Fixed-Point Analysis of FLRW Cosmology Through the Einstein Tensor Cycle Transformation

We define the iterative map from the metric \( g_{\mu\nu} \) to the Einstein tensor \( G_{\mu\nu} \) as the Einstein Tensor Cycle (ETC) transformation, \( g^{(n+1)}_{\mu\nu} := G_{\mu\nu}[g^{(n)}] \), and geometrically characterize Einstein spaces containing the cosmological constant \( \Lambda \) through its fixed points \( G_{\mu\nu}=\lambda g_{\mu\nu} \). 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 \( k=\pm1,0 \), analyzing how distinct spatial geometries are uniformly derived through a single iteration procedure. For the de Sitter family (\( H_{0}=\sqrt{\Lambda/3} \)), we confirm that \( G_{00}=\Lambda \) and corresponding spatial components are realized in the first transformation and remain invariant in subsequent iterations for both k=+1 with \( a(t)=a_{0}\cosh(H_{0}t) \) and k=-1 with \( a(t)=a_{0}\sinh(H_{0}t) \). For the flat case (k=0), the Friedmann equation \( G_{00}=8\pi G\rho/c^{2} \) 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.