SPECIAL RIEMANNIAN STRUCTURES ON FOUR-MANIFOLDS: FROM TWISTOR SPACES TO UNOBSTRUCTED CONDITIONS

This thesis deals with special Riemannian metrics on smooth manifolds: in particular, we study the relations between the existence of metrics satisfying curvature properties and the topology of the manifold they are defined on. In the first part of the work, we introduce Riemannian twistor spaces, which are 6-dimensional manifolds which can be associated to any Riemannian four-manifold: these spaces can be endowed with two natural almost Hermitian structures, which are strictly linked to the geometry of the underlying manifold. We prove rigidity and characterization results for special metrics on four-manifolds in twistorial terms and we study curvature conditions holding on the twistor spaces: in particular, we show a non-existence result regarding locally conformally flat twistor spaces, we classify Bochner-parallel and Ricci-parallel twistor spaces, we generalize the integrability condition of the celebrated Atiyah-Hitchin-Singer almost complex structure, which occurs if and only if the underlying manifold is half conformally flat, and we characterize manifolds such that the almost complex structures on their twistor spaces satisfy a PDE in their first covariant derivatives. We also focus on Einstein four-manifolds, finding two local characterizations in terms of the Nijenhuis tensors defined on their twistor spaces; moreover, we classify Einstein four-manifolds with positive scalar curvature, under the assumption that the scalar curvature of their twistor spaces is constant on the fibers of the twistor bundle. In the second part of the thesis, we adapt a construction method introduced by Aubin in order to produce Riemannian metrics which are not obstructed by the topology of the manifold and which satisfy non-vanishing curvature conditions involving the Weyl tensor and the geometric quantities derived from it. In particular, we prove that, on any compact four-manifold, there exists a Riemannian metric whose (anti-)self-dual Weyl tensor has constant squared norm equal to 1. We show that this result also implies the existence, in a given conformal class of metrics, of minimizers, called weak half harmonic Weyl metrics, of the L^2-norm of the divergence of this tensor; we also show that, on any compact manifold of dimension at least 4, there exists a Riemannian metric such that the Cotton tensor vanishes at finitely many points. Finally, we prove the existence of Riemannian metrics with nowhere vanishing Bach tensor on any compact four-manifold.