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Let $p:E\longrightarrow B$ be a surjective submersion and $\sigma: p^*(TB)\longrightarrow TE$ a complete connection. Given a path $\gamma: [a, b]\longrightarrow B$ and $s, t\in [a, b]$ such that $s<t$ define $$\textrm{Hol}^\sigma_{\gamma_{t, s}}:=\textrm{Hol}^\sigma_{\gamma|_{[s, t]}}:[s, t]\longrightarrow E.$$ Notice $$\textrm{Hol}^\sigma_{\gamma_{t, 0}}(e)=\textrm{Hor}^\sigma_{\gamma, e}(t)\quad\quad\quad (1)$$ for every $e\in E_{\gamma(0)}$. We can then define a curve $$\alpha_e(t):=\textrm{Hol}^\sigma_{\gamma_{t, 0}}(e).$$ By $(1)$ we get: $$\alpha_e^\prime(t)=\frac{d}{ds}\biggr|_{s=t} \textrm{Hol}^\sigma_{\gamma_{s, 0}}(e)=\frac{d}{ds}\biggr|_{s=t} \textrm{Hor}^\sigma_{\gamma, e}(s)=\textrm{Hor}^\sigma_{\gamma^h_e(t)}(\gamma^\prime(t)).$$ Fine, we could also have defined $$\beta_e(t):=\textrm{Hol}^\sigma_{\gamma_{1, t}}(e).$$ What will be the analogous of $(1)$ and what will $\beta_e^\prime(t)$ be?

Notations.

  • $\textrm{Hol}^\sigma_\gamma$ stands for the holonomy of the connection $\sigma$ along the curve $\gamma$;

  • $\textrm{Hor}^\sigma_{\gamma, e}$ stands for the horizontal lift of the curve at the point $e\in E_{\gamma(0)}$;

  • $\textrm{Hor}^\sigma_x: T_{p(x)} B\longrightarrow H_x E$ stands for the horizontal lift of tangent vectors with respect to $\sigma$ where $H_x E$ is the horizontal subspace of $T_x E$ induced by $\sigma$.

Sketch. I'm almost getting to a conclusion but there is a missing step. The idea goes as follows:

Notice for every $s$ we can write $$(\gamma\circ \tau)_{1-s, 0}=\gamma_{1, s}\circ \tau_s,$$ where $\tau_s: [0, 1-s]\longrightarrow [s, 1]$ is given by $\tau_s(x):=x+s$. This diffeomorphism has the property $\tau_s(0)=s$ and the first equality below is justified: $$\textrm{Hol}^\sigma_{\gamma_{1, s}}=\textrm{Hol}^\sigma_{\gamma_{1, s}\circ \tau_s}=\textrm{Hol}^\sigma_{(\gamma\circ \tau)_{1-s, 0}}.$$ This should enable us to compute the derivative starting from what we know: $$\frac{d}{ds}\biggr|_{s=t} \textrm{Hol}^\sigma_{\gamma_{1, s}}(e)=\frac{d}{ds}\biggr|_{s=t} \textrm{Hol}^\sigma_{(\gamma\circ \tau_s)_{1-s, 0}}(e).$$ And next? How to finish this computation? The problem is the $s$ in $\gamma\circ \tau_s$ above.

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