The Skew-Torsion Holonomy Theorem

This chapter is based on [261,262] and related to the work of Agricola and Friedrich in [5, 7]. We develop the theory of the so-called skew-torsion holonomy systems, which extend Simons’ holonomy systems from Chapter 8. Our approach is geometric and based on Euclidean submanifold geometry and normal holonomy. The main result in this chapter is the Skew-Torsion Holonomy Theorem, which extends the Simons Holonomy Theorem. In contrast to the latter case, the only transitive group that can occur is the full special orthogonal group.