2023/07/08 by Stefan Ivanov, Ivanov, Stefan, Nikola Stanchev +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.2307.03986
openalex publication_date 2023/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, dT=δT=0 or the curvature of the torsion connection R∈ S2Λ2 then the scalar curvature of a ∇-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.