Variations of gwistor space

Abstract

We study natural variations of the $\mathrm{G}_2$ structure $\sigma_0\in\Lambda^3_+$ existing on the unit tangent sphere bundle $SM$ of any oriented Riemannian 4-manifold $M$. We find a circle of structures for which the induced metric is the usual one, the so-called Sasaki metric, and prove how the original structure has a preferred role in the theory. We deduce the equations of calibration and cocalibration, as well as those of $W_3$ pure type and nearly-parallel type.