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 19 May 2018


Published Paper

Inserted: 11 jan 2018
Last Updated: 19 may 2018

Journal: Annals of Global Analysis and Geometry
Volume: 53
Number: 4
Pages: 467--501
Year: 2018
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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