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Spinorial description of \(\mathrm{SU}(3)\)- and \(\mathrm{G}_2\)-manifolds. (English) Zbl 1333.53037

Using the spinor theory, a systematic and uniform description of \(\mathrm{SU}(3)\)-structures in dimension 6 and of \(\mathrm G_2\)-structures in dimension 7 is given. \(\mathrm{SU}(3)\)-manifolds and \(\mathrm G_2\)-manifolds are considered as Riemannian spin manifolds of dimension 6 or 7, respectively, equipped with a real spinor field of length one. A method to construct \(\mathrm G_2\)-structures on cones over an \(\mathrm{SU}(3)\)-manifold is also given.

MSC:

53C10 \(G\)-structures
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C27 Spin and Spin\({}^c\) geometry
53C29 Issues of holonomy in differential geometry
53C80 Applications of global differential geometry to the sciences
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References:

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