Riemannian Holonomy
The holonomy of a Riemannian manifold (M, g) is just the holonomy group of the Levi-Civita connection on the tangent bundle to M. A 'generic' n-dimensional Riemannian manifold has an O(n) holonomy, or SO(n) if it is orientable. Manifolds whose holonomy groups are proper subgroups of O(n) or SO(n) have special properties.
One of the earliest fundamental results on Riemannian holonomy is the theorem of Borel & Lichnerowicz (1952), which asserts that the holonomy group is a closed Lie subgroup of O(n). In particular, it is compact.
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