  A metric projective structure is a manifold equipped with the unparametrised
geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of
such structures in the case that there is a projective Weyl curvature 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 metrisability 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 equation studied by Gallot
et al.
  We apply the results to study the metrisability 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 K\"ahler manifold do not admit nontrivial solutions. We also prove that, on
a closed manifold, two nontrivially projectively equivalent metrics cannot have
the same tracefree Ricci tensor. We show that on a closed manifold a metric
having a nontrivial solution of the metrisablity equation cannot have
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 nontrivially projectively equivalent Riemannian metrics on closed
manifolds with nonconstant $B$.

Differential Geometry;General Relativity and Quantum Cosmology