Einstein $\mathrm{SU}(3)$ and $\mathrm{G}_2$ structures

TL;DR

This paper presents a method to construct Einstein metrics on 6 and 7-dimensional manifolds using warped products of lower-dimensional structures, advancing the understanding of special geometric structures in differential geometry.

Contribution

It introduces a novel approach to generate Einstein $ ext{SU}(3)$ and $ ext{G}_2$-structures from lower-dimensional Einstein structures via warped products.

Findings

Constructed new classes of Einstein $ ext{SU}(3)$-structures.

Constructed new classes of Einstein $ ext{G}_2$-structures.

Demonstrated the method's applicability to different structure classes.

Abstract

We describe a method to obtain $\mathrm{SU}(3)$-structures and $\mathrm{G}_2$-structures on 6 and 7-dimensional manifolds respectively, such that its associated metric is Einstein. More concretely, we have that different classes of $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$-structures, on 5 and 6-dimensional manifolds whose induced metric is Einstein can produce, via warped products, different classes $\mathrm{SU}(3)$ and $\mathrm{G}_2$-structures such that its associated metric is also Einstein.