Holonomy group

A characteristic of a connection on a fibre bundle. The holonomy group is defined for a principal fibre bundle $P$ with a Lie structure group $G$ and (second countable) base $B$ on which an infinitesimal connection $\Gamma$ is given. It is also defined for any fibre bundle $E$ associated to $P$ whose fibres are copies of some representation space $F$ of $G$.

The connection $\Gamma$ on $P$ (or, respectively, on $E$) defines, for any piecewise-smooth curve $L$ in $B$, an isomorphic mapping $\Gamma L$ between the fibres corresponding to the beginning and the end of $L$. To each piecewise-smooth closed curve $L$ in $B$ beginning and ending at a point $x\in B$ corresponds an automorphism of the fibre $G_x$ (or, respectively, $F_x$) over the point $x$. These automorphisms form a Lie group $\Phi_x$, which is called the holonomy group of the connection $\Gamma$ at $x$.

If the base is (pathwise) connected, then $\Phi_x$ and $\Phi_{x'}$ are isomorphic for any $x$ and $x'$ in $B$. One may accordingly speak of the holonomy group of a bundle $P$ (or $E$) with connection $\Gamma$ and with (pathwise) connected base.

The holonomy group $\Phi_x$ is a subgroup of the structure group $G$. In the case of a linear connection on $P$ this subgroup may be defined directly. Let a point $p\in P$ in the fibre $G_x$ over a point $x$ be given. The set of elements $g\in G$ such that the points $p$ and $pg^{-1}$ can be connected by horizontal curves in $P$ forms a subgroup $\Phi_p$ of $G$, which is isomorphic to $\Phi_x$.

The limited (restricted) holonomy group $\Phi_x^0$ is the subgroup of the holonomy group $\Phi_x$ generated by the closed curves that are homotopic to zero. It coincides with the pathwise-connected component of the unit element of $\Phi_x$; moreover, $\Phi/\Phi^0$ is at most countable.

The role of holonomy groups in the differential geometry of fibre bundles is explained by the following theorems on connections on $P$.

Reduction theorem. Let $P(B,G)$ be a principal fibre bundle satisfying the second axiom of countability; let $\Phi$ be the holonomy group of the connection $\Gamma$ defined on $P$. Then the structure group $G$ is reducible to its subgroup $\Phi$, and the connection $\Gamma$ is reducible to a connection on the reduced fibre bundle $P'(B,\Phi)$, the holonomy group of which coincides with $\Phi$.

Holonomy theorem. The holonomy algebra (the algebra of the restricted holonomy group) is a subalgebra of the algebra of $G$ generated by all vectors $\Omega_y(Y,Y')$, where $\Omega_y$ is the curvature form at the point $y$, where $y$ runs through the set of points which may be connected with the beginning point $y_0$ by a horizontal path, and $Y$ and $Y'$ are arbitrary horizontal vectors.

References

[1]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2]	R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964)
[3]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) Zbl 0129.13102