Say I have a Hermitian vector bundle $E \to M$ where $M$ is some manifold. If I choose a connection $D_0$ on $E$, (and assuming this connection has no globally covariantly constant sections) then I can define a Sobolev space by completion under
$ |D_0 f|_{L^2} $
Now assume I perturb the connection a small amount to $D_0 + A$, where $A$ is an $L^2$ section of the adjoint bundle of $E$. Now I could instead complete under
$ |(D_0+A) f|_{L^2} $
How does this change the resulting Sobolev space?
I would like to show that these define equivalent metrics on the resulting Sobolev space.
Edit: I have since learned that the above statement is not true, (specifically you can construct counter examples on $\mathbb{C}^2 \times \mathbb{R} \to \mathbb{R}$) but am still looking for resources that go in this direction.