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The Ricci tensor of almost parahermitian manifolds

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

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via Cozzi 55, 20125, Milan, Italy

    Diego Conti & Federico A. Rossi

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  1. Diego Conti
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  2. Federico A. Rossi
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Correspondence to Federico A. Rossi.

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This work was partially supported by FIRB 2012 “Geometria differenziale e teoria geometrica delle funzioni” and GNSAGA of INdAM.

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Conti, D., Rossi, F.A. The Ricci tensor of almost parahermitian manifolds. Ann Glob Anal Geom 53, 467–501 (2018). https://doi.org/10.1007/s10455-017-9584-y

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