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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.

MSC:

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.)

Citations:

Zbl 0889.53029
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References:

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