I am not familiar with the formal compute about the invariant under diffeomorphism (isometric),so I want a detail example. For example,$M,N$ are Riemannian manifolds, $\Phi :M\rightarrow N$ is diffeomorphism.How to formally show $$ \int_MR(\Phi^*g_{ij}) dV(\Phi^*g_{ij})=\int_N R(g_{ij})dV(g_{ij}) $$
I always don't know how to compute the integration,when the metric and manifold be changed.So, I really want a detail example to imitate.
Change of variable implies that $$ \int_NRdV(g)=\int_M R\circ f dV(f^\ast g) $$ where $f: M\rightarrow (N,g)$
Hence we must show that $ R(g,f(x))=R(f^\ast g,x) $ But scalar curvature is invariant under isometry so that this is clear.