Formula for curvature of dual connection

From Diffgeom
Revision as of 22:13, 24 July 2009 by Vipul (talk | contribs) (Applications)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search


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)).

Related facts