Ambrose–Singer holonomy theorem

The Lie algebra of the restricted holonomy group is generated by parallel transports of curvature.

Ambrose–Singer holonomy theorem

Let $M$ be a connected smooth manifold and let $\pi:P\to M$ be a principal G-bundle with structure group a Lie group $G$ . Fix a principal connection $\omega$ on $P$ with curvature form $\Omega$ .

For $x\in M$ and $p\in P_x$ , let $\mathrm{Hol}_p(\omega)\subseteq G$ denote the holonomy group based at $p$ , and let $\mathrm{Hol}^0_p(\omega)$ be its identity component. Write

\mathfrak{hol}_p(\omega):=\mathrm{Lie}\bigl(\mathrm{Hol}^0_p(\omega)\bigr),

the corresponding Lie algebra .

Theorem (Ambrose–Singer)

Let $p\in P_x$ . Consider all points $q\in P$ that can be reached from $p$ by a piecewise smooth horizontal curve (equivalently, by parallel transport from $x$ along some curve in $M$ ). For each such $q$ , and for each pair of tangent vectors $U,V\in T_{\pi(q)}M$ , choose their horizontal lifts $U^h,V^h\in T_qP$ .

Then $\mathfrak{hol}_p(\omega)$ is the smallest Lie subalgebra of $\mathfrak{g}$ containing all elements

\Omega_q(U^h,V^h)\in\mathfrak{g}

as $q,U,V$ vary as above. Equivalently, $\mathfrak{hol}_p(\omega)$ is generated (as a Lie algebra) by the curvature values “seen” along horizontal transport from the basepoint.

A common equivalent formulation uses an associated vector bundle: for any associated bundle $E=P\times_G V$ with its induced connection on a vector bundle $\nabla$ , the holonomy Lie algebra at $x$ is generated by endomorphisms obtained by transporting curvature operators back to $x$ :

\mathrm{span\;Lie}\left\{ \mathrm{PT}_{\gamma}^{-1}\circ R_y(u,v)\circ \mathrm{PT}_{\gamma} \right\},

where $y\in M$ , $u,v\in T_yM$ , and $\mathrm{PT}_\gamma$ is parallel transport along a curve $\gamma$ from $x$ to $y$ ; here $R$ denotes the curvature of $\nabla$ .

Examples

Flat connections have zero holonomy Lie algebra.
If $\Omega=0$ , then the generating set above is empty, so $\mathfrak{hol}_p(\omega)=\{0\}$ . Thus the restricted holonomy group is trivial; any remaining holonomy comes from monodromy of the base’s fundamental group.
Round sphere has full orthogonal holonomy (Levi-Civita).
For the Levi-Civita connection on the tangent bundle of the round $n$ -sphere ( $n\ge2$ ), curvature at a point spans $\mathfrak{so}(n)$ (in orthonormal frames). By Ambrose–Singer, the restricted holonomy Lie algebra is $\mathfrak{so}(n)$ , hence the restricted holonomy group is $\mathrm{SO}(n)$ .
Direct sums/products give direct-sum holonomy algebras.
For a direct-sum connection on a bundle $E=E_1\oplus E_2$ (or a product connection on a product manifold), the curvature is block-diagonal, so the generated holonomy Lie algebra is (at most) a direct sum of the holonomy Lie algebras of the factors.