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 (called Riemannian holonomy), 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 : [0,1] → M based at x in M, the connection defines a parallel transport map Pγ : Ex → Ex 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(Ex). 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(k, R). Explicitly, if γ is a path from x to y in M, then
Choosing different identifications of Ex with Rk also gives conjugate subgroups. Sometimes, particularly in general or informal discussions (such as below), 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(k, R). is the identity component of If M is simply connected, then ∇ is flat (i.e. has vanishing curvature) 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 : [0,1] → 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 (i.e. has vanishing curvature) 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(p) be the set of points in P which can be joined to p by a horizontal curve. Then it can be shown that H(p), with the evident projection map, is a principal bundle over M with structure group This principal bundle is called the holonomy bundle (through p) of the connection. The connection ω restricts to a connection on H(p), since its parallel transport maps preserve H(p). Thus H(p) is a reduced bundle for the connection. Furthermore, since no subbundle of H(p) 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 (since, by assumption, M is path-connected). Hence H(q) = H(p) 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(p) is a principal bundle for and so also admits an action of the restricted holonomy group (which is a normal subgroup of the full 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 (resp. restricted holonomy) of this bundle will be denoted by (resp. ) for each p with π(p) ∈ U.
If U ⊂ V are two open sets containing π(p), 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 Uk 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 Uk 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. (Note that, by property 1, the local holonomy group is a connected Lie subgroup of G, so the adjoint is well-defined.)
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:
If the dimension of the local holonomy group is constant, then the local and restricted holonomy agree:
Ambrose–Singer theorem The Ambrose–Singer theorem (due to ) 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 (or a connection in the tangent bundle the Levi-Civita connection, for example). The curvature arises when one travels around an infinitesimal parallelogram.
In detail, if σ: [0, 1] × [0, 1] → 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 (x, 0), then along (1, y), followed by (x, 1) going in the negative direction, and then (0, y) 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 [0, x] × [0, y]. 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 (the infinitesimal parallelogram). More formally, the curvature is the differential of the holonomy action at the identity of the holonomy group. In other words, R(X, Y) 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:
The Lie algebra of is spanned by all the elements of g of the form as q ranges over all points which can be joined to p by a horizontal curve (q ~ p), and X and Y are horizontal tangent vectors at q.
Alternatively, the theorem can be restated in terms of the holonomy bundle:
The Lie algebra of is the subspace of g spanned by elements of the form where q ∈ H(p) and X and Y are horizontal vectors at q.
Riemannian holonomy The holonomy of a Riemannian manifold (M, g) is the holonomy group of the Levi-Civita connection on the tangent bundle to M. A 'generic' n-dimensional Riemannian manifold has an O(n) holonomy, or SO(n) if it is orientable. Manifolds whose holonomy groups are proper subgroups of O(n) or SO(n) have special properties.
One of the earliest fundamental results on Riemannian holonomy is the theorem of , which asserts that the restricted holonomy group is a closed Lie subgroup of O(n). In particular, it is compact.
Reducible holonomy and the de Rham decomposition Let x ∈ M be an arbitrary point. Then the holonomy group Hol(M) acts on the tangent space TxM. This action may either be irreducible as a group representation, or reducible in the sense that there is a splitting of TxM into orthogonal subspaces TxM = T′xM ⊕ T″xM, each of which is invariant under the action of Hol(M). In the latter case, M is said to be reducible.
Suppose that M is a reducible manifold. Allowing the point x to vary, the bundles T′M and T″M formed by the reduction of the tangent space at each point are smooth distributions which are integrable in the sense of Frobenius. The integral manifolds of these distributions are totally geodesic submanifolds. So M is locally a Cartesian product M′ × M″. The (local) de Rham isomorphism follows by continuing this process until a complete reduction of the tangent space is achieved:
Let M be a simply connected Riemannian manifold,This theorem generalizes to non-simply connected manifolds, but the statement is more complicated. and TM = T(0)M ⊕ T(1)M ⊕ ⋯ ⊕ T(k)M be the complete reduction of the tangent bundle under the action of the holonomy group. Suppose that T(0)M consists of vectors invariant under the holonomy group (i.e., such that the holonomy representation is trivial). Then locally M is isometric to a product