Let (E,∇) be a fiber bundle with connection ∇, we can define the dual fiber bundle with connection (E∗,∇∗):
(E,∇)↔(E∗,∇∗)
when the vector bundle is viewed as a 1-cocycle of Cech:
E=(gij)↔((gtij)−1)=E∗
locally:
∇=d+A↔d−At=∇∗
with gauge change:
g∗∇↔(gt)−1∗∇∗
d+g−1Ag+g−1dg↔d−(g−1Ag+g−1dg)t
When is a connection metric ? Is the dual connection in this case also metric ?