Manifolds with vectorial torsion

TL;DR

This paper studies metric connections with vectorial torsion on semi-Riemannian manifolds, exploring their curvature, construction, and spinor fields, revealing conditions for symmetry, conformal equivalence, and implications for general relativity.

Contribution

It provides new insights into the properties of connections with vectorial torsion, including curvature conditions, construction methods, and spinor field analysis, extending understanding in differential geometry and relativity.

Findings

Curvature symmetry iff $V^{lat}$ is closed.

Construction of $V$-torsion connections on warped products.

Existence and properties of $V$-parallel spinor fields.

Abstract

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.