In this paper we look at the question of integrability, or not, of the two natural almost complex structures J(del)(+/-) defined on the twistor space J(M, g) of an even-dimensional manifold M with additional structures g and del a g-connection. We measure their non-integrability by the dimension of the span of the values of N-J del +/-. We also look at the question of the compatibility of J(del)(+/-) with a natural closed 2-form omega(J(M,g,del)) defined on J(M, g). For (M, g) we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection del. In all cases J(M, g) is a bundle of complex structures on the tangent spaces of M compatible with g. In the case M is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.
NSTL
Amer Inst Mathematical Sciences-Aims
