Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 17 / 5 / a / Solution

A connection on a vector bundle is an $\mathbb{R}$-linear mapwhich satisfies $\nabla (fs)=df\otimes s+f\nabla s$ for every smooth function $f$ and section $s$. Here ${\Omega }^1(M;E)=\Gamma (T^{\ast }M\otimes E)$. Equivalently, writing ${\nabla }_{X}s=(\nabla s)(X)$, it is linear over smooth functions in $X$, linear over $\mathbb{R}$ in $s$, and obeysChoose a trivializing open cover and a local frame $e_{i1},\dots ,e_{ir}$ on each $U_i$. Differentiating the coefficient functions gives a local connection on a vector bundle:No compatibility of these local connections is assumed. Choose a smooth locally finite partition of unity $\{{\varphi }_i\}$ subordinate to the cover and defineextending each weighted term by zero outside $U_i$. Its support lies inside $U_i$, so each extension is smooth, and local finiteness makes the whole sum smooth. The operation is linear over smooth functions in $X$. For the Leibniz rule,using ${\sum }_i{\varphi }_i=1$. This proves every smooth vector bundle admits a connection by the construction of a vector bundle connection by a partition of unity.