A flow of dual connections

Question

Let $(E,h)$, a metric fibre bundle. I define a flow of dual connections:

$$\frac{\partial \nabla}{\partial t}=\nabla-\nabla^*$$

where $\nabla^*$ is the dual connection when the dual fibre bundle is identified with the fibre bundle via the metric.

Have we solutions for the flow of dual connections? 

Answer 1

Locally, with the metric, we have $\nabla=d+A$, $\nabla^*=d-A^t$ so that:

$$\frac{\partial A}{\partial t}= A+A^t$$

$$A(t)=e^{2t} A_0+B_0$$

$A_0^t=A_0$ and $B_0^t=-B_0$.