Let $\pi: P \to M$ be a smooth principal $G$-bundle equipped with a connection, and let $H \subset TP$ denote its horizontal distribution. For each piecewise smooth path $\gamma: [0,1] \to M$, let
denote parallel transport along $\gamma$, defined by $\operatorname{PT}_{\gamma}(p) = \widetilde{\gamma}_p(1)$, where $\widetilde{\gamma}_p: [0,1] \to P$ is the unique piecewise smooth horizontal lift satisfying $\pi \circ \widetilde{\gamma}_p = \gamma$ and $\widetilde{\gamma}_p(0) = p$.
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Then the following hold.
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1. If $x \in M$ and $c_x: [0,1] \to M$ is the constant path $c_x(t) = x$, then
3. If $\gamma_1: [0,1] \to M$ and $\gamma_2: [0,1] \to M$ are piecewise smooth paths satisfying $\gamma_1(1) = \gamma_2(0)$, and if $\gamma_2 * \gamma_1: [0,1] \to M$ denotes the piecewise smooth concatenation obtained by first following $\gamma_1$ and then following $\gamma_2$, then