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A possible answer would be to invoke Ambrose-Singer Holonomy theorem:

Theorem (Ambrose-Singer) Let $M$ be a (smooth) connected manifold, $E\to M$ a vector bundle over $M$, and $\nabla$ a connection on $E$. Then, for each $x\in M$, $\mathfrak{hol}_x(\nabla)$ is a Lie subalgebra of $\text{End}(E_x)$ which, as a vector space, is spanned by all elements of $\text{End}(E_x)$ of the form $P_{\gamma}^{-1}[ (F_{\nabla})_y\cdot{(v\wedge w)}] P_{\gamma}$, where $\gamma:[0,1]\to M$ is a piece-wise smooth path with end points $\gamma(0) = x$ and $\gamma(1) = y$, $P_{\gamma}: E_x \to E_y$ is the associated parallel translation map, and $v,w\in T_yM$.

In particular, if the connection $\nabla$ is flat (i.e. if $F_{\nabla}=0$), then $\mathfrak{hol}_x(\nabla)=0$, and therefore the restricted holonomy group $\text{Hol}_x^0(\nabla)$ is trivial, for each $x\in M$. Recalling $$ \text{Hol}_x^0(\nabla)=\{P_{\gamma}: \gamma\text{ is a null-homotopic loop based at }x\}, $$ the conclusion you need follows. Indeed, given two homotopic paths $\gamma_1$ and $\gamma_2$ (I'm assuming this homotopy preserves the base-points), say with $\gamma_1(0)=\gamma_2(0)=x$, then you can consider the null-homotopic concatenation $\gamma_2^{-1}\cdot{\gamma_1}$, which is a (piece-wise smooth) loop based at $x$. By the above, we have that $1=P_{\gamma_2^{-1}\cdot{\gamma_1}} = P_{\gamma_2}^{-1}\circ P_{\gamma_1}=1$$1=P_{\gamma_2^{-1}\cdot{\gamma_1}} = P_{\gamma_2}^{-1}\circ P_{\gamma_1}$, i.e. $P_{\gamma_1}=P_{\gamma_2}$, as we wanted.

A principal bundle version of this theorem (together with a proof) can be found e.g. in Kobayashi&Nomizu's book Foundations of differential geometry, vol. I, Theorem 8.1, p. 89. The statement I give here was based on Theorem 2.4.3. (a) of Joyce's book Riemannian Holonomy Groups and Calibrated Geometry.

A possible answer would be to invoke Ambrose-Singer Holonomy theorem:

Theorem (Ambrose-Singer) Let $M$ be a (smooth) connected manifold, $E\to M$ a vector bundle over $M$, and $\nabla$ a connection on $E$. Then, for each $x\in M$, $\mathfrak{hol}_x(\nabla)$ is a Lie subalgebra of $\text{End}(E_x)$ which, as a vector space, is spanned by all elements of $\text{End}(E_x)$ of the form $P_{\gamma}^{-1}[ (F_{\nabla})_y\cdot{(v\wedge w)}] P_{\gamma}$, where $\gamma:[0,1]\to M$ is a piece-wise smooth path with end points $\gamma(0) = x$ and $\gamma(1) = y$, $P_{\gamma}: E_x \to E_y$ is the associated parallel translation map, and $v,w\in T_yM$.

In particular, if the connection $\nabla$ is flat (i.e. if $F_{\nabla}=0$), then $\mathfrak{hol}_x(\nabla)=0$, and therefore the restricted holonomy group $\text{Hol}_x^0(\nabla)$ is trivial, for each $x\in M$. Recalling $$ \text{Hol}_x^0(\nabla)=\{P_{\gamma}: \gamma\text{ is a null-homotopic loop based at }x\}, $$ the conclusion you need follows. Indeed, given two homotopic paths $\gamma_1$ and $\gamma_2$ (I'm assuming this homotopy preserves the base-points), say with $\gamma_1(0)=\gamma_2(0)=x$, then you can consider the null-homotopic concatenation $\gamma_2^{-1}\cdot{\gamma_1}$, which is a (piece-wise) loop based at $x$. By the above, we have that $1=P_{\gamma_2^{-1}\cdot{\gamma_1}} = P_{\gamma_2}^{-1}\circ P_{\gamma_1}=1$, i.e. $P_{\gamma_1}=P_{\gamma_2}$, as we wanted.

A principal bundle version of this theorem (together with a proof) can be found e.g. in Kobayashi&Nomizu's book Foundations of differential geometry, vol. I, Theorem 8.1, p. 89. The statement I give here was based on Theorem 2.4.3. (a) of Joyce's book Riemannian Holonomy Groups and Calibrated Geometry.

A possible answer would be to invoke Ambrose-Singer Holonomy theorem:

Theorem (Ambrose-Singer) Let $M$ be a (smooth) connected manifold, $E\to M$ a vector bundle over $M$, and $\nabla$ a connection on $E$. Then, for each $x\in M$, $\mathfrak{hol}_x(\nabla)$ is a Lie subalgebra of $\text{End}(E_x)$ which, as a vector space, is spanned by all elements of $\text{End}(E_x)$ of the form $P_{\gamma}^{-1}[ (F_{\nabla})_y\cdot{(v\wedge w)}] P_{\gamma}$, where $\gamma:[0,1]\to M$ is a piece-wise smooth path with end points $\gamma(0) = x$ and $\gamma(1) = y$, $P_{\gamma}: E_x \to E_y$ is the associated parallel translation map, and $v,w\in T_yM$.

In particular, if the connection $\nabla$ is flat (i.e. if $F_{\nabla}=0$), then $\mathfrak{hol}_x(\nabla)=0$, and therefore the restricted holonomy group $\text{Hol}_x^0(\nabla)$ is trivial, for each $x\in M$. Recalling $$ \text{Hol}_x^0(\nabla)=\{P_{\gamma}: \gamma\text{ is a null-homotopic loop based at }x\}, $$ the conclusion you need follows. Indeed, given two homotopic paths $\gamma_1$ and $\gamma_2$ (I'm assuming this homotopy preserves the base-points), say with $\gamma_1(0)=\gamma_2(0)=x$, then you can consider the null-homotopic concatenation $\gamma_2^{-1}\cdot{\gamma_1}$, which is a (piece-wise smooth) loop based at $x$. By the above, we have that $1=P_{\gamma_2^{-1}\cdot{\gamma_1}} = P_{\gamma_2}^{-1}\circ P_{\gamma_1}$, i.e. $P_{\gamma_1}=P_{\gamma_2}$, as we wanted.

A principal bundle version of this theorem (together with a proof) can be found e.g. in Kobayashi&Nomizu's book Foundations of differential geometry, vol. I, Theorem 8.1, p. 89. The statement I give here was based on Theorem 2.4.3. (a) of Joyce's book Riemannian Holonomy Groups and Calibrated Geometry.

Source Link
Compactness
Compactness
  • 26
  • 5

A possible answer would be to invoke Ambrose-Singer Holonomy theorem:

Theorem (Ambrose-Singer) Let $M$ be a (smooth) connected manifold, $E\to M$ a vector bundle over $M$, and $\nabla$ a connection on $E$. Then, for each $x\in M$, $\mathfrak{hol}_x(\nabla)$ is a Lie subalgebra of $\text{End}(E_x)$ which, as a vector space, is spanned by all elements of $\text{End}(E_x)$ of the form $P_{\gamma}^{-1}[ (F_{\nabla})_y\cdot{(v\wedge w)}] P_{\gamma}$, where $\gamma:[0,1]\to M$ is a piece-wise smooth path with end points $\gamma(0) = x$ and $\gamma(1) = y$, $P_{\gamma}: E_x \to E_y$ is the associated parallel translation map, and $v,w\in T_yM$.

In particular, if the connection $\nabla$ is flat (i.e. if $F_{\nabla}=0$), then $\mathfrak{hol}_x(\nabla)=0$, and therefore the restricted holonomy group $\text{Hol}_x^0(\nabla)$ is trivial, for each $x\in M$. Recalling $$ \text{Hol}_x^0(\nabla)=\{P_{\gamma}: \gamma\text{ is a null-homotopic loop based at }x\}, $$ the conclusion you need follows. Indeed, given two homotopic paths $\gamma_1$ and $\gamma_2$ (I'm assuming this homotopy preserves the base-points), say with $\gamma_1(0)=\gamma_2(0)=x$, then you can consider the null-homotopic concatenation $\gamma_2^{-1}\cdot{\gamma_1}$, which is a (piece-wise) loop based at $x$. By the above, we have that $1=P_{\gamma_2^{-1}\cdot{\gamma_1}} = P_{\gamma_2}^{-1}\circ P_{\gamma_1}=1$, i.e. $P_{\gamma_1}=P_{\gamma_2}$, as we wanted.

A principal bundle version of this theorem (together with a proof) can be found e.g. in Kobayashi&Nomizu's book Foundations of differential geometry, vol. I, Theorem 8.1, p. 89. The statement I give here was based on Theorem 2.4.3. (a) of Joyce's book Riemannian Holonomy Groups and Calibrated Geometry.