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Let $p:E\longrightarrow B$ be a smooth surjective submersion and $\sigma, \sigma^\prime: p^*(TB)\longrightarrow TE$ be two complete connections. Given a path $\gamma:I\longrightarrow B$ we can consider the holonomomies: $$\textrm{Hol}^\sigma_{\gamma}, \textrm{Hol}^{\sigma^\prime}_{\gamma}: E_{\gamma(0)}\longrightarrow E_{\gamma(1)},$$ which are diffeomorphisms between the fibers $E_{\gamma(j)}:=p^{-1}(\gamma(j))$. Is there a way to compare those maps? It has to do with this question.

Thanks.

Remark. The connections $\sigma, \sigma^\prime$ induce a bundle map $\theta^{\sigma, \sigma^\prime}: p^*(TB)\longrightarrow TE$ given by $$\theta^{\sigma, \sigma^\prime}(x, v):=(x, (\sigma_x-\sigma_x^\prime)(v)),$$ where $\sigma_x$ and $\sigma_x^\prime$ are uniquely determined by $\sigma(x, v)=(x, \sigma_x(v))$ and $\sigma^\prime(x, v)=(x, \sigma_x^\prime(v))$ so that $\sigma_x, \sigma_x^\prime: T_{p(x)} B\longrightarrow T_x E$ are linear. Indeed $\theta^{\sigma, \sigma^\prime}$ takes its values on the vertical bundle $VE$ of $E$.

Recall $p^*(TB)$ is the bundle $$p^*(TB)=\{(x, v)\in E\times TB: p(x)=\pi_{TB}(v)\}\longrightarrow E, (x, v)\longmapsto x.$$ So we could write something like $$\sigma=\sigma^\prime+_{TE} \theta^{\sigma, \sigma^\prime},$$ where $+_{TE}$ stands for the fiberwise adition on $TE$. This sugests we could relate the horizontal lifts of curves/vector fields with respect to $\sigma$ and $\sigma^\prime$ and consequently the holonomies.

Also, notice $\theta^{\sigma, \sigma^\prime}$ induces a map $$\theta^{\sigma, \sigma^\prime}_*: \Gamma(TB)\longrightarrow \Gamma(VE)\subset \Gamma(E), X\longmapsto (x\longmapsto \theta^{\sigma, \sigma^\prime}(x, X_{p(x)})).$$

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  • $\begingroup$ You could compose one with the inverse of the other, a kind of ``ratio'', a diffeomorphism of the fiber. But it is not clear what sort of comparison is relevant. $\endgroup$
    – Ben McKay
    Commented Aug 2, 2016 at 13:32
  • $\begingroup$ Indeed, I'd like to find a path on $E_{\gamma(1)}$ connecting $\textrm{Hol}^\sigma_\gamma(e)$ and $\textrm{Hol}^{\sigma^\prime}_{\gamma}(e)$ for every $e\in E_{\gamma(0)}$ fixed. $\endgroup$
    – PtF
    Commented Aug 2, 2016 at 13:38

1 Answer 1

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The expression $\sigma_t=(1-t)\sigma+t\sigma'$ is a connection, for every $t \in \mathbb{R}$, giving a path between the holonomies.

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    $\begingroup$ Subject to completeness, clearly. For example, this works for principal bundles or vector bundles. $\endgroup$
    – Ben McKay
    Commented Aug 2, 2016 at 13:54
  • $\begingroup$ Cool!! I guess I'll have to start working under more restrictive hypothesis, Thanks =) $\endgroup$
    – PtF
    Commented Aug 2, 2016 at 14:06
  • $\begingroup$ I've added some thoughts, could you check it? $\endgroup$
    – PtF
    Commented Aug 2, 2016 at 16:18
  • $\begingroup$ You could also write this as $\sigma_t=\sigma+t(\sigma'-\sigma)$ in terms of the difference between the two connections. Therefore we can canonically deform horizontal lifts and consequently deform holonomies. $\endgroup$
    – Ben McKay
    Commented Aug 2, 2016 at 16:22

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