Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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D. Conti - F. A. Rossi

The Ricci tensor of almost parahermitian manifolds

created by rossi on 11 Jan 2018
modified on 12 Jan 2018


Ann Glob Anal Geom

Inserted: 11 jan 2018
Last Updated: 12 jan 2018

Journal: 10.1007/s10455-017-9584-y
Year: 2016
Doi: 10.1007/s10455-017-9584-y

ArXiv: 1605.01890 PDF


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.

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