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Diffgeom β

Formula for curvature of dual connection

Statement

Suppose M is a differential manifold, E is a vector bundle over M, and \nabla is a connection on E. Suppose E^* is the dual bundle and \nabla^* is the dual connection to \nabla. If R_\nabla and R_{\nabla^*} denote respectively the Riemann curvature tensors of \nabla and \nabla^*, then we have:

R_{\nabla^*}(l) = s \mapsto -l(R(X,Y)(s)).

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