Dual connection

Definition

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 $s \in \Gamma(E)$

Motivation

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:

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