Some theorems on holonomy groups of Riemannian manifolds

Abstract

For Riemannian manifolds there are four kinds of holonomy groups: the (nonrestricted) holonomy group H, the restricted holonomy group HO, the (nonrestricted) homogeneous holonomy group h, and the restricted homogeneous holonomy group h?. It is known that all of these are Lie groups of transformations and HO and hf are the connected components of the identity of H and h respectively. 1. Relations among invanant linear subspaces of h? and h. If the restricted homogeneous holonomy group h? is reducible (in the real number field) it is completely reducible, for h is a subgroup of the orthogonal group. If the holonomy group hf is reducible, we can take a repere in the tangent space ER(O) at the base point 0 of the holonomy group so that all elements of the group h? can be represented by matrices of the following type: