This note examines, from a purely geometric viewpoint, the deformation of the induced metric tensor that arises when 4-dimensional Euclidean space is mapped onto the surface of a 5-dimensional hypersphere of curvature radius R via the gnomonic (central) projection. In particular, we present step-by-step derivations of: (i) the exact induced metric tensor g_μν (including the contribution of the fifth component), and (ii) the geometric relationship between the Einstein tensor and the constant Λ = 3/R². Physical interpretation is deferred to future work; no cosmological assumptions are made in this note.
Noriaki Kihara (Sun,) studied this question.