Curvature conditions for spatial isotropy

In the context of mathematical cosmology, study necessary and sufficient conditions for a semi-Riemannian manifold to be (generalized) Robertson-Walker space-time is important. particular, it requirement development initial data reproduce or approximate standard cosmological model. Usually these involve Einstein field equations, which change if one considers alternative theories gravity coupling matter fields change. Therefore, derivation do not depend on equations an advantage. this work we present geometric such condition. We require existence unit vector distinguish at each point space two (non-equal) sectional curvatures. This equivalent Riemann tensor adopt specific form. Our geometrical approach yields local isometry between same dimension, curvature metric sign (the dimension largest subspace negative definite). Remarkably, simply-connected, global. result generalizes class spaces non-constant theorem that constant curvature, must locally isometric. Because make any assumptions regarding sign, can readily use models within with different fields.