Let (M,g) be an oriented n-dimensional Riemannian manifold. Parallel transport around closed loops based at a point p induces linear transformations of the tangent space T_pM. The holonomy group is the resulting subgroup of SO(T_pM) = SO(n). For a generic metric, the holonomy group is equal to SO(n), but for special metrics we get strict subgroups of SO(n). In this talk I will discuss which subgroups can arise, and how the holonomy of a metric manifests itself through additional structures on the manifold.

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