Differential geometry Connections and curvature Theorem

Ambrose–Singer theorem

Builds on Holonomy Parallel transport Riemann curvature tensor Curvature form Connection on a vector bundle

Statement

Let $M$ be a connected smooth manifold, let $E \to M$ be a smooth vector bundle with a connection $\nabla$ , and let $p \in M$ . Let $R$ be the curvature 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 along a piecewise smooth curve $\gamma$ . Then the Lie algebra of the holonomy group $\mathrm{Hol}_p(\nabla) \subset \mathrm{GL}(E_p)$ 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 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 $\set{\pm 1}$ .

Going deeper

For a student

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.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`T61AW674XD`
URI	`m2://T61AW674XD`
Kind	statement, theorem
Writing	W0 stub
MSC	53C29, 53C05
Status	This copy of the site has no build data. Run `m2 status T61AW674XD`.
About	Holonomy
Uses	Holonomy, Parallel transport, Riemann curvature tensor, Curvature form, Connection on a vector bundle
Changed	2 November 2026