Geometric structures of vectorial type

We study geometric structures of W4W4-type in the sense of A. Gray on a Riemannian manifold. If the structure group G⊂SO(n) preserves a spinor or a non-degenerate differential form, its intrinsic torsion ΓΓ is a closed 1-form (Proposition 2.1 and Theorem 2.1). Using a G-invariant spinor we prove a splitting theorem (Proposition 2.2). The latter result generalizes and unifies a recent result obtained in [S. Ivanov, M. Parton, P. Piccinni, Locally conformal parallel G2- and Spin(7)-structures, mathdg/0509038], where this splitting has been proved in dimensions n=7,8n=7,8 only. Finally we investigate geometric structures of vectorial type and admitting a characteristic connection ∇c. An interesting class of geometric structures generalizing Hopf structures are those with a ∇c-parallel intrinsic torsion ΓΓ. In this case, ΓΓ induces a Killing vector field (Proposition 4.1) and for some special structure groups it is even parallel.