Nijenhuis geometry of parallel tensors

A tensor—meaning here a tensor field \(\,\Theta \) $\mathrm{\Theta }$ of any type (p, q) on a manifold—may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of integrability for differential q-forms, \(q=0,1,2,n-1,n\) $q=0,1,2,n-1,n$ (in dimension n), vectors, bivectors, symmetric \(\,(2,0)\,\) $(2,0)$ and \(\,(0,2)\,\) $(0,2)$ tensors, as well as complex-diagonalizable and nilpotent tensors of type \(\,(1,1)\) $(1,1)$ . In most cases, integrability is equivalent to algebraic constancy of \(\,\Theta \,\) $\mathrm{\Theta }$ coupled with the vanishing of one or more suitably defined Nijenhuis-type tensors, depending on \(\,\Theta \,\) $\mathrm{\Theta }$ via a quasilinear first-order differential operator. For \(\,(p,q)=(1,1)\) $(p,q)=(1,1)$ , they include the ordinary Nijenhuis tensor.