Projectively related metrics, Weyl nullity, and metric projectively invariant equations

TL;DR

This paper develops a comprehensive theory of metric projective structures with Weyl nullity, introducing new invariants and connections, and applies these to classify solutions and geometries, including results on special manifolds and metric equivalences.

Contribution

It introduces a fundamental 2-tensor invariant, a new canonical tractor connection, and invariant differential operators for metric projective structures with Weyl nullity, advancing the understanding of metrisability equations.

Findings

Strong local and global classification results for solutions.

Closed Sasakian and Kähler manifolds do not admit nontrivial solutions.

On closed manifolds, nontrivially projectively equivalent metrics cannot share the same tracefree Ricci tensor.

Abstract

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 $B$.