Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 469From the Riemann curvature tensor ˆR µνρ σ we get the Ricci curvaturescalar by contracting the indicesˆR := Ĝµν ˆRνµρ ρ .ˆR indeed transforms as a scalar which may be checked explicitly by takingthe deformed coproduct (3.285) into account.To get an integral which is invariant with respect to the Hopf algebraof deformed infinitesimal diffeomorphisms we need a measure function Ê.We demand the transformation propertyˆδξ Ê = − ˆX ξ Ê − ˆX (∂µξ µ)Ê. (3.290)Then it follows with the deformed coproduct (3.285) that for any scalarfield Ŝˆδξ ÊŜ = − ˆ∂ µ ( ˆX ξ µ(ÊŜ)).Hence, transforming the product of an arbitrary scalar field with a measurefunction Ê we get a total derivative which vanishes under the integral. Asuitable measure function with the desired transformation property (3.290)is for instance given by the determinant of the vierbein ʵ aÊ = det(ʵ a ) := 1 4! εµ 1···µ 4 εa1···a 4Ê µ1a 1Ê µ2a 2Ê µ3a 3Ê µ4a 4.That Ê transforms correctly can be shown by using that the product offour ʵ a iitransforms as a tensor of fourth rank and some combinatorics.Now we have all ingredients to write down the Einstein–Hilbert action.Note that having chosen a differential calculus as in (3.271), the integral isuniquely determined up to a normalization factor by requiring 16 [Douglasand Nekrasov (2001)]∫ˆ∂ µ ˆf = 0for all ˆf ∈ Â. Then we define the Einstein–Hilbert action on  as∫Ŝ EH := det(ʵ a ) ˆR + complex conjugate.16 We consider functions that “vanish at infinity”.