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 Élie Cartan (1926) 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
Hol x ( ∇ ) = { P γ ∈ G L ( E x ) ∣ γ is a loop based at x } . {\displaystyle \operatorname {Hol} _{x}(\nabla )=\{P_{\gamma }\in \mathrm {GL} (E_{x})\mid \gamma {\text{ is a loop based at }}x\}.}
The restricted holonomy group based at x is the subgroup Hol x 0 ( ∇ ) {\displaystyle \operatorname {Hol} _{x}^{0}(\nabla )} 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
Hol y ( ∇ ) = P γ Hol x ( ∇ ) P γ − 1 . {\displaystyle \operatorname {Hol} _{y}(\nabla )=P_{\gamma }\operatorname {Hol} _{x}(\nabla )P_{\gamma }^{-1}.}
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:
Hol 0 ( ∇ ) {\displaystyle \operatorname {Hol} ^{0}(\nabla )} is a connected Lie subgroup of GL(k, R).
Hol 0 ( ∇ ) {\displaystyle \operatorname {Hol} ^{0}(\nabla )} is the identity component of Hol ( ∇ ) . {\displaystyle \operatorname {Hol} (\nabla ).}
If M is simply connected, then Hol ( ∇ ) = Hol 0 ( ∇ ) . {\displaystyle \operatorname {Hol} (\nabla )=\operatorname {Hol} ^{0}(\nabla ).}
… excerpt ends here. Continue reading the full article.
