Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence of the curvature of the connection. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.
Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry. Important examples include: holonomy of the Levi-Civita connection in Riemannian geometry, holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the holonomy of the connection can be identified with a Lie group, the holonomy group. The holonomy of a connection is closely related to the curvature of the connection, via the Ambrose–Singer theorem.
The study of Riemannian holonomy has led to a number of important developments. Holonomy was introduced by in order to study and classify symmetric spaces. It was not until much later that holonomy groups would be used to study Riemannian geometry in a more general setting. In 1952 Georges de Rham proved the de Rham decomposition theorem, a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible spaces under the action of the local holonomy groups. Later, in 1953, Marcel Berger classified the possible irreducible holonomies. The decomposition and classification of Riemannian holonomy has applications to physics and to string theory.

Definitions

Holonomy of a connection in a vector bundle

Let E be a rank-k vector bundle over a smooth manifold M, and let ∇ be a connection on E. Given a piecewise smooth loop γ : → M based at x in M, the connection defines a parallel transport map P_γ : E_x → E_x on the fiber of E at x. This map is both linear and invertible, and so defines an element of the general linear group GL. The holonomy group of ∇ based at x is defined as
The restricted holonomy group based at x is the subgroup coming from contractible loops γ.
If M is path-connected, then the holonomy group depends on the basepoint x only up to conjugation in GL. Explicitly, if γ is a path from x to y in M, then
Choosing different identifications of E_x with R^k also gives conjugate subgroups. Sometimes, particularly in general or informal discussions, one may drop reference to the basepoint, with the understanding that it is defined uniquely only up to conjugation.
Some important properties of the holonomy group include:

is a connected Lie subgroup of GL.
is the identity component of
If M is simply connected, then
∇ is flat if and only if is trivial. In this case, may still be nontrivial.
There is a natural, surjective group homomorphism where is the fundamental group of M, which sends the homotopy class to the coset
Holonomy of a connection in a principal bundle

The definition for holonomy of connections on principal bundles proceeds in parallel fashion. Let G be a Lie group and P a principal G-bundle over a smooth manifold M which is paracompact. Let ω be a connection on P. Given a piecewise smooth loop γ : → M based at x in M and a point p in the fiber over x, the connection defines a unique horizontal lift such that The end point of the horizontal lift,, will not generally be p but rather some other point p·''g in the fiber over x''. Define an equivalence relation ~ on P by saying that p ~ q if they can be joined by a piecewise smooth horizontal path in P.
The holonomy group of ω based at p is then defined as
The restricted holonomy group based at p is the subgroup coming from horizontal lifts of contractible loops γ.
If M and P are connected then the holonomy group depends on the basepoint p only up to conjugation in G. Explicitly, if q is any other chosen basepoint for the holonomy, then there exists a unique g ∈ G such that q ~ p·''g. With this value of g'',
In particular,
Moreover, if p ~ q then As above, sometimes one drops reference to the basepoint of the holonomy group, with the understanding that the definition is good up to conjugation.
Some important properties of the holonomy and restricted holonomy groups include:

is a connected Lie subgroup of G.
is the identity component of
There is a natural, surjective group homomorphism
If M is simply connected then
ω is flat if and only if is trivial.
Holonomy bundles

Let M be a connected paracompact smooth manifold and P a principal G-bundle with connection ω, as above. Let p ∈ P be an arbitrary point of the principal bundle. Let H be the set of points in P which can be joined to p by a horizontal curve. Then it can be shown that H, with the evident projection map, is a principal bundle over M with structure group This principal bundle is called the holonomy bundle of the connection. The connection ω restricts to a connection on H, since its parallel transport maps preserve H. Thus H is a reduced bundle for the connection. Furthermore, since no subbundle of H is preserved by parallel transport, it is the minimal such reduction.
As with the holonomy groups, the holonomy bundle also transforms equivariantly within the ambient principal bundle P. In detail, if q ∈ P is another chosen basepoint for the holonomy, then there exists a unique g ∈ G such that q ~ p ''g. Hence H'' = H ''g. As a consequence, the induced connections on holonomy bundles corresponding to different choices of basepoint are compatible with one another: their parallel transport maps will differ by precisely the same element g''.

Monodromy

The holonomy bundle H is a principal bundle for and so also admits an action of the restricted holonomy group . The discrete group is called the monodromy group of the connection; it acts on the quotient bundle There is a surjective homomorphism so that acts on This action of the fundamental group is a monodromy representation of the fundamental group.

Local and infinitesimal holonomy

If π: P → M is a principal bundle, and ω is a connection in P, then the holonomy of ω can be restricted to the fibre over an open subset of M. Indeed, if U is a connected open subset of M, then ω restricts to give a connection in the bundle π⁻¹U over U. The holonomy of this bundle will be denoted by for each p with π ∈ U.
If U ⊂ V are two open sets containing π, then there is an evident inclusion
The local holonomy group at a point p is defined by
for any family of nested connected open sets U_k with.
The local holonomy group has the following properties:

It is a connected Lie subgroup of the restricted holonomy group
Every point p has a neighborhood V such that In particular, the local holonomy group depends only on the point p, and not the choice of sequence U_k used to define it.
The local holonomy is equivariant with respect to translation by elements of the structure group G of P; i.e., for all g ∈ G.

The local holonomy group is not well-behaved as a global object. In particular, its dimension may fail to be constant. However, the following theorem holds:

Ambrose–Singer theorem

The Ambrose–Singer theorem relates the holonomy of a connection in a principal bundle with the curvature form of the connection. To make this theorem plausible, consider the familiar case of an affine connection. The curvature arises when one travels around an infinitesimal parallelogram.
In detail, if σ: × → M is a surface in M parametrized by a pair of variables x and y, then a vector V may be transported around the boundary of σ: first along, then along, followed by going in the negative direction, and then back to the point of origin. This is a special case of a holonomy loop: the vector V is acted upon by the holonomy group element corresponding to the lift of the boundary of σ. The curvature enters explicitly when the parallelogram is shrunk to zero, by traversing the boundary of smaller parallelograms over × . This corresponds to taking a derivative of the parallel transport maps at x = y = 0:
where R is the curvature tensor. So, roughly speaking, the curvature gives the infinitesimal holonomy over a closed loop. More formally, the curvature is the differential of the holonomy action at the identity of the holonomy group. In other words, R is an element of the Lie algebra of
In general, consider the holonomy of a connection in a principal bundle P → M over P with structure group G. Let g denote the Lie algebra of G, the curvature form of the connection is a g-valued 2-form Ω on P. The Ambrose–Singer theorem states:
Alternatively, the theorem can be restated in terms of the holonomy bundle: