This paper analyzes one and two-point correlation functions of the energy-momentum tensor on homogeneous spaces of constant curvature, focusing on the possibility of proving a c-theorem. The authors derive Ward identities relating these correlation functions and obtain explicit expressions for free scalar, spinor, and vector fields in general dimensions and four dimensions, respectively. They use a geometric formalism independent of coordinate choices and analyze the role of conformal symmetries on these spaces. The results are constrained by the operator product expansion. For negative curvature, the spectral representation of two-point functions of vector currents is derived in detail, and an extension to the energy-momentum tensor is discussed. The paper demonstrates that two-point functions at non-coincident points are not directly related to the coefficient \(a\) and shows that the irreversibility of RG flow of any function of couplings reducing to \(a\) at fixed points cannot be straightforwardly demonstrated in this framework.This paper analyzes one and two-point correlation functions of the energy-momentum tensor on homogeneous spaces of constant curvature, focusing on the possibility of proving a c-theorem. The authors derive Ward identities relating these correlation functions and obtain explicit expressions for free scalar, spinor, and vector fields in general dimensions and four dimensions, respectively. They use a geometric formalism independent of coordinate choices and analyze the role of conformal symmetries on these spaces. The results are constrained by the operator product expansion. For negative curvature, the spectral representation of two-point functions of vector currents is derived in detail, and an extension to the energy-momentum tensor is discussed. The paper demonstrates that two-point functions at non-coincident points are not directly related to the coefficient \(a\) and shows that the irreversibility of RG flow of any function of couplings reducing to \(a\) at fixed points cannot be straightforwardly demonstrated in this framework.