Geodesic
On Conformally Flat Spaces with Commuting Curvature and Ricci Transformations
Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 799-804

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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. 
Bishop, R. L.; Goldberg, S.I. On Conformally Flat Spaces with Commuting Curvature and Ricci Transformations. Canadian journal of mathematics, Tome 24 (1972) no. 5, pp. 799-804. doi: 10.4153/CJM-1972-077-6
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     title = {On {Conformally} {Flat} {Spaces} with {Commuting} {Curvature} and {Ricci} {Transformations}},
     journal = {Canadian journal of mathematics},
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