Difference between revisions of "Formula for curvature of dual connection"

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