I am reading Jost's Riemannian Geometry and Geometric Analysis. I have a few questions when deriving the formal dual of the covariant exterior derivative, arising from the following pictures from the book.
First of all, what is the definition of the innner product $(\cdot,\cdot)$ in the space $\Omega^p(\text{Ad} E)$? Then how to derive the formal dual $D^*$ in $(4.2.6)$? And finally, what is the meaning of the last sentence: $A$ operates on the form part by contraction and not by multiplication with $dx^i$"?
I am totally confused here... Any hints or comments will be appreciated! TIA...
