A circle quotient of a \(G_2\) cone (Q1995641)

The work is a fundamental study on the manifold $\mathbb{R}^6$ as a singular quotient of the conical space $\mathbb{R}^+\times \mathbb{CP}^3$ with $g$ holonomy $G_2$ with respect to the action of the group $\mathrm{U}(1)$ on $\mathbb{CP}^3$. Some important expressions for induced metrics, symplectic 2-forms and similar objects that characterize the reduction, are found. The invariance of the tensors with respect to the action of the Lie group $\operatorname{SO}(3)$ on $\mathbb{R}^6$ is studied. The twistor fibration $\pi:\mathbb{CP}^3\to S^4$ is considered also with respect to the action of $\operatorname{SO}(3)$. Families of Riemannian metrics on some submanifolds are studied. Important results on the restrictions of these metrics are obtained.