Manifolds with vectorial torsion

作者：Agricola Ilka^*; Kraus Margarita

来源：Differential Geometry and its Applications, 2016, 45: 130-147.

DOI：10.1016/j.difgeo.2016.01.004

• Metric connection with vectorial torsion
• Dirac operator
• Warped product
• Parallel spinor
• Killing spinor
• Weyl manifold

The present note deals with the properties of metric connections del with vectorial torsion V on semi-Riemannian manifolds (M-n, g). We show that the del-curvature is symmetric if and only if V-b is closed, and that V-perpendicular to 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-fiat 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(b) = 0 for n = 3 or n >= 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 Ric(V) = 0 for n not equal 4 and Ricv(X) = X right perpendicular dV(b) 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 R and an Einstein space of positive scalar curvature. We also prove that if dV(b) = 0, there are no non-trivial del-Killing spinor fields.

• 出版日期2016-4