Annals of Global Analysis and Geometry

, Volume 53, Issue 4, pp 467–501 | Cite as

The Ricci tensor of almost parahermitian manifolds

  • Diego Conti
  • Federico A. Rossi


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 \(\mathrm {SL}(n,\mathbb {R})\)-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.


Einstein pseudoriemannian metrics Almost parahermitian structures Intrinsic torsion Ricci tensor 

Mathematics Subject Classification

53C15 53C10 53C29 53C50 


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Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanItaly

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