I got a question regarding the total scalar curvature / Einstein-Hilbert-functional.
I found it in Peter Topping's book "Lectures on the Ricci flow" (you can download it for free here: http://homepages.warwick.ac.uk/~maseq/RFnotes.html ) at the very beginning of chapter 6.
I'm fine with the calculations he does on p. 67. Then in the next page he defines the Gradient of E, which looks... pretty random to me and so the following question arose:
Is it possible to make the domain of E (i.e. the set of smooth Riemannian metrics on a closed Riemannian manifold) into a Riemannian manifold, then look at E as a (differentiable?) function on this manifold and show that its gradient is equal to the definition that Peter Topping gave?
Thanks in advance for any help!