Tensor product of metric connections is metric

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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').

Proof

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