Strong G₂-structures with torsion: a survey and new examples

This article has two main parts. The first consists of a brief survey of G₂-structures with torsion, namely G₂-structures admitting a G₂-connection with totally skew-symmetric torsion. We focus primarily on the strong condition that the associated torsion $3$-form is closed. We review their basic properties and their connections with the heterotic G₂ system, generalized geometry, and six-dimensional geometry. In the second part, we construct two complete cohomogeneity-one strong G₂T-structures on R× S³× S³ with non-zero closed and co-closed characteristic torsion, and non-vanishing characteristic Ricci tensor. Both have an asymptotically cylindrical end; at the other end, the first is asymptotically conical (AC), with a torsion-free G₂ cone as its asymptotic model, while the second is asymptotically locally conical (ALC), with a constant-length circle bundle over a Calabi--Yau cone as its asymptotic model. We investigate the induced $SU(3)$-structures on their circle quotients and show that they are globally conformally almost Kähler and globally conformally Kähler, respectively. To the best of our knowledge, these provide the first complete strong G₂T-structures whose underlying metrics are not locally Riemannian products.