On nearly parallel \(G_2\)-structures. (English) Zbl 0898.53038

Nearly parallel \(G_2\)-manifolds are seven-dimensional manifolds admitting a spin structure with a Killing spinor. Excluding the case of the seven-dimensional sphere, there exist three types of such manifolds, depending on the dimension of the space \(KS\) of all Killing spinors. Nearly parallel \(G_2\)-manifolds with \(\dim (KS)=3\) are 3-Sasakian manifolds, and nearly parallel \(G_2\)-manifolds with \(\dim (KS)=2\) are Einstein-Sasakian manifolds. Examples of nearly parallel \(G_2\)-manifolds with \(\dim (KS)=1\) are the squashed 7-sphere, Wallach spaces and an Einstein metric on \(SO(5)/SO(3)\).
In this paper, the authors construct new examples of nearly parallel \(G_2\)-manifolds with \(\dim (KS)\) \(=1\) from the examples of 3-Sasakian manifolds due to C. P. Boyer, K. Galicki and B. M. Mann [J. Reine Angew. Math. 455, 183-220 (1994; Zbl 0889.53029)]. In fact, the authors prove that a seven-dimensional manifold with a 3-Sasakian structure has a second Einstein metric obtained from the given one by scaling the metric in the directions of the orbits of the Spin(3)-action. Furthermore, the authors study the automorphism group of compact nearly parallel \(G\)-manifolds, and they classify such manifolds with a large symmetry group.


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)


Zbl 0889.53029
Full Text: DOI


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