Paper · Jul 8, 2023

The Riemannian Bianchi identities of metric connections with skew torsion and generalized Ricci solitons

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Abstract

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form $T$ is harmonic, $d T = δ T = 0$ or the curvature of the torsion connection $R \in S^{2} Λ^{2}$ then the scalar curvature of a $\nabla$ -Einstein manifold is determined by the norm of the torsion up to a constant. It is proved that a compact generalized gradient Ricci soliton with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature are constants. In this case the torsion 3-form is harmonic and the gradient function has to be constant. Necessary and sufficient conditions a metric connection with skew torsion to satisfy the Riemannian first Bianchi identity as well as the contracted Riemannian second Binachi identity are presented. It is shown that if the torsion connection satisfies the Riemannian first Bianchi identity then it satisfies the contracted Riemannian second Bianchi identity. It is also proved that a metric connection with skew torsion satisfying the curvature identity $R (X, Y, Z, V) = R (Z, Y, X, V)$ must be flat.

Comments: exposition improved, typos corrected, 14 pages, accepted in Results in Math

Generalized & G₂ flows Ricci solitons Einstein metrics Ricci-flat & Calabi–Yau Scalar curvature

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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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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The Riemannian Bianchi identities of metric connections with skew torsion and generalized Ricci solitons

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