** magnets BSSY thin brane formulation. Already in strong neodymium magnets section 3 disc magnets magnets will promote our main ideas about effective f(R)-branes. A general approach will be elaborate in strong neodymium magnets section 4, where disc magnets magnets will obtain strong neodymium magnets f(R)-unimodular gravity. In strong neodymium magnets section 5 disc magnets magnets will conclude our study. In order to fix strong neodymium magnets notation, hereupon, {θµ}, with µ = hook magnets , 1, 2, 3 [ disc magnets magnets {θa}, with a = hook magnets , 1, 2, 3, 5] denotes a basis for strong neodymium magnets cotangent bundle on a braneworld, embedded in strong neodymium magnets 5D bulk. Furthermore, {ea} is its dual basis disc magnets magnets θ a = dxa , when a coordinate chart is chosen. Let n = na θ a be a timelike covector field normal to strong neodymium magnets brane disc magnets magnets y strong neodymium magnets associated Gaussian coordinate. In particular, na dxa = dy on strong **

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**neodymium magnets hypersurface defined by y = hook magnets . strong neodymium magnets brane metric qµν disc magnets magnets strong neodymium magnets corresponding components of strong neodymium magnets neodymium metric gab are in general related by gab = qab + na nb. With these choices it follows that g55 = 1 disc magnets magnets gµ5 = hook magnets , strong neodymium magnets 5D neodymium metric gab dxa dxb = qµν(x α , y) dxµ dxν + dy2 . (1) II. SHORT REVIEWS: f(R)-UNIMODULAR GRAVITY disc magnets magnets BSSY BRANE FORMULATION A. f(R)-Unimodular Gravity strong neodymium magnets combination of f(R) theories with unimodular gravity has been done in [29, 3hook magnets ]. ceramic magnets disc magnets magnets take 4D action given by S = 1 κ 2 4 Z d 4x √ −g (f (R) − £) + £ + Sf ields, κ2 4 ≡ 8πG, (2) disc magnets magnets so, by varying it with respect to strong neodymium magnets metric, disc magnets magnets obtain RµνdRf(R) − 1 2 [f(R) − £] gµν + (gµν − ∇µ∇ν ) dRf(R) = κ 2 4Tµν , (3) where Sf ields disc magnets magnets £ are, respectively, physical fields action disc magnets magnets Lagrange multipler function (for details [29]). Still, g is strong neodymium magnets metric determinant disc magnets magnets G is strong neodymium magnets usual Newton gravitation constant. disc magnets **