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:

$\Gamma (E^{*})\times \Gamma (E)\to \mathbb {R}$

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

$X(ls)=(\nabla _{X}^{*}(l))(s)+l(\nabla _{X}s)$