Affine connections with W=0

Abstract

For 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 the Weyl curvature vanishes if and only if the manifold is (locally) diffeomorphic to a real projective space with the connection, when transported to the projective space, in the projective class of the Levi-Civita connection of the Fubini-Study metric. Associated to a connection on an even-dimensional M is an almost complex structure on J(M) the bundle of all complex structures on the tangent spaces of M, c.f. [O'Brian-Rawnsley]. We show that this structure is a projective invariant, and when integrable can be obtained from a torsionless connection which must then have W=0. We also show that two torsionless connections define the same almost complex structure if and only if they are projectively equivalent.