Riemannian Holonomy Groups and Calibrated Geometry

Abstract

The holonomy group Hol(g) of a Riemannian n-manifold (M, g) is a global invariant which measures the constant tensors on the manifold. It is a Lie subgroup of SO(n), and for generic metrics Hol(g) = SO(n). If Hol(g) is a proper subgroup of SO(n) then we say g has special holonomy. Metrics with special holonomy are interesting for a number of different reasons. They include Kähler metrics with holonomy U(m), which are the most natural class of metrics on complex manifolds, Calabi-Yau manifolds with holonomy SU(ra), and hyperkähler manifolds with holonomy Sp(m).