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A tour of exceptional geometry. (English) Zbl 1055.53039

Berger’s classification of holonomy groups [Bull. Soc. Math. Fr. 83, 279–330 (1955; Zbl 0068.36002)], includes \(SU(3)\) and the exceptional holonomy groups \( G_{2}\), Spin(7). After a discussion of \(G_{2}\) and its manifestation and the definition of various groups the author shows that in certain simple situations it is possible to calculate the Levi-Civita connection \(\nabla\) directly from knowledge of the exterior derivative \(d\) and to calculate it for specific metrics on \(7\)-manifolds. Before the last section the author studies the \(G_{2}\), Spin(7), \(SU(3)\) structures respectively and in the last section he gives some topological observations that are relevant for the construction of orbifolds.

MSC:

53C29 Issues of holonomy in differential geometry
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds

Citations:

Zbl 0068.36002
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