The Ricci tensor of SU(3)-manifolds

TL;DR

The paper develops explicit relationships between the curvature of a 6-dimensional SU$(3)$-manifold and its intrinsic torsion encoded by six torsion forms. Using a principal SU$(3)$-bundle framework and representation-theoretic decompositions, it derives concrete expressions for the Ricci tensor and scalar curvature in terms of the torsion data, and specializes these formulas to GCY, SGCY, and half-flat cases. A key result is that GCY manifolds have non-positive scalar curvature, vanishing only when the SU$(3)$-structure is Calabi–Yau, while in the SGCY setting the Einstein condition forces Calabi–Yau. The paper also provides an explicit left-invariant example on a nilmanifold demonstrating the interplay between torsion and curvature, and includes technical proofs in an appendix. Overall, it advances the program of understanding Riemannian invariants of SU$(3)$-structures through intrinsic torsion and torsion forms.

Abstract

Following the approach of Bryant we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing of the intrinsic torsion.

Theorems & Definitions (31)

• Remark 1.1
• Definition 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Corollary 2.2
• Lemma 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6