Projectively related metrics, Weyl nullity and metric projectively invariant equations

A metric projective structure is a manifold equipped with the unparametrized geodesics of some pseudo-Riemannian metric. We make a comprehensive treatment of such structures in the case that there is a certain (well-motivated) algebraic restriction on the projective Weyl curvature, a nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invariant that we discover. It leads to a new canonical tractor connection for these geometries, which is defined on a rank ( n + 1 ) -bundle. We show this connection is linked to the metrizability equations that govern the existence of metrics compatible with the structure. The fundamental 2-tensor also leads to a new class of invariant linear differential operators that are canonically associated to these geometries; included is a third-order equation studied by Gallot et al. and via the new connection we show its equivalence (on suitable geometries and classes of solutions) to the first-order metrizability equation. We apply the results to study the metrizability equation, in the nullity setting described. We obtain strong local and global results on the nature of solutions and also on the nature of the geometries admitting such solutions, obtaining classification results in some cases. We show that closed Sasakian and Kähler manifolds do not admit non-trivial solutions. We also prove that, on a closed manifold, two non-trivially projectively equivalent metrics cannot have the same trace-free Ricci tensor. We show that on a closed manifold, a metric having a non-trivial solution of the metrizability equation cannot have a two-dimensional nullity space at every point. In these statements, the meaning of trivial solution is dependent on the context. There is a function B naturally appearing if a metric projective structure has nullity. We analyse in detail the case when this is not a constant, and describe all non-trivially projectively equivalent Riemannian metrics on closed manifolds with non-constant B.