Let (M~(p,q),g) be a connected pseudo-Riemannian manifold of signature (p,q). There is a unique torsion-free metric connection D, called the Levi-Civita connection, giving rise to a parallel transport along each curve. Given a point x in M, the holonomy group H_x is the subgroup of O(T_XM, g_x) generated by parallel transport along all the closed curves starting at x. If we only consider the closed curves starting at x which are null-homotopic, we define the restricted holonomy group H_x~0. The (restricted) holonomy group at points of M are all isometric, and we may talk about the (restricted) holonomy of M, up to conjugacy. If (M~(p,q), g) is simply connected, then the holonomy group is equal to the restricted holonomy group. To simplify our course, we only consider this case. The holonomy group H is one of the fundamental algebraic objects associated to a pseudo-Riemannian manifold (M~(p,q),g). It is a Lie subgroup of O(p, q) measuring the parallel tensors on the manifolds. For example, H is reduced to the identity <=> (M~(p,q),g) is flat H is contained in U(p, q) <=> (M~(p,q),g) is a Kaehler manifold H is contained in SU(p,q) <=> (M~(p,q),g) is a special Kaehler manifold H is contained in Sp(p,q) • Sp(1) <=> (M~(p,q),g) is a quaternionic Kaehler manifold. H is decomposable into the direct product of normal subgroups <=> (M~(p,q),g) is at least locally isometric to a product of pseudo-Riemannian manifolds.
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