Examples of "holonomy"
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".
Some important properties of the holonomy and restricted holonomy groups include:
The theory of isoparametric submanifolds is deeply related to the theory of holonomy groups. Actually, any isoparametric submanifold is foliated by the holonomy tubes of a submanifold with constant principal curvatures i.e. a focal submanifold. The paper "Submanifolds with constant principal curvatures and normal holonomy groups" is a very good introduction to such theory. For more detailed explanations about holonomy tubes and focalizations see the book "Submanifolds and Holonomy".
as the holonomy of KZ equation. Oppositely, a KZ equation gives the linear representation of braid groups as its holonomy.
The holonomy of a Riemannian manifold ("M", "g") is just 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.
Affine holonomy groups are the groups arising as holonomies of torsion-free affine connections; those which are not Riemannian or pseudo-Riemannian holonomy groups are also known as non-metric holonomy groups. The deRham decomposition theorem does not apply to affine holonomy groups, so a complete classification is out of reach. However, it is still natural to classify irreducible affine holonomies.
The study of Riemannian holonomy has led to a number of important developments. The 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, M. Berger classified the possible irreducible holonomies. The decomposition and classification of Riemannian holonomy has applications to physics and to string theory.
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 Hol(ω).
Some important properties of the holonomy group include:
The local holonomy group has the following properties:
with finite holonomy, the space of leaves is Hausdorff.
In symbols, the holonomy angle mod 2π is given by
As parallel transport supplies a local realization of the connection, it also supplies a local realization of the curvature known as holonomy. The Ambrose-Singer theorem makes explicit this relationship between curvature and holonomy.
We need the notion of a holonomy. A holonomy is a measure of how much the initial and final values of a spinor or vector differ after parallel transport around a closed loop; it is denoted
Theorem: "Let formula_1 be a complete conformal foliation of codimension formula_36 of a connected manifold formula_4. If formula_1 has a compact leaf with finite holonomy group, then all the leaves of formula_1 are compact with finite holonomy group."
Theorem: "Let formula_1 be a holomorphic foliation of codimension formula_3 in a compact complex Kähler manifold. If formula_1 has a compact leaf with finite holonomy group then every leaf of formula_1 is compact with finite holonomy group."
H.P. Zeiger (1967) proved an important variant called the holonomy decomposition (Eilenberg 1976). The holonomy method appears to be relatively efficient and has been implemented computationally by A. Egri-Nagy (Egri-Nagy & Nehaniv 2005).
We need the notion of a holonomy. A holonomy is a measure of how much the initial and final values of a spinor or vector differ after parallel transport around a closed loop; it is denoted
G is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G holonomy are also called G-manifolds.
One of the earliest fundamental results on Riemannian holonomy is the theorem of , which asserts that the holonomy group is a closed Lie subgroup of O("n"). In particular, it is compact.