View source for Isotropy group

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529701.png" /> of elements of a given group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529702.png" />, acting on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529703.png" /> as a group of transformations, that leave a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529704.png" /> fixed. This set turns out to be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529705.png" /> and is called the isotropy group of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529706.png" />. The following terminology is used with the same meaning: stationary subgroup, stabilizer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529708.png" />-centralizer. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529709.png" /> is a Hausdorff space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297010.png" /> is a topological group acting continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297012.png" /> is a closed subgroup. If, furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297014.png" /> are locally compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297015.png" /> has a countable base and acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297016.png" />, then there exists a unique homeomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297017.png" /> into the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297019.png" /> is one of the isotropy groups; all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297021.png" />, are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297022.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297023.png" /> be a smooth manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297024.png" /> a Lie group acting smoothly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297025.png" />. Then the isotropy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297026.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297027.png" /> induces a group of linear transformations of the tangent vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297028.png" />; the latter is called the linear isotropy group at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297029.png" />. On passing to tangent spaces of higher order at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297030.png" /> 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|Isotropy representation]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Sulanke,   P. Wintgen,   "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft.  (1972)</TD></TR></table>