Dual connection

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Suppose E is a vector bundle over a differential manifold M and \nabla is a connection on E. The dual connection to \nabla, denoted \nabla^*, is a connection on the dual vector bundle E^*, defined as follows.

For any l \in \Gamma(E^*) and X \in \Gamma(TM), we have:

\nabla^*_X(l) := s \mapsto X(ls) - l(\nabla_X s)

where s \in \Gamma(E)


The definition of a dual connection is chosen in such a way that the bilinear form for evaluation:

\Gamma(E^*) \times \Gamma(E) \to \R

satisfies the Leibniz rule. In other wors, we need to ensure that for l \in \gamma(E^*) and s \in \Gamma(E), we have:

X(ls) = (\nabla^*_X(l))(s) + l(\nabla_X s)