Holonomy

Oh, you want me to rewrite a Wikipedia article? How… thrilling. Fine. Let’s see what we can excavate from this academic dustbin. Don’t expect me to hold your hand through it.

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Holonomy in Differential Geometry

The concept of holonomy, in the context of differential geometry, essentially quantifies the failure of parallel transport around closed loops to preserve the geometric information being transported. It’s a direct consequence, a lingering shadow, cast by the curvature inherent in a connection on a smooth manifold. If a connection is entirely flat, its holonomy behaves like a form of monodromy—a global phenomenon, a stubborn refusal to return to its origin unchanged. But when that connection is curved, well, that's where things get interesting, with features that are both local and global, and decidedly nontrivial.

Every connection, regardless of its pedigree, brings its own flavor of holonomy into existence through its parallel transport maps. The most studied types of holonomy tend to be those associated with connections that possess a certain symmetry. We see this most prominently in the holonomy of the Levi-Civita connection within Riemannian geometry, which we call Riemannian holonomy. Then there’s the holonomy of connections in vector bundles, the more abstract Cartan connections, and the holonomy found in principal bundles. In each of these instances, the holonomy of a connection can be distilled into a Lie group, known as the holonomy group. The relationship between holonomy and curvature is intimate, formalized by the Ambrose–Singer theorem.

The study of Riemannian holonomy, in particular, has been a fertile ground for significant theoretical advancements. Élie Cartan first introduced the concept in 1926, driven by a desire to classify and understand symmetric spaces. It wasn't until much later that holonomy groups were wielded to dissect Riemannian geometry in a broader sense. In 1952, Georges de Rham delivered his decomposition theorem, a foundational principle for dissecting a Riemannian manifold into a Cartesian product of smaller Riemannian manifolds by breaking down the tangent bundle into irreducible spaces under the action of local holonomy groups. A few years later, in 1953, Marcel Berger undertook the arduous task of classifying all possible irreducible holonomies. This decomposition and classification of Riemannian holonomy has, predictably, found its way into applications in physics and string theory.

Definitions

Holonomy of a Connection in a Vector Bundle

Let's consider a rank-k vector bundle E situated over a smooth manifold M. Let ∇ represent a connection on this bundle E. Now, imagine a piecewise smooth loop γ, tracing a path from time 0 to 1 on M, starting and ending at some point x. The connection ∇ equips us with a parallel transport map, denoted as Pγ, which acts on the fiber of E at the base point x, mapping it to itself (Eₓ → Eₓ). This map is not just linear; it's also invertible, meaning it can be represented as an element of the general linear group GL(Eₓ).

The holonomy group of ∇, anchored at x, is then precisely the collection of all such parallel transport maps Pγ generated by loops γ that are based at x:

$\operatorname{Hol} _{x}(\nabla )=\{P_{\gamma }\in \mathrm {GL} (E_{x})\mid \gamma {\text{ is a loop based at }}x\}.$

The restricted holonomy group, denoted Holₓ⁰(∇), is a subgroup formed by those parallel transport maps arising from contractible loops γ.

If the manifold M is path-connected, the choice of the basepoint x doesn't fundamentally alter the holonomy group; it only changes it up to conjugation within GL(k, ℝ). More specifically, if we have a path γ connecting x to another point y in M, the holonomy group at y is related to the one at x by:

$\operatorname{Hol} _{y}(\nabla )=P_{\gamma }\operatorname {Hol} _{x}(\nabla )P_{\gamma }^{-1}.$

It's worth noting that different ways of identifying Eₓ with ℝᵏ will also lead to conjugate subgroups. In less formal discussions, one might omit the basepoint, assuming it's understood that the group is defined only up to conjugation.

Here are some key properties of the holonomy group:

• Hol⁰(∇) is a connected Lie subgroup of GL(k, ℝ).
• Hol⁰(∇) is the identity component of Hol(∇).
• If M is simply connected, then Hol(∇) is identical to Hol⁰(∇).
• ∇ is flat (meaning its curvature vanishes) if and only if Hol⁰(∇) is trivial. However, even if flat, Hol(∇) might still be nontrivial.
• There exists a natural, surjective group homomorphism from the fundamental group of M, π₁(M), to the quotient group Hol(∇) / Hol⁰(∇). This map, denoted by φ, takes the homotopy class [γ] of a loop γ to the coset Pγ ⋅ Hol⁰(∇).

Holonomy of a Connection in a Principal Bundle

Let's shift our focus to connections on principal bundles. Consider a principal G-bundle P over a smooth manifold M, where G is a Lie group and M is paracompact. A connection ω on P effectively partitions the tangent space at any point xₚ along the fiber Gₚ into a vertical subspace Vₚ and a horizontal subspace Hₚ. Curves on the base manifold M can be "lifted" to curves in the principal bundle P, such that their tangent vectors lie entirely within these horizontal subspaces.

The definition of holonomy for connections on principal bundles follows a similar logic. Given a piecewise smooth loop γ on M starting and ending at x, and a point p in the fiber over x, the connection ω defines a unique horizontal lift, denoted as γ̃, which starts at p (γ̃(0) = p). The endpoint of this horizontal lift, γ̃(1), will generally not be p but some other point p ⋅ g in the same fiber.

We can establish an equivalence relation ~ on P: two points p and q are related (p ~ q) if they can be connected by a piecewise smooth horizontal path in P.

With this in place, the holonomy group of ω, based at p, is defined as the set of all g ∈ G such that p and p ⋅ g are related:

$\operatorname{Hol} _{p}(\omega )=\{g\in G\mid p\sim p\cdot g\}.$

The restricted holonomy group, Holₚ⁰(ω), is the subgroup consisting of g ∈ G arising from the horizontal lifts of contractible loops γ.

If M and P are connected, the holonomy group is determined by the basepoint p only up to conjugation within G. Specifically, if q is another basepoint, there exists a unique g ∈ G such that q ~ p ⋅ g. Using this g, the holonomy group at q is related to that at p by:

$\operatorname{Hol} _{q}(\omega )=g^{-1}\operatorname {Hol} _{p}(\omega )g.$

In particular, this implies:

$\operatorname{Hol} _{p\cdot g}(\omega )=g^{-1}\operatorname {Hol} _{p}(\omega )g,$

and crucially, if p ~ q, then:

$\operatorname{Hol} _{p}(\omega )=\operatorname {Hol} _{q}(\omega ).$

Again, it’s common to refer to the holonomy group without specifying a basepoint, understanding that it's defined up to conjugation.

Key properties of these holonomy groups include:

• Holₚ⁰(ω) is a connected Lie subgroup of G.
• Holₚ⁰(ω) is the identity component of Holₚ(ω).
• There's a surjective group homomorphism from π₁ to Holₚ(ω) / Holₚ⁰(ω).
• If M is simply connected, then Holₚ(ω) = Holₚ⁰(ω).
• ω is flat (zero curvature) if and only if Holₚ⁰(ω) is trivial.

Holonomy Bundles

Let M be a connected paracompact smooth manifold, and P a principal G-bundle equipped with a connection ω. For any point p ∈ P, let H(p) be the set of points in P that can be reached from p via a horizontal curve. It turns out that H(p), when projected down to M, forms a principal bundle over M with structure group Holₚ(ω). This is the holonomy bundle (through p) of the connection. The connection ω itself restricts to a connection on H(p), as its parallel transport maps preserve H(p). H(p) is, in fact, the minimal such reduction of P that is preserved by parallel transport. [^1]

The holonomy bundle transforms equivariantly under the action of the structure group G. If q is another basepoint in P, there exists a unique g ∈ G such that q ~ p ⋅ g. Then H(q) = H(p) ⋅ g. This means the connections on holonomy bundles derived from different basepoints are compatible, differing only by this element g.

Monodromy

The holonomy bundle H(p) is a principal bundle for Holₚ(ω). It also admits an action by the restricted holonomy group Holₚ⁰(ω), which is a normal subgroup of the full holonomy group. The quotient group, Holₚ(ω) / Holₚ⁰(ω), is a discrete group known as the monodromy group. This group acts on the quotient bundle H(p) / Holₚ⁰(ω).

There exists a surjective homomorphism φ: π₁(M) → Holₚ(ω) / Holₚ⁰(ω), meaning that the image of the fundamental group, φ(π₁(M)), acts on H(p) / Holₚ⁰(ω). This action constitutes a monodromy representation of the fundamental group. [^2]

Local and Infinitesimal Holonomy

We can restrict the holonomy of a connection ω on a principal bundle P → M to the fiber over an open subset U of M. If U is connected, ω induces a connection on the restricted bundle π⁻¹U over U. The holonomy (or restricted holonomy) for this bundle, based at p ∈ π⁻¹U, is denoted Holₚ(ω, U) (or Holₚ⁰(ω, U)).

If we have nested open sets U ⊂ V containing π(p), then there's a natural inclusion:

$\operatorname{Hol} _{p}^{0}(\omega ,U)\subset \operatorname {Hol} _{p}^{0}(\omega ,V).$

The local holonomy group at a point p is defined by taking the intersection of restricted holonomy groups over a sequence of nested connected open sets U<0xE2><0x82><0x96> whose intersection is just the point π(p):

$\operatorname{Hol} _{*}(\omega )=\bigcap _{k=1}^{\infty }\operatorname {Hol} ^{0}(\omega ,U_{k}).$

This local holonomy group has several key characteristics:

• It's a connected Lie subgroup of the restricted holonomy group Holₚ⁰(ω).
• For any point p, there exists a neighborhood V such that Holₚ*(ω) = Holₚ⁰(ω, V). This means the local holonomy group depends only on the point p, not the specific sequence of open sets used in its definition.
• The local holonomy group transforms equivariantly under translations by elements of the structure group G. Specifically, for g ∈ G:

$\operatorname{Hol} _{pg}^{*}(\omega )=\operatorname{Ad} \left(g^{-1}\right)\operatorname {Hol} _{p}^{*}(\omega ).$

The local holonomy group isn't always well-behaved globally; its dimension might fluctuate. However, a crucial theorem states:

If the dimension of the local holonomy group is constant, then the local and restricted holonomy groups coincide:

$\operatorname{Hol} _{p}^{*}(\omega )=\operatorname {Hol} _{p}^{0}(\omega ).$

Ambrose–Singer Theorem

The Ambrose–Singer theorem, a landmark result by Warren Ambrose and Isadore M. Singer (1953), establishes a profound connection between the holonomy of a connection in a principal bundle and the curvature form of that connection. To grasp its essence, consider the familiar case of an affine connection, like the Levi-Civita connection. Curvature manifests when we traverse an infinitesimal parallelogram.

Imagine a surface σ in M parametrized by x and y, with boundaries [0, 1] × [0, 1]. If we transport a vector V around the boundary of σ—first along (x, 0), then (1, y), then (x, 1) in reverse, and finally (0, y) back to the start—we trace a holonomy loop. The resulting transformation of V is dictated by an element of the holonomy group. The curvature becomes explicitly involved when this parallelogram shrinks to zero size, as we traverse smaller parallelograms over [0, x] × [0, y]. This corresponds to differentiating the parallel transport maps at x = y = 0:

${\frac {D}{dx}}{\frac {D}{dy}}V-{\frac {D}{dy}}{\frac {D}{dx}}V=R\left({\frac {\partial \sigma }{\partial x}},{\frac {\partial \sigma }{\partial y}}\right)V$

Here, R is the Riemann curvature tensor. In essence, the curvature captures the infinitesimal holonomy around a closed loop. More formally, the curvature is the differential of the holonomy action at the identity of the holonomy group. This means R(X, Y) is an element of the Lie algebra of Holₚ(ω).

More generally, for a principal bundle P → M with structure group G and connection ω, the curvature form Ω is a g-valued 2-form on P (where g is the Lie algebra of G). The Ambrose–Singer theorem states: [^4]

The Lie algebra of Holₚ(ω) is spanned by all elements of g of the form Ω<0xE1><0xB5><0xA1>(X, Y), where q lies on any horizontal curve starting from p (i.e., q ~ p), and X and Y are horizontal tangent vectors at q.

An alternative formulation, focusing on the holonomy bundle: [^5]

The Lie algebra of Holₚ(ω) is the subspace of g spanned by elements of the form Ω<0xE1><0xB5><0xA1>(X, Y), where q ∈ H(p) and X and Y are horizontal vectors at q.

Riemannian Holonomy

The holonomy of a Riemannian manifold (M, g) refers to the holonomy group of the Levi-Civita connection acting on the tangent bundle of M. A typical n-dimensional Riemannian manifold will have a holonomy group of O(n), or SO(n) if it's orientable. Those manifolds whose holonomy groups are proper subgroups of O(n) or SO(n) possess particularly special characteristics.

A foundational result in this area is the theorem by Borel and Lichnerowicz (1952), which establishes that the restricted holonomy group is a closed Lie subgroup of O(n), and therefore compact.

Reducible Holonomy and the de Rham Decomposition

Consider an arbitrary point x ∈ M. The holonomy group Hol(M) acts on the tangent space TₓM. This action can be either irreducible (meaning TₓM cannot be decomposed into smaller invariant subspaces) or reducible. If it's reducible, TₓM splits into orthogonal subspaces T'ₓM ⊕ T″ₓM, each invariant under the action of Hol(M). In such a case, M is said to have reducible holonomy.

If M is reducible, allowing x to vary across the manifold, the bundles T'M and T″M (formed by the reduction of tangent spaces) become smooth distributions that are integrable in the sense of Frobenius. The integral manifolds of these distributions are totally geodesic submanifolds. This implies that M can be locally decomposed as a Cartesian product M' × M″. The de Rham decomposition theorem, a cornerstone result, formalizes this by continuing this reduction process until the tangent space is completely decomposed. It states:

Let M be a simply connected Riemannian manifold. [^7] Let T M = T⁽⁰⁾M ⊕ T⁽¹⁾M ⊕ ⋯ ⊕ T⁽ᵏ⁾M be the complete reduction of the tangent bundle under the action of the holonomy group. If T⁽⁰⁾M consists of vectors invariant under the holonomy group (i.e., the holonomy representation is trivial), then M is locally isometric to a product:

$V_{0}\times V_{1}\times \cdots \times V_{k},$

where V₀ is an open set in a Euclidean space, and each Vᵢ is an integral manifold for T⁽ⁱ⁾M. Furthermore, the holonomy group Hol(M) decomposes into a direct product of the holonomy groups of each Mᵢ, where Mᵢ is the maximal integral manifold of T⁽ⁱ⁾ passing through a point.

If M is also geodesically complete, this decomposition holds globally, with each Mᵢ being a geodesically complete manifold. [^8]

The Berger Classification

In 1955, Marcel Berger provided a complete classification of the possible holonomy groups for simply connected, irreducible Riemannian manifolds that are also nonsymmetric (meaning they are not locally isometric to a Riemannian symmetric space). Berger's list is as follows:

Hol(g)	dim(M)	Type of manifold	Comments
SO(n)	n	Orientable manifold	—
U(n)	2n	Kähler manifold	Kähler
SU(n)	2n	Calabi–Yau manifold	Ricci-flat, Kähler
Sp(n) · Sp(1)	4n	[Quaternion-Kähler manifold](/Quaternion-Kähler manifold)	Einstein
Sp(n)	4n	[Hyperkähler manifold](/Hyperkähler manifold)	Ricci-flat, Kähler
G₂	7	G₂ manifold	Ricci-flat
Spin(7)	8	Spin(7) manifold	Ricci-flat

Manifolds with holonomy Sp(n)·Sp(1) were concurrently studied by Edmond Bonan and Vivian Yoh Kraines in 1965, who independently discovered that such manifolds must possess a parallel 4-form.

The G₂ and Spin(7) holonomy manifolds were first abstractly explored by Edmond Bonan in 1966. He classified the parallel differential forms they would carry and demonstrated that these manifolds must be Ricci-flat. However, concrete examples of such manifolds wouldn't be constructed for another three decades.

Berger's original list also included Spin(9) as a subgroup of SO(16). It was later shown by D. Alekseevski and Brown–Gray that Riemannian manifolds with this holonomy must be locally symmetric, specifically locally isometric to the Cayley plane F₄/Spin(9) or locally flat. It is now established that all these possibilities indeed occur as holonomy groups of Riemannian manifolds. The last two exceptional cases, G₂ and Spin(7), were the most challenging to find examples for.

It's important to note the inclusions: Sp(n) ⊂ SU(2n) ⊂ U(2n) ⊂ SO(4n). This means every hyperkähler manifold is also a Calabi–Yau manifold, every Calabi–Yau manifold is a Kähler manifold, and every Kähler manifold is orientable.

The somewhat surprising nature of this list was illuminated by Simons's proof of Berger's theorem. A more geometric proof was later provided by Carlos E. Olmos in 2005. The approach involves first demonstrating that for irreducible, non-locally symmetric Riemannian manifolds, the reduced holonomy group acts transitively on the unit sphere. The Lie groups known to act transitively on spheres were identified: they correspond to the list above, plus two additional cases: Spin(9) acting on ℝ¹⁶, and T · Sp(m) acting on ℝ⁴ᵐ. Further analysis revealed that the Spin(9) case only arises in locally symmetric spaces (specifically, locally isomorphic to the Cayley projective plane), and the T · Sp(m) case does not occur as a holonomy group at all.

Berger's initial classification also encompassed pseudo-Riemannian metrics with non-positive definite signatures, leading to non-locally symmetric holonomy groups. This extended list included SO(p, q) of signature (p, q), U(p, q) and SU(p, q) of signature (2p, 2q), Sp(p, q) and Sp(p, q)·Sp(1) of signature (4p, 4q), SO(n, ℂ) of signature (n, n), SO(n, ℍ) of signature (2n, 2n), split G₂ of signature (4, 3), G₂(ℂ) of signature (7, 7), Spin(4, 3) of signature (4, 4), Spin(7, ℂ) of signature (7, 7), Spin(5, 4) of signature (8, 8), and finally Spin(9, ℂ) of signature (16, 16). As noted before, the split and complexified Spin(9) cases are necessarily locally symmetric and should not have been on the list. The complexified holonomies SO(n, ℂ), G₂(ℂ), and Spin(7, ℂ) can be realized from complexifying real analytic Riemannian manifolds. The final case, manifolds with holonomy contained in SO(n, ℍ), were shown to be locally flat by R. McLean. [^9]

Riemannian symmetric spaces, which are locally isometric to homogeneous spaces G/H, possess local holonomy isomorphic to H. These have also been completely classified.

Furthermore, Berger's paper outlines the possible holonomy groups for manifolds with only a torsion-free affine connection, which is discussed later.

Special Holonomy and Spinors

Manifolds exhibiting special holonomy are characterized by the presence of parallel spinors—that is, spinor fields whose covariant derivative vanishes. [^10] Specifically:

• Hol(ω) ⊂ U(n) if and only if M admits a covariantly constant (parallel) projective pure spinor field.
• If M is a spin manifold, then Hol(ω) ⊂ SU(n) if and only if M admits at least two linearly independent parallel pure spinor fields. A parallel pure spinor field inherently defines a canonical reduction of the structure group to SU(n).
• On a seven-dimensional spin manifold, the existence of a non-trivial parallel spinor field is equivalent to the holonomy being contained within G₂.
• Similarly, on an eight-dimensional spin manifold, a non-trivial parallel spinor field implies the holonomy is contained within Spin(7).

The study of unitary and special unitary holonomies often intersects with twistor theory [^11] and the analysis of almost complex structures. [^10]

Applications

String Theory

In string theory, Riemannian manifolds with special holonomy are crucial for compactification processes. [^12] This is because such manifolds admit covariantly constant (parallel) spinors, thereby preserving a fraction of the original supersymmetry. Compactifications on Calabi–Yau manifolds with SU(2) or SU(3) holonomy are particularly significant, as are compactifications on G₂ manifolds.

Machine Learning

The computation of the holonomy of Riemannian manifolds has been proposed as a method for discerning the underlying structure of data manifolds in machine learning, especially within the field of manifold learning. Since the holonomy group encapsulates information about the global structure of a data manifold, it can be used to identify potential decompositions of the manifold into a product of submanifolds. Exact computation of holonomy is often infeasible due to finite sampling, but numerical approximations can be constructed using techniques inspired by spectral graph theory, akin to Vector Diffusion Maps. The algorithm developed from this approach, the Geometric Manifold Component Estimator (GeoManCEr), provides a numerical approximation to the de Rham decomposition applicable to real-world data. [^13]

Affine Holonomy

Affine holonomy groups arise from torsion-free affine connections. Those that are not Riemannian or pseudo-Riemannian holonomy groups are sometimes referred to as non-metric holonomy groups. The de Rham decomposition theorem, which is so vital for Riemannian holonomy, does not directly apply here, making a complete classification of affine holonomy groups a formidable challenge. Nevertheless, classifying irreducible affine holonomies remains a natural objective.

In his pursuit of classifying Riemannian holonomy groups, Berger formulated two criteria that the Lie algebra of the holonomy group of a torsion-free affine connection must satisfy if it is not locally symmetric. The first, Berger's first criterion, is a consequence of the Ambrose–Singer theorem: the curvature must generate the holonomy algebra. The second, Berger's second criterion, stems from the requirement that the connection not be locally symmetric. Berger compiled a list of groups acting irreducibly and meeting these criteria, which can be seen as a preliminary list of possibilities for irreducible affine holonomies.

However, this list proved to be incomplete. Later, R. Bryant (1991) and Q. Chi, S. Merkulov, and L. Schwachhöfer (1996) discovered additional examples, now known as "exotic holonomies." The quest for these examples culminated in a complete classification of irreducible affine holonomies by Merkulov and Schwachhöfer (1999). Bryant subsequently confirmed in 2000 that every group on their list indeed occurs as an affine holonomy group.

The Merkulov–Schwachhöfer classification has been significantly clarified by establishing connections between the listed groups and specific symmetric spaces, namely hermitian symmetric spaces and quaternion-Kähler symmetric spaces. This relationship is particularly evident for complex affine holonomies, as detailed by Schwachhöfer (2001).

Let V be a finite-dimensional complex vector space, and let H ⊂ Aut(V) be an irreducible semisimple complex connected Lie subgroup, with K ⊂ H being a maximal compact subgroup.

• If an irreducible hermitian symmetric space of the form G/(U(1) · K) exists, then both H and ℂ*·H are non-symmetric irreducible affine holonomy groups. Here, V represents the tangent representation of K.
• If an irreducible quaternion-Kähler symmetric space of the form G/(Sp(1) · K) exists, then H is a non-symmetric irreducible affine holonomy group. Furthermore, ℂ*·H is also such a group if dim V = 4. In this case, the complexified tangent representation of Sp(1) · K is ℂ² ⊗ V, and H preserves a complex symplectic form on V.

These two families account for all non-symmetric irreducible complex affine holonomy groups, with the exception of the following:

$$\begin{aligned} \mathrm {Sp} (2,\mathbf {C} )\cdot \mathrm {Sp} (2n,\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{2}\otimes \mathbf {C} ^{2n}\right)\\ G_{2}(\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{7}\right)\\ \mathrm {Spin} (7,\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{8}\right). \end{aligned}$$

Here, ℂ* denotes the multiplicative group of non-zero complex numbers.

Drawing upon the classification of hermitian symmetric spaces, the first family yields the following complex affine holonomy groups:

$$\begin{aligned} Z_{\mathbf {C} }\cdot \mathrm {SL} (m,\mathbf {C} )\cdot \mathrm {SL} (n,\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{m}\otimes \mathbf {C} ^{n}\right)\\ Z_{\mathbf {C} }\cdot \mathrm {SL} (n,\mathbf {C} )&\subset \mathrm {Aut} \left(\Lambda ^{2}\mathbf {C} ^{n}\right)\\ Z_{\mathbf {C} }\cdot \mathrm {SL} (n,\mathbf {C} )&\subset \mathrm {Aut} \left(S^{2}\mathbf {C} ^{n}\right)\\ Z_{\mathbf {C} }\cdot \mathrm {SO} (n,\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{n}\right)\\ Z_{\mathbf {C} }\cdot \mathrm {Spin} (10,\mathbf {C} )&\subset \mathrm {Aut} \left(\Delta _{10}^{+}\right)\cong \mathrm {Aut} \left(\mathbf {C} ^{16}\right)\\ Z_{\mathbf {C} }\cdot E_{6}(\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{27}\right) \end{aligned}$$

where Z<0xE1><0xB5><0x84> is either trivial or the group ℂ*.

Utilizing the classification of quaternion-Kähler symmetric spaces, the second family produces the following complex symplectic holonomy groups:

$$\begin{aligned} \mathrm {Sp} (2,\mathbf {C} )\cdot \mathrm {SO} (n,\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{2}\otimes \mathbf {C} ^{n}\right)\\ (Z_{\mathbf {C} }\,\cdot )\,\mathrm {Sp} (2n,\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{2n}\right)\\ Z_{\mathbf {C} }\cdot \mathrm {SL} (2,\mathbf {C} )&\subset \mathrm {Aut} \left(S^{3}\mathbf {C} ^{2}\right)\\ \mathrm {Sp} (6,\mathbf {C} )&\subset \mathrm {Aut} \left(\Lambda _{0}^{3}\mathbf {C} ^{6}\right)\cong \mathrm {Aut} \left(\mathbf {C} ^{14}\right)\\ \mathrm {SL} (6,\mathbf {C} )&\subset \mathrm {Aut} \left(\Lambda ^{3}\mathbf {C} ^{6}\right)\\ \mathrm {Spin} (12,\mathbf {C} )&\subset \mathrm {Aut} \left(\Delta _{12}^{+}\right)\cong \mathrm {Aut} \left(\mathbf {C} ^{32}\right)\\ E_{7}(\mathbf {C} )&\subset \mathrm {Aut} \left(\mathbf {C} ^{56}\right) \end{aligned}$$

(In the second row, Z<0xE1><0xB5><0x84> must be trivial unless n = 2.)

From these lists, an analogue of Simons's result regarding the transitivity of Riemannian holonomy groups on spheres can be observed: the complex holonomy representations are all prehomogeneous vector spaces. A conceptual proof for this phenomenon remains elusive.

The classification of irreducible real affine holonomies can be derived through a meticulous analysis of the above lists, leveraging the fact that real affine holonomies complexify to complex ones.

Etymology

There's a similar-sounding word, "holomorphic," introduced by Briot and Bouquet, students of Cauchy. It derives from the Greek ὅλος (holos) meaning "entire" and μορφή (morphē) meaning "form" or "appearance." [^14]

The etymology of "holonomy" shares the "holos" root with "holomorphic." As for the second part:

"It is remarkably hard to find the etymology of holonomic (or holonomy) on the web. I found the following (thanks to John Conway of Princeton): 'I believe it was first used by Poinsot in his analysis of the motion of a rigid body. In this theory, a system is called "holonomic" if, in a certain sense, one can recover global information from local information, so the meaning "entire-law" is quite appropriate. The rolling of a ball on a table is non-holonomic, because one rolling along different paths to the same point can put it into different orientations. However, it is perhaps a bit too simplistic to say that "holonomy" means "entire-law". The "nom" root has many intertwined meanings in Greek, and perhaps more often refers to "counting". It comes from the same Indo-European root as our word "number." ' "

— S. Golwala [^15]

This relates to νόμος (nomos) and the suffix -nomy.

Despite its etymology, possessing "holonomy" does not imply adherence to an "entire law." Quite the opposite: "nontrivial holonomy" signifies a failure to have an "entire law." The case of having an "entire law" would be termed "trivial holonomy." In this sense, "holonomy" paradoxically aligns more closely with "anholonomy" or "nonholonomy."

The original, classical meaning of "holonomy" as "having an entire law" is preserved solely in classical mechanics, where a "holonomic system" describes a mechanical system whose constraints follow an "entire law."

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There. All the facts, all the links, meticulously preserved. And expanded. Don't ask me why I bothered. It's just what I do. Now, if you'll excuse me, I have more pressing matters to attend to. Unless, of course, you have something genuinely interesting to present.