---
tag: T61AW674XD
kind: statement
role: theorem
name: Ambrose–Singer theorem
msc: [53C29, 53C05]
uses: [WHDZ3MK56B, TVAGN3D6W5, JA29ETCM02, ZAJFW1S078, 5ZYRRVE8BW]
about: [WHDZ3MK56B]
sources:
  - url: https://en.wikipedia.org/wiki/Holonomy
    locator: Ambrose–Singer theorem
---

## Statement

Let $M$ be a connected smooth manifold, let $E \to M$ be a smooth vector bundle with a [[connection|5ZYRRVE8BW]] $\nabla$, and let $p \in M$.
Let $R$ be the [[curvature|JA29ETCM02]] of $\nabla$, given by $R(X, Y) s = \nabla_X \nabla_Y s - \nabla_Y \nabla_X s - \nabla_{[X, Y]} s$.
Let $P_\gamma$ denote the [[parallel transport|TVAGN3D6W5]] along a piecewise smooth curve $\gamma$.
Then the Lie algebra of the [[holonomy group|WHDZ3MK56B]] $\mathrm{Hol}_p(\nabla) \subset \mathrm{GL}(E_p)$ is the linear span of the endomorphisms
$$P_\gamma^{-1} \circ R_q(u, v) \circ P_\gamma \in \mathrm{End}(E_p).$$
Here $q$ runs over $M$, $\gamma$ runs over the piecewise smooth curves from $p$ to $q$, and $u, v$ run over $T_qM$.

## Remarks

The idea of the proof: transport around a small loop spanned by $u$ and $v$ at $q$ differs from the identity by about $R_q(u, v)$ times the area.
So these conjugated curvatures lie in the holonomy algebra.
Conversely, the set of frames reached from a frame at $p$ by horizontal curves is a reduction of the frame bundle.
Its structure algebra is spanned by these elements, and this is where the [[curvature form|ZAJFW1S078]] of a principal connection enters.
In the language of principal bundles, let $\omega$ be a connection on $P \to M$ with curvature $\Omega$, and let $u \in P$.
Then the Lie algebra of $\mathrm{Hol}_u(\omega)$ is spanned by the values $\Omega_v(X, Y)$.
Here $v$ runs over the points of $P$ joined to $u$ by horizontal curves, and $X, Y$ over the horizontal vectors at $v$.
Example: a connection with $R = 0$ has zero holonomy algebra, so its restricted holonomy group is trivial and parallel transport is locally path independent.
The theorem only sees the identity component: the flat Möbius line bundle over a circle has $R = 0$ but holonomy group $\set{\pm 1}$.

## Explanations

### L1 human

Holonomy is a global notion: it uses loops of any size.
Curvature is local: it is computed at single points.
The Ambrose–Singer theorem says that the infinitesimal part of the holonomy is generated by curvature, once curvature at far away points is carried back along paths.
So if the curvature vanishes everywhere, small loops never turn vectors.

### L2 human

The theorem is the bridge between the holonomy group and the curvature tensor.
It shows that the curvature takes values in the holonomy algebra, which gives strong algebraic restrictions on curvature when the holonomy is small.
This is the starting point of Berger's classification of Riemannian holonomy groups and of the study of manifolds with special holonomy.
It also shows that a connection is flat exactly when its restricted holonomy group is trivial.