# Applications of Ambrose-Singer theorem on holonomy

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.

• One result in one answer please. – Praphulla Koushik Aug 28 '18 at 20:23
• Use Google Scholar to search for citations to the Ambrose-Singer paper? I don't know any off-hand, so that is one thing I would do. Also, if you are wanting a list, with no one correct answer, the community norms here is that this question should probably be converted to CW. – David Roberts Sep 2 '18 at 20:13
• I find many hits here: scholar.google.com/scholar?cites=13701757514937985358 – David Roberts Sep 3 '18 at 0:08
• @DavidRoberts I was thinking it is better if someone who already has experience give a crux of where this can be/is used than checking google... – Praphulla Koushik Sep 3 '18 at 4:09
• I found arxiv.org/abs/1007.3543 which refers to a generalisation of Ambrose-Singer that helps understand the KP hierarchy (scholarpedia.org/article/Kadomtsev-Petviashvili_equation) and there seems to be a bunch of integrable systems work that uses the theorem. – David Roberts Sep 3 '18 at 4:47