On Conformally Flat Spaces with Commuting Curvature and Ricci Transformations

Abstract

Let (M, g) be a C∞ Riemannian manifold and A be the field of symmetric endomorphisms corresponding to the Ricci tensor S; that is,We consider a condition weaker than the requirement that A be parallel (▽ A = 0), namely, that the “second exterior covariant derivative” vanish ( ▽x▽YA — ▽Y ▽XA — ▽[X,Y]A = 0), which by the classical interchange formula reduces to the propertywhere R(X, Y) is the curvature transformation determined by the vector fields X and Y.The property (P) is equivalent toTo see this we observe first that a skew symmetric and a symmetric endomorphism commute if and only if their product is skew symmetric.