The Ricci tensor of almost parahermitian manifolds

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 parakähler version of the Goldberg conjecture, and obtain the first compact examples of a non-flat, Ricci-flat nearly parakähler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parakähler metrics.

36 pages; v2, minor corrections, two references added, presentation improved; v3, clarified definition of \overline{F}; corrected coefficients in Proposition 12, fixed typos in statements of Lemma 13 and Theorem 14