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

TL;DR

This paper explores the geometric properties of compact almost complex Calabi-Yau with torsion 6-manifolds, revealing conditions under which the torsion connection is Ricci-flat and satisfies Riemannian curvature identities.

Contribution

It establishes new curvature identities and conditions for Ricci-flatness in almost Calabi-Yau with torsion 6-manifolds, linking torsion properties to Riemannian curvature.

Findings

Nijenhuis tensor is parallel with respect to torsion connection.

If torsion is closed, the manifold is a generalized gradient Ricci soliton.

Torsion connection is Ricci-flat when torsion norm or scalar curvature is constant.

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.