The Geometry of Complex Space-Times with Torsion

TL;DR

This paper investigates the conditions for the existence of special surfaces called α-surfaces in complex space-times with torsion, revealing how torsion influences their integrability and extending results to real manifolds.

Contribution

It derives the necessary and sufficient conditions for α-surfaces in complex space-times with torsion, incorporating torsion effects into integrability conditions.

Findings

Derived integrability condition involving torsion spinor

Identified a class of conformally right-flat, right-torsion-free space-times

Extended results to real manifolds with positive-definite metrics

Abstract

The necessary and sufficient condition for the existence of $\alpha$-surfaces in complex space-time manifolds with nonvanishing torsion is derived. For these manifolds, Lie brackets of vector fields and spinor Ricci identities contain explicitly the effects of torsion. This leads to an integrability condition for $\alpha$-surfaces which does not involve just the self-dual Weyl spinor, as in complex general relativity, but also the torsion spinor, in a nonlinear way, and its covariant derivative. A similar result also holds for four-dimensional, smooth real manifolds with a positive-definite metric. Interestingly, a particular solution of the integrability condition is given by conformally right-flat and right-torsion-free space-times.