I am planning to introduce to a group of Graduate students the notion of connections on Principal bundle, curvature of connection, Holonomy. I want to conclude with the statement of Ambrose-Singer theorem which is as follows.

Let $P(M,G)$ be a Principal bundle where $M$ is connected and paracompact. 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 $\Gamma$. Then the Lie algebra of holonomy group $\Phi(u)$ is equal to the subspace of $\mathfrak{g}$, Lie algebra of $G$, generated by all elements of the form $\Omega_v(X,Y)$ where $v\in P(u)$ and $X$ and $Y$ are arbitrary horizantal vectors at $v$.

I am not planning to prove the theorem but just state it. It looks beautiful in itself but, if I can say some applications of this theorem, it would bring out the power of this theorem.

Can you suggest some applications of this Ambrose-Singer theorem on Holonomy.

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