Dual connection

Definition

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

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

$\underset{X}{\overset{*}{\mathrm{\nabla }}}(l):=s\mapsto X(ls)-l({\mathrm{\nabla }}_{X}s)$

where $s\in \mathrm{\Gamma }(E)$

Motivation

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

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

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

$X(ls)=(\underset{X}{\overset{*}{\mathrm{\nabla }}}(l))(s)+l({\mathrm{\nabla }}_{X}s)$