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Let $\Gamma$ be a connection in $P$, $\Omega$ the curvature form, $\Phi(u)$ the holonomy group with reference point $u\in P$ and $P(u)$ the holonomy bundle through $u$ of $\Gamma$. … Can you suggest some applications of this Ambrose-Singer theorem on Holonomy. …

Recall that a choice of a connection $1$-form $A\in \Omega^1(P,\mathfrak{g})^G$ on a principal $G$-bundle $P$ over the manifold $M$ and a choice of base point $x\in M$ gives rise to the holonomy … Question is from another line in the paper : For a connected manifold $M$ holonomy map uniquely determines the connection $A$, and infact the bundle $P$ itself. …

Given $u\in P$ we have defined what is called a Holonomy bundle $P(u)=\{v\in P:v\sim u\}$ based at $u$ and what is called a Holonomy group $\Phi(u)=\{a\in G:u\sim ua\}$ based at $u$. … Let $\Gamma$ be a connection in $P$, $\Omega$ the curvature form, $\Phi(u)$ the holonomy group with reference point $u\in P$ and $P(u)$ the holonomy bundle through $u$ of $\Gamma$. …

Any reference for concept of holonomy on gerbes would be useful. EDIT : I thank user Tsemo for proving the equality that I said I was not able to prove. … Any thoughts on motivation behind calling this holonomy is welcome. Is this collection $\{c_{\alpha\beta\gamma}\}$ restircted to some subset is holonomy of some (line) bundle? …