Ambrose-Singer Holonomy Theorem

The Ambrose-Singer theorem describes the holonomy group of a connection in terms of its curvature, providing the fundamental link between the infinitesimal (curvature) and global (holonomy) aspects of a connection.

Holonomy Groups

Definition7.13Holonomy group

Let $\nabla$ be a connection on a vector bundle $E \to M$ (or equivalently on a principal $G$ -bundle $P \to M$ ). Fix a point $p \in M$ . The holonomy group $\mathrm{Hol}_p(\nabla) \subset \mathrm{GL}(E_p)$ consists of all parallel transport maps around piecewise smooth loops based at $p$ :

\mathrm{Hol}_p(\nabla) = \{P_\gamma : E_p \to E_p \mid \gamma \text{ is a loop at } p\}.

The restricted holonomy group $\mathrm{Hol}^0_p(\nabla)$ uses only contractible loops.

The Theorem

Theorem7.4Ambrose-Singer holonomy theorem

Let $P \to M$ be a principal $G$ -bundle with connection $\omega$ and curvature $\Omega$ . For $u \in P$ with $\pi(u) = p$ , the Lie algebra of the holonomy group $\mathrm{Hol}_u(\omega)$ is

\mathfrak{hol}_u(\omega) = \operatorname{span}\{\Omega_v(X, Y) : v \in P(u), \, X, Y \in H_v\},

where $P(u)$ is the holonomy bundle through $u$ (all points reachable from $u$ by horizontal paths) and $H_v$ is the horizontal subspace at $v$ .

Proof Outline

Proof

Part 1: Curvature lies in the holonomy algebra. Consider an infinitesimal parallelogram at $p$ . Parallel transport around it differs from the identity by $F(X,Y)\epsilon^2 + O(\epsilon^3)$ , so $F(X,Y) \in \mathfrak{hol}_p$ .

More precisely, for tangent vectors $X, Y$ at $v$ , consider the loop obtained by flowing along $X$ , then $Y$ , then $-X$ , then $-Y$ for time $\epsilon$ . The holonomy of this loop is $\mathrm{id} + \epsilon^2 \Omega_v(X,Y) + O(\epsilon^3)$ , showing $\Omega_v(X,Y) \in \mathfrak{hol}_u$ .

Similarly, $\Omega_v(X,Y)$ for any $v$ in the holonomy bundle contributes, since parallel transporting to $v$ and back conjugates the infinitesimal holonomy into $\mathfrak{hol}_u$ .

Part 2: Nothing else lies in the holonomy algebra. This requires showing the subbundle of $P$ defined by the span of curvature endomorphisms is integrable (by the Frobenius theorem applied to the distribution $H \oplus \mathfrak{h}$ where $\mathfrak{h}$ is the curvature span). The Bianchi identity ensures the integrability condition. $\blacksquare$

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Applications

ExampleFlat connections have trivial restricted holonomy

If $\Omega = 0$ everywhere, then $\mathfrak{hol}^0 = 0$ , so $\mathrm{Hol}^0 = \{e\}$ . The full holonomy group is then discrete, corresponding to the monodromy representation $\pi_1(M) \to G$ . This recovers the classical result that flat connections correspond to representations of the fundamental group.

Theorem7.5Berger's classification

If $(M^n, g)$ is a simply connected, irreducible (not a product), non-symmetric Riemannian manifold, then $\mathrm{Hol}(g)$ is one of:

| Holonomy | Dimension | Geometry | |----------|-----------|----------| | $SO(n)$ | $n$ | Generic | | $U(n/2)$ | $n$ (even) | Kahler | | $SU(n/2)$ | $n$ (even) | Calabi-Yau | | $Sp(n/4)$ | $n$ ( $4|n$ ) | Hyperkahler | | $Sp(n/4) \cdot Sp(1)$ | $n$ ( $4|n$ ) | Quaternionic Kahler | | $G_2$ | 7 | Exceptional | | $\mathrm{Spin}(7)$ | 8 | Exceptional |

RemarkHolonomy in physics

In string theory, compactifications on manifolds with special holonomy (particularly $SU(3)$ for Calabi-Yau threefolds and $G_2$ for $M$ -theory) preserve supersymmetry. Berger's classification thus constrains the possible internal geometries in string compactifications.