Holonomy of a loop

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Definition

Let M be a differential manifold, E a vector bundle over M and \nabla a connection on E. Let m \in M and \gamma:[0,1]\to M be a loop at m (viz \gamma(0) = \gamma(1) = m. The holonomy of \gamma is defined as the following linear transformation at E_p: it sends v \in E_p to the element \phi_1(v) where \phi_t(v) is the transport of v along \gamma, with respect to the connection \nabla.

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