# Difference between revisions of "Curvature of direct sum of connections equals direct sum of curvatures"

Suppose $E,E'$ are vector bundles over a differential manifold $M$. Suppose $\nabla,\nabla'$ are connections on $E,E'$ respectively. Let $R_\nabla$ denote the Riemann curvature tensor of $\nabla$ and $R_\nabla'$ denote that Riemann curvature tensor of $\nabla'$. Then if $E \oplus E'$ is the direct sum of vector bundles, $\nabla \oplus \nabla'$ is the direct sum of connections, and $R_{\nabla \oplus \nabla'}$ is its Riemann curvature tensor, we have:
$R_{\nabla \oplus \nabla'}(X,Y)(s ,s') = \left(R_\nabla(X,Y)(s),R_\nabla(X,Y)s'\right)$.