Formula for curvature of tensor product of connections

Statement

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 $E \otimes E'$ is the tensor product of vector bundles, $\nabla \otimes \nabla'$ is the tensor product of connections, and $R_{\nabla \otimes \nabla'}$ is its Riemann curvature tensor, we have:

$R_{\nabla \otimes \nabla'}(X,Y)(s \otimes s') = R_\nabla(X,Y)(s) \otimes s' + s \otimes R_\nabla(X,Y)s'$.

Related facts

Applications

• Tensor product of flat connections is flat