Isotropy group
The set of elements of a given group
, acting on a set
as a group of transformations, that leave a point
fixed. This set turns out to be a subgroup of
and is called the isotropy group of the point
. The following terminology is used with the same meaning: stationary subgroup, stabilizer,
-centralizer. If
is a Hausdorff space and
is a topological group acting continuously on
, then
is a closed subgroup. If, furthermore,
and
are locally compact,
has a countable base and acts transitively on
, then there exists a unique homeomorphism from
into the quotient space
, where
is one of the isotropy groups; all the
,
, are isomorphic to
.
Let be a smooth manifold and
a Lie group acting smoothly on
. Then the isotropy group
of a point
induces a group of linear transformations of the tangent vector space
; the latter is called the linear isotropy group at
. On passing to tangent spaces of higher order at the point
one obtains natural representations of the isotropy group in the structure groups of the corresponding tangent bundles of higher order; they are called the higher-order isotropy groups (see also Isotropy representation).
References
[1] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) |
[2] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) |
[3] | R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft. (1972) |
Isotropy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotropy_group&oldid=17780