  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.
