On Conformally Flat Spaces with Commuting Curvature and Ricci Transformations

Extract

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 property

where R(X, Y) is the curvature transformation determined by the vector fields X and Y.

The property (P) is equivalent to

To see this we observe first that a skew symmetric and a symmetric endomorphism commute if and only if their product is skew symmetric.