Restricted holonomy group

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Definition

Let M be a differential manifold, E a vector bundle over M, and \nabla a connection on E. The restricted holonomy group at m \in M is defined as the subgroup of GL(E_m) comprising all those linear transformations that occur as the holonomy of a loop that is homotopic to the constant loop.

Facts

The restricted holonomy group is a normal subgroup of the holonomy group, which comprises all the linear transformations that occur as the holonomy of a loop (not necessarily homotopic to the constant loop).