Noncompactness and maximum mobility of type III Ricci-flat self-dual neutral Walker four-manifolds

TL;DR

This paper classifies certain four-dimensional Ricci-flat self-dual neutral manifolds of Petrov type III, showing they cannot be compact or homogeneous, and explores their geometric properties, symmetries, and explicit examples using hyperbolic geometry.

Contribution

It provides a canonical coordinate form, classifies manifolds with maximum degree of mobility, and describes the kernel of the Killing operator for specific surface connections.

Findings

Such manifolds cannot be compact or locally homogeneous.

Maximum degree of mobility for these manifolds is 3.

Explicit examples of foliations with special connections are constructed.

Abstract

It is shown that a self-dual neutral Einstein four-manifold of Petrov type III, admitting a two-dimensional null parallel distribution compatible with the orientation, cannot be compact or locally homogeneous, and its maximum possible degree of mobility is 3. Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo found a general coordinate form of such manifolds. The present paper also provides a modified version of that coordinate form, valid in a suitably defined generic case and, in a sense, "more canonical" than the usual formulation. Moreover, the local-isometry types of manifolds as above having the degree of mobility equal to 3 are classified. Further results consist in explicit descriptions, first, of the kernel and image of the Killing operator for any torsionfree surface connection with everywhere-nonzero, skew-symmetric Ricci tensor, and, secondly, of a moduli curve for surface connections with the properties just mentioned that are, in addition, locally homogeneous. Finally, hyperbolic plane geometry is used to exhibit examples of codimension-two foliations on compact manifolds of dimensions greater than 2 admitting a transversal torsionfree connection, the Ricci tensor of which is skew-symmetric and nonzero everywhere. No such connection exists on any closed surface, so that there are no analogous examples in dimension 2.