Paper · Jul 11, 2023

The Riemannian curvature identities for the torsion connection on -manifold and generalized Ricci solitons

Stefan Ivanov, Alexander Petkov

math.DGarXiv:2307.06438v4
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Abstract

It is shown that on compact –manifold with exterior derivative of the Lee form lying in the Lie algebra the curvature of the –torsion connection with vanishing Ricci tensor if and only if the -form torsion is parallel with respect to the Levi-Civita connection. It is also proved that satisfies the Riemannian first Bianchi identity exactly when the -form torsion is parallel with respect to the Levi-Civita and to the –torsion connections simultaneously. Precise conditions for a compact –manifold to has closed torsion are given in terms of the Ricci tensor of the –torsion connection. It is shown that a compact –manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact –manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the –structure.

Comments: 17 pages, exposition improved, final version to appear in Math. Nachrichten

Generalized & G₂ flowsRicci solitonsRicci-flat & Calabi–YauScalar curvature

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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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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