Let $(E,\nabla)$ be a fiber bundle with connection $\nabla$, we can define the dual fiber bundle with connection $(E^*,\nabla^*)$:

$$(E,\nabla) \leftrightarrow ( E^*, \nabla^*)$$

when the vector bundle is viewed as a 1-cocycle of Cech:

$$E=(g_{ij}) \leftrightarrow ((g_{ij}^t)^{-1})=E^*$$

locally:

$$\nabla=d+A \leftrightarrow d - A^t=\nabla^*$$

with gauge change:

$$g_* \nabla \leftrightarrow (g^t)^{-1}_* \nabla^*$$

$$d+g^{-1}Ag+ g^{-1}dg \leftrightarrow d-(g^{-1}Ag+g^{-1}dg)^t$$

When is a connection metric ? Is the dual connection in this case also metric ?