Tensor product of metric connections is metric

Statement

A differential manifold $M$. Two metric bundles $(E,g)$ and $(E',g')$ (i.e., $E,E'$ are vector bundles and $g,g'$ are Riemannian metrics or pseudo-Riemannian metrics on these). $\nabla ,\nabla '$ are metric connections on $(E,g)$ and $(E',g')$ respectively. Then, the tensor product of connections $\nabla \otimes \nabla '$ is a metric connection on $(E\otimes E',g\otimes g')$.

Related facts

• Tensor product of torsion-free connections is torsion-free
• Tensor product of flat connections is flat