Suppose gab is a metric on a manifold M, ∇ is the derivative operator on M compatible with gab, and Rabcd is associated with ∇. Then Rabcd (= gam Rmbcd) satisfies the following conditions.
(1) Rab(cd) = 0.
(2) Ra[bcd] = 0.
(3) R(ab)cd = 0.
(4) Rabcd = Rcdab.
Conditions (1) and (2) follow directly from clauses (2) and (3) of proposition, which goes like
Suppose ∇ is a derivative operator on the manifold M. Then the curvature tensor field Rabcd associated with ∇ satisfies the following conditions:
(1) For all smooth tensor fields αa1…arb1 …bs on M,
2∇[c∇d] αa1…arb1 …bs = αa1…arnb2…bs Rnb1cd +…+ αa1…arb1…bs-1n Rnbscd – αna2…arb1…bs Ra1ncd -…- αa1…ar-1nb1…bs Rarncd.
(2) Rab(cd) = 0.
(3) Ra[bcd] = 0.
(4) ∇[mRa|b|cd (Bianchi’s identity).
And by clause (1) of that proposition, we have, since ∇agbc = 0,
0 = 2∇[c∇d]gab = gnbRnacd + ganRnbcd = Rbacd + Rabcd.
That gives us (3). So it will suffice for us to show that clauses (1) – (3) jointly imply (4). Note first that
0 = Rabcd + Radbc + Racdb
= Rabcd − Rdabc − Racbd.
(The first equality follows from (2), and the second from (1) and (3).) So anti-symmetrization over (a, b, c) yields
0 = R[abc]d −Rd[abc] −R[acb]d.
The second term is 0 by clause (2) again, and R[abc]d = −R[acb]d. So we have an intermediate result:
R[abc]d = 0 —– (1)
Now consider the octahedron in the figure below.
Using conditions (1) – (3) and equation (1), one can see that the sum of the terms corresponding to each triangular face vanishes. For example, the shaded face determines the sum
Rabcd + Rbdca + Radbc = −Rabdc − Rbdac − Rdabc = −3R[abd]c = 0
So if we add the sums corresponding to the four upper faces, and subtract the sums corresponding to the four lower faces, we get (since “equatorial” terms cancel),
4Rabcd −4Rcdab = 0
This gives us (4).
We say that two metrics gab and g′ab on a manifold M are projectively equivalent if their respective associated derivative operators are projectively equivalent – i.e., if their associated derivative operators admit the same geodesics up to reparametrization. We say that they are conformally equivalent if there is a map : M → R such that
g′ab = Ω2gab
is called a conformal factor. (If such a map exists, it must be smooth and non-vanishing since both gab and g′ab are.) Notice that if gab and g′ab are conformally equivalent, then, given any point p and any vectors ξa and ηa at p, they agree on the ratio of their assignments to the two; i.e.,
(g′ab ξa ξa)/(gab ηaηb) = (gab ξa ξb)/(g′ab ηaηb)
(if the denominators are non-zero).
If two metrics are conformally equivalent with conformal factor, then the connecting tensor field Cabc that links their associated derivative operators can be expressed as a function of Ω.