A circle quotient of a $G_2$ cone

TL;DR

This paper investigates the geometric structure of a $G_2$ cone quotient, deriving explicit formulas for the induced metric, curvature, and symplectic forms, revealing symmetry properties and invariants in the reduced space.

Contribution

It provides explicit expressions for the metric, curvature, and symplectic forms of a $G_2$ cone quotient, highlighting symmetry invariants and geometric features.

Findings

Explicit formulas for the induced metric, curvature, and symplectic forms.

Identification of $SO(3)$ invariance in the geometric tensors.

Characterization of the quotient space's geometric structure.

Abstract

A study is made of $R^6$ as a singular quotient of the conical space $R^+\times CP^3$ with holonomy $G_2$ with respect to an obvious action by $U(1)$ on $CP^3$ with fixed points. Closed expressions are found for the induced metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of $SO(3)$ on $R^6$, which can be used effectively to describe the resulting geometrical features.