# 34. The holonomy algebra and the Ambrose-Singer Holonomy Theorem

Let $ \pi \colon E \to M$ be a vector bundle with connection $ \nabla$. We know from Lecture 32 that each holonomy group $ \operatorname{Hol}^{\nabla}(x)$ is a Lie subgroup of $ \operatorname{GL}(E_x)$. Thus its Lie algebra, denoted by $ \mathfrak{hol}^{\nabla}(x)$, is a Lie subalgebra of $ \mathfrak{gl}(E_x)$.

We form the *holonomy algebra*** **of $ \nabla$ as the vector bundle

$$ \mathfrak{hol}^{\nabla} := \bigsqcup_{x \in M} \mathfrak{hol}^{\nabla}(x).$$

We prove that the holonomy algebra is a (Lie algebra) subbundle of $\operatorname{Hom}E,E)$. Next, we observe that the curvature at a point $x$ is itself an element of the holonomy algebra:

$$ R^{\nabla}(v,w) \in \mathfrak{hol}^{\nabla}(x), \qquad \forall \, v,w \in T_x M.$$

This leads us to the famous *Ambrose-Singer Holonomy Theorem*, which says that the *entire* holonomy algebra is generated by parallel transporting curvature elements. We will prove this theorem in Lecture 41, after we have defined connections on principal bundles.

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