# Given a parallel field, what information can get about a connection?

Setup

Having as base manifold $M$ the Minkowski space $\mathbf{R}^{(1,3)}$, a $G$-principal bundle $P\to{M}$ and an associated vector bundle $E\equiv{P}\times_{\rho}{V}\to{M}$ are both trivial, since there exist global sections in $\Gamma(P)$. Assuming a fixed gauge $s\in\Gamma(P)$, I can pull-back a connection 1-from on $P$ and define a connection 1-form $A\in\Omega^1(M,\mathfrak{g})$. Also, for any section $e\in\Gamma(E)$ there exists a unique $\varphi\in{C}^{\infty}(M,V)$ such that $e=[s,\varphi]$.

Remark

Assuming that $\varphi$ is parallel on some $\Sigma\subset{M}$ such that $\Sigma\cong{S^2}$, i.e. $\left.\nabla^{A}\varphi\right|_\Sigma=0$, I can construct $\left.\varphi\right|_\Sigma$ by parallel transporting some initial $v\in{V}$ along a $\gamma\colon{I}\to\Sigma$ w.r.t. to $A$.

The question

Under the above setup, is the coverse of the above remark possible? That is, given the restriction $\left.\varphi\right|_\Sigma$ and the condition that $\varphi$ is parallel on $\Sigma$, what can I infer about the connection restricted on $\Sigma$? Is there a way to explicitly construct it, perhaps up to gauge-equivalence? Or at least assure that, given $\left.\varphi\right|_\Sigma$ and the parallel condition, an $A\in\Omega^1(\Sigma,\mathfrak{g})$ exists, such that the parallel transport some $v\in\varphi(\Sigma)$ w.r.t. to that $A$ will reproduce the given $\left.\varphi\right|_\Sigma$?