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Manifolds with vectorial torsion

The present note deals with the properties of metric connections $\nabla$ with vectorial torsion $V$ on semi-Riemannian manifolds $(M^n,g)$. We show that the $\nabla$-curvature is symmetric if and only if $V^{\flat}$ is closed, and that $V^\perp$ then defines an $(n-1)$-dimensional integrable distribution on $M^n$. 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 $dV^\flat=0$ for $n=3$ or $n\geq 5$; for $n=4$, 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 $\mathrm{Ric}^V=0$ for $n\neq 4$ and $\mathrm{Ric}^V(X)=X\lrcorner dV^\flat$ for $n=4$. 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 $\mathbb{R}$ and an Einstein space of positive scalar curvature. We also prove that if $dV^\flat=0$, there are no non-trivial $\nabla$-Killing spinor fields.