Bochner formulae for orthogonal \(G\)-structures on compact manifolds. (Q1608954)

In this article, a general approach is developed to obtain an obstruction to the existence of reductions of an \(\text{O}(n)\)-structure on a compact manifold to a subgroup \(G\subset \text{O}(n)\). This obstruction is given in the form of an integral formula connecting the \(G\)-irreducible components of the intrinsic torsion tensor of the \(G\)-structure with \(G\)-invariants of the curvature tensor of the associated Riemannian metric. The approach is illustrated for the groups \(\text{U}_n\), \(n\geq 2\); \(\text{SU}_n\), \(n\geq 3\); \(G_2\) and \(\text{Spin}_7\).