Holonomy group
A characteristic of a connection on a fibre bundle. The holonomy group is defined for a principal fibre bundle with a Lie structure group
and (second countable) base
on which an infinitesimal connection
is given. It is also defined for any fibre bundle
associated to
whose fibres are copies of some representation space
of
.
The connection on
(or, respectively, on
) defines, for any piecewise-smooth curve
in
, an isomorphic mapping
between the fibres corresponding to the beginning and the end of
. To each piecewise-smooth closed curve
in
beginning and ending at a point
corresponds an automorphism of the fibre
(or, respectively,
) over the point
. These automorphisms form a Lie group
, which is called the holonomy group of the connection
at
.
If the base is (pathwise) connected, then and
are isomorphic for any
and
in
. One may accordingly speak of the holonomy group of a bundle
(or
) with connection
and with (pathwise) connected base.
The holonomy group is a subgroup of the structure group
. In the case of a linear connection on
this subgroup may be defined directly. Let a point
in the fibre
over a point
be given. The set of elements
such that the points
and
can be connected by horizontal curves in
forms a subgroup
of
, which is isomorphic to
.
The limited (restricted) holonomy group is the subgroup of the holonomy group
generated by the closed curves that are homotopic to zero. It coincides with the pathwise-connected component of the unit element of
; moreover,
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 .
Reduction theorem. Let be a principal fibre bundle satisfying the second axiom of countability; let
be the holonomy group of the connection
defined on
. Then the structure group
is reducible to its subgroup
, and the connection
is reducible to a connection on the reduced fibre bundle
, the holonomy group of which coincides with
.
Holonomy theorem. The holonomy algebra (the algebra of the restricted holonomy group) is a subalgebra of the algebra of generated by all vectors
, where
is the curvature form at the point
, where
runs through the set of points which may be connected with the beginning point
by a horizontal path, and
and
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) |
Holonomy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holonomy_group&oldid=12095