Formula for curvature of dual connection

Statement

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

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

Related facts

Applications

• Duall connection to flat connection is flat