## 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

### Keywords:

special holonomy; differential forms; Hermitian manifold

Zbl 0068.36002
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