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\begin{document}

A Yang-Mills gauge field is a lie algebra valued one-form

\( A_{ij}=A^{a}_{u}(\lambda _{a})_{ij}dx^{u} \) in \( T^{*}M\times G \) , \( u\in [1,..,dim(M)] \), \( a\in [1,...,dim(G)] \), \( i,j\in [1,..dim(\rho )] \)

Thes connections can be used to define a horizontal life of a curve.

\( g_{ij}'(\tau )+(A_{u})_{ik}x'^{u}(\tau )g_{kj}(\tau )=0 \). This is a first order differential equation. \( g(\tau )=Pe^{-\int _{0}^{\tau }A_{\mu }(\tau )x'^{u}(\tau )d\tau }g(0) \).

\( \frac{d}{d\tau }=x'^{u}\frac{\partial }{\partial x^{u}}+g_{ij}'\frac{\partial }{\partial g_{ij}} \)\( =x'^{u}(\frac{\partial }{\partial x^{u}}-A^{a}_{u}(\lambda _{a}g)_{ij}\frac{\partial }{\partial g_{ij}} \)

\( =x'^{u}(\frac{\partial }{\partial x^{u}}+A_{u}^{a}\overline{L}_{a})=x'^{u}D_{u} \) where \( \overline{L}_{a} \) are the canonical basis and \( D_{u} \)is the covariant derivative.

\end{document}