cap H$, hence is closed in the Hausdorff manifold and contained in the chart; zero extension is smooth. In dimension zero use the indicator of the open-and-closed singleton. Each construction is at a fixed point, requiring only finitely many choices.

Fix $s$ and write $L(X)=D_{X}s$. If $X$ vanishes near $p$, take a cutoff supported there with $\chi (p)=1$. Then $\chi X=0$ and $0=L(\chi X)(p)=L(X)(p)$. Thus $L(X)(p)$ depends only on the germ of $X$. Local fields may now be multiplied by a cutoff equal to one near $p$ and extended by zero; applying $L$ and evaluating at $p$ is independent of that extension. On a neighbourhood where one fixed cutoff is one, the result is the restriction of a smooth global output, so these local values are smooth.

In a coordinate neighbourhood write $X={\sum }_i{X}^i{\mathrm{\partial }}_i$. The extended local operator satisfies $L(X)(p)={\sum }_i{X}^i(p)L({\mathrm{\partial }}_i)(p)$: extend all fields and coefficient functions using a common cutoff equal to one near $p$ and apply global function-linearity, then use step 2.1. Hence $L(X)(p)$ depends only on $X(p)$. Define ${\eta }_s(p)(v)={\sum }_i{v}^{i}L({\mathrm{\partial }}_i)(p)$. Germ independence shows this definition is independent of coordinates, and the smoothness in step 2.1 shows ${\eta }_s$ is a smooth Hom section. Every vector at $p$ has a local constant-coordinate extension and then a cutoff global extension, so its value is uniquely forced by $D$.

Real linearity in $s$ gives real linearity of $s\mapsto {\eta }_s$. Testing on a tangent vector using a global field with that value, the third law gives ${\eta }_{fs}(p)(v)=d{f}_p(v)s(p)+f(p){\eta }_s(p)(v)$. Thus $\mathrm{\nabla }s={\eta }_s$ is the unique connection inducing $D$. If $M$ is empty there are no pointwise tests and both operators are unique zero maps; if $\dim M=0$ the coordinate sum is empty. Rank zero gives zero outputs, while in rank one the same finite calculation has one output component. No simultaneous family of cutoff choices was made: uniqueness defines the global section from locally proved values.