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Diffgeom β

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'.

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