Almost Paracomplex Structures on 4-Manifolds

TL;DR

This paper investigates conditions under which almost paracomplex structures on 4-manifolds are parallel, revealing links between curvature, topology, and geometric structures, especially in conformally flat and Einstein manifolds.

Contribution

It characterizes when almost paracomplex structures are parallel on 4-manifolds, linking their properties to scalar curvature, Einstein metrics, and topological constraints.

Findings

Parallel structures imply zero scalar curvature on conformally flat manifolds.

Parallelism of $j$ corresponds to tangent eigenplanes with orthogonal foliations.

Certain topological conditions, like vanishing Hirzebruch signature, are necessary for Einstein metrics with parallel structures.

Abstract

Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics of any real 3-dimensional space form admits both isometric and anti-isometric paracomplex structures. This paper considers the existence or otherwise of isometric and anti-isometric almost paracomplex structures $j$ on a pseudo-Riemannian 4-manifold $(M,g)$, such that $j$ is parallel with respect to the Levi-Civita connection of $g$. It is shown that if an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, then the scalar curvature of the metric must be zero. In addition, it is found that $j$ is parallel iff the eigenplanes are tangent to a pair of mutually orthogonal foliations by totally geodesic surfaces. The composition of a Riemannian metric with an isometric almost paracomplex structure $j$ yields a neutral metric $g'$. It is proven that if $j$ is parallel, then $g$ is Einstein iff $g'$ is conformally flat and scalar flat. The vanishing of the Hirzebruch signature is found to be a necessary topological condition for a closed 4-manifold to admit an Einstein metric with a parallel isometric paracomplex structure. Thus, while the K3 manifold admits an Einstein metric with an isometric paracomplex structure, it cannot be parallel. The same holds true for certain connected sums of complex projective 2-space and its conjugate.