This paper analyzes the one and two point functions of the energy momentum tensor on homogeneous spaces of constant curvature, with a focus on their implications for the c-theorem in four-dimensional field theories. The authors derive Ward identities and explicit expressions for correlation functions in free scalar, spinor, and vector field theories. 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 and show that two-point functions on negative curvature spaces are not directly related to the a coefficient in any straightforward way. The paper discusses the spectral representation of two-point functions for vector currents and the energy momentum tensor on spaces of constant curvature, and explores the implications for the c-theorem in four dimensions. It also examines the consistency conditions for the energy momentum tensor trace anomaly and renormalization group equations on curved backgrounds. The authors derive the general form of the energy momentum tensor two-point function on homogeneous spaces of constant curvature, showing that it can be expressed in terms of two independent tensor structures representing spin 2 and spin 0 contributions. The paper concludes with a discussion of the c-theorem and the challenges in deriving it for field theories on spaces of constant curvature.This paper analyzes the one and two point functions of the energy momentum tensor on homogeneous spaces of constant curvature, with a focus on their implications for the c-theorem in four-dimensional field theories. The authors derive Ward identities and explicit expressions for correlation functions in free scalar, spinor, and vector field theories. 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 and show that two-point functions on negative curvature spaces are not directly related to the a coefficient in any straightforward way. The paper discusses the spectral representation of two-point functions for vector currents and the energy momentum tensor on spaces of constant curvature, and explores the implications for the c-theorem in four dimensions. It also examines the consistency conditions for the energy momentum tensor trace anomaly and renormalization group equations on curved backgrounds. The authors derive the general form of the energy momentum tensor two-point function on homogeneous spaces of constant curvature, showing that it can be expressed in terms of two independent tensor structures representing spin 2 and spin 0 contributions. The paper concludes with a discussion of the c-theorem and the challenges in deriving it for field theories on spaces of constant curvature.