GTA Seminar | Speaker: Henrik Winther (Univ. Tromsø, Noruega)

Title: Metric connections with cyclic torsion

Abstract: The study of affine metric connections dates back to Cartan who proved that the torsion tensor of a metric connection splits under the action of the orthogonal group into three modules: vectorial torsion, skew torsion and traceless cyclic torsion. In the context of G-structures, the machinery developed by D. Spencer and others in the 60's, provides a framework for studying connections via certain G-equivariant complexes. We will review this theory and show how it specializes to the Riemannian setting. This is closely related to the famous work of Gray and Hervella on the 16 classes of almost Hermitian manifolds, which has been generalized to most of the popular Riemannian G-structures. For producing adapted connections, one can ask that the torsion takes values in the orthogonal complement to the image of the Spencer delta map. This choice is standard and leads to good compatibility with the classes in the sense of Gray and Hervella, however not necessarily with Cartan's decomposition of the torsion tensors. We will show that sometimes there are natural connections which arise from a different choice of complement (called a "normalization condition"), which are aligned with Cartan's decomposition. We illustrate this by showing that for any almost Hermitian manifold in dimension 4, there is a unique Hermitian connection with self-dual traceless-cyclic torsion. 
Joint work with Ana Cristina Ferreira and Ilka Agricola.