The Ricci tensor of almost parahermitian manifolds

TL;DR

This paper derives a formula for the Ricci tensor of almost parahermitian manifolds using intrinsic torsion, constructs examples of Einstein and Ricci-flat structures, and explores obstructions to Ricci-flat parak"ahler metrics.

Contribution

It provides a new Ricci tensor formula for almost parahermitian manifolds and constructs novel Einstein and Ricci-flat examples, disproving a conjecture and analyzing topological obstructions.

Findings

Derived Ricci tensor formula using intrinsic torsion.

Constructed Einstein and Ricci-flat examples on Lie groups.

Disproved the parak"ahler Goldberg conjecture.

Abstract

We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi-Civita connection. The formula uses the intrinsic torsion of an underlying SL(n,R)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parak\"ahler version of the Goldberg conjecture, and obtain the first compact examples of a non-flat, Ricci-flat nearly parak\"ahler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parak\"ahler metrics.