  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.
