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.


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


Zbl 0068.36002
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