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

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

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.

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

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.

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.