Direct sum of connections

Definition

Suppose $M$ is a differential manifold and $E,E^{\prime }$ are vector bundles on $M$. Suppose $\mathrm{\nabla },{\mathrm{\nabla }}^{\prime }$ are connections on $E$ and $E^{\prime }$ respectively. Then, we define $\mathrm{\nabla }\oplus {\mathrm{\nabla }}^{\prime }$ as a connection on $E\oplus E^{\prime }$ given by:

$(\mathrm{\nabla }\oplus {\mathrm{\nabla }}^{\prime })(s,s^{\prime })=\mathrm{\nabla }(s)\oplus {\mathrm{\nabla }}^{\prime }(s^{\prime })$.