Paper · Jul 11, 2023

The Riemannian curvature identities on almost Calabi-Yau with torsion 6-manifold and generalized Ricci solitons

Stefan Ivanov, Nikola Stanchev

math.DGarXiv:2307.05001v5
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Abstract

It is observed that on a compact almost complex Calabi-Yau with torsion 6-manifold the Nijenhuis tensor is parallel with respect to the torsion connection. If the torsion is closed then the space is a compact generalized gradient Ricci soliton. In this case, the torsion connection is Ricci-flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. On a compact almost complex Calabi-Yau with torsion 6-manifold it is shown that the curvature of the torsion connection is symmetric on exchange of the first and the second pairs and has vanishing Ricci tensor if and only if it satisfies the Riemannian first Bianchi identity.

Comments: title slightly changed, typos corrected, exposition improved, 27 pages, no figures, online published in Annali di Matematica Pura ed Applicata (1923 -), https://doi.org/10.1007/s10231-024-01494-4. arXiv admin note: substantial text overlap with arXiv:2307.05001

Generalized & G₂ flowsRicci solitonsRicci-flat & Calabi–YauKähler geometryScalar curvature

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    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Scalar curvature · Same authors Stefan Ivanov, Nikola Stanchev · Shared terms “torsion connection”, “Bianchi identity”, “curvature identity”

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    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Ricci solitons · Same authors Stefan Ivanov, Nikola Stanchev · Shared terms “curvature identity”, “Riemannian curvature”, “vanishing Ricci”

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  8. Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds

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    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Scalar curvature · Shared terms “Riemannian scalar”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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  3. 11B.1
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    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.