Let \(\tilde{\gamma}\) be a horizontal lift of the curve \(\gamma : [0,1] \to M\) on a principal bundle \(P(M,G)\), with the projection \( \pi : P \to M\). The parallel transport is defined by the map:

\(\Gamma(\tilde{\gamma}^{-1}) : \pi^{-1}(\gamma(1)) \to \pi^{-1} (\gamma(0)) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \tag{1}\)

where \(\tilde{\gamma}^{-1}(t) = \tilde{\gamma}(1-t) \). Apparently, this means that \(\Gamma(\tilde{\gamma}^{-1}) = \Gamma (\tilde{\gamma})^{-1}\), but I don't see how this is true. As far as I understand, equation (1) implies that:

\(\Gamma(\tilde{\gamma}) : \pi^{-1}(\gamma(1)) \to \pi^{-1} (\gamma(0))\)

because \(\Gamma(\tilde{\gamma})\) sends the fibre at \(t=0\) (i.e.\(\gamma(1-0) = \gamma(1)\)) to the fibre at \(t=1\) (i.e. \(\gamma(1-1)=\gamma(0)\)). But this is clearly wrong because it implies that \(\Gamma(\tilde{\gamma}^{-1}) = \Gamma (\tilde{\gamma}) \implies \Gamma(\tilde{\gamma}^{-1}) \neq \Gamma (\tilde{\gamma})^{-1}\). Anybody knows where I'm making a mistake?