Symmetry-Preserving Fixed Points in FLRW Cosmology: The Einstein Tensor Cycle Transformation and Gravitational Invariance

We define the iterative map from the metric gμν to the Einstein tensor Gμν as the Einstein Tensor Cycle (ETC) transformation, gμν(n+1):=Gμν[g(n)], and geometrically characterize Einstein spaces containing the cosmological constant Λ through its fixed points Gμν=λgμν. The FLRW metric’s fundamental symmetries—spatial isotropy (SO(3)) and spacetime homogeneity—are preserved under the ETC transformation and manifest as a fixed-point structure. We apply the ETC transformation to the FLRW metric with curvature parameters k=±1,0, analyzing how distinct spatial geometries are uniformly derived through a single iteration procedure. For the de Sitter family (H0=Λ/3), we confirm that G00=Λ and corresponding spatial components are realized in the first transformation and remain invariant in subsequent iterations for both k=+1 with a(t)=a0cosh(H0t) and k=−1 with a(t)=a0sinh(H0t). For the flat case (k=0), the Friedmann equation G00=8πGρ/c2 is reproduced under exponential expansion. The ETC transformation functions as a unified framework that simultaneously provides solution identification and stability evaluation in cosmological models, clarifying the deep relationship between spacetime symmetry and fixed-point structure.