Nearly parallel \(G_2\)-structures with large symmetry group. (English) Zbl 1462.53016
Summary: We prove the existence of a one-parameter family of nearly parallel \(G_2\)-structures on the manifold \(\text{S}^3\times\mathbb{R}^4\), which are mutually non-isomorphic and invariant under the cohomogeneity one action of the group \(\text{SU}(2)^3\). This family connects the two locally homogeneous nearly parallel \(G_2\)-structures that are induced by the homogeneous ones on the sphere \(S^7\).
MSC:
| 53C10 | \(G\)-structures |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C30 | Differential geometry of homogeneous manifolds |
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