The Intrinsic Torsion of Su(3) and G2 Structures

Let G be a subgroup of SO(n). A G-structure on a smooth manifold M of dimension n induces a Riemannian metric g on M . The failure of the holonomy group of the Levi-Civita connection of g to reduce to G is measured by the so-called intrinsic torsion τ . It is known that the latter is a tensor which takes values at each point in T ∗ ⊗ g where T ∗ is the cotangent space and g is the orthogonal complement of g in so(n) ∼= ∧2 T ∗ . This note is concerned with the cases