Affine Connections with W = 0

If ∇ is a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl’s result that W vanishes if and only if (M,∇) is (locally) diffeomorphic to RP with ∇, when transported to RP, in the projective class of ∇LC , the Levi-Civita connection of the Fubini–Study metric on RP. If M is even dimensional and J(M) denotes the bundle of all endomorphisms j of the tangent spaces of M , a connection∇ determines an almost complex structure J on J(M) [9]. We show that J is a projective invariant, that an integrable J can be obtained from a torsionless connection and that we must then have W = 0. We also show for torsionless connections ∇, ∇ that J = J ′ if and only if ∇ and ∇ are projectively equivalent.