  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.
