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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