To any Riemannian manifold of dimension n is associated a closed subgroup of SO(n), the holonomy
group; this is one of the most basic invariants of the metric. A famous theorem of Berger gives
a complete (and rather small) list of the groups that can appear. The construction of compact
manifolds with holonomy smaller than SO(n) leads to the study of special algebraic varieties
(Calabi–Yau, complex symplectic or complex contact manifolds) for which Riemannian geometry
raises interesting questions.