On a Riemannian manifold M, tangent vectors can be moved along a path by parallel transport, which preserves vector addition and scalar multiplication. So a closed loop at a base point p, gives rise to a invertible linear map of TM_p, the tangent vectors at p. It is possible to compose closed loops by following one after the other, and to invert them by going backwards. Hence, the set of linear transformations arising from parallel transport along closed loops is a group, called the holonomy group.
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