Manifolds with vectorial torsion

The present note deals with the properties of metric connections ∇ with vectorial torsion V   on semi-Riemannian manifolds (Mn,g)(Mn,g). We show that the ∇-curvature is symmetric if and only if V♭V♭ is closed, and that V⊥V⊥ then defines an (n−1)(n−1)-dimensional integrable distribution on MnMn. If the vector field V is exact, we show that the V-curvature coincides up to global rescaling with the Riemannian curvature of a conformally equivalent metric. We prove that it is possible to construct connections with vectorial torsion on warped products of arbitrary dimension matching a given Riemannian or Lorentzian curvature—for example, a V-Ricci-flat connection with vectorial torsion in dimension 4, explaining some constructions occurring in general relativity. Finally, we investigate the Dirac operator D of a connection with vectorial torsion. We prove that for exact vector fields, the V-Dirac spectrum coincides with the spectrum of the Riemannian Dirac operator. We investigate in detail the existence of V-parallel spinor fields; several examples are constructed. It is known that the existence of a V  -parallel spinor field implies dV♭=0dV♭=0 for n=3n=3 or n≥5n≥5; for n=4n=4, this is only true on compact manifolds. We prove an identity relating the V-Ricci curvature to the curvature in the spinor bundle. This result allows us to prove that if there exists a nontrivial V  -parallel spinor, then RicV=0RicV=0 for n≠4n≠4 and RicV(X)=XdV♭RicV(X)=XdV♭ for n=4n=4. We conclude that the manifold is conformally equivalent either to a manifold with Riemannian parallel spinor or to a manifold whose universal cover is the product of RR and an Einstein space of positive scalar curvature. We also prove that if dV♭=0dV♭=0, there are no non-trivial ∇-Killing spinor fields.