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A characteristic of a connection on a fibre bundle. The holonomy group is defined for a principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475901.png" /> with a Lie structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475902.png" /> and (second countable) base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475903.png" /> on which an infinitesimal connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475904.png" /> is given. It is also defined for any fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475905.png" /> associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475906.png" /> whose fibres are copies of some representation space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475907.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475908.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h0475909.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759010.png" /> (or, respectively, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759011.png" />) defines, for any piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759013.png" />, an isomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759014.png" /> between the fibres corresponding to the beginning and the end of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759015.png" />. To each piecewise-smooth closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759017.png" /> beginning and ending at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759018.png" /> corresponds an automorphism of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759019.png" /> (or, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759020.png" />) over the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759021.png" />. These automorphisms form a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759022.png" />, which is called the holonomy group of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759023.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759024.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759026.png" /> are isomorphic for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759029.png" />. One may accordingly speak of the holonomy group of a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759030.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759031.png" />) with connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759032.png" /> and with (pathwise) connected base.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759033.png" /> is a subgroup of the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759034.png" />. In the case of a linear connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759035.png" /> this subgroup may be defined directly. Let a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759036.png" /> in the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759037.png" /> over a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759038.png" /> be given. The set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759039.png" /> such that the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759041.png" /> can be connected by horizontal curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759042.png" /> forms a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759044.png" />, which is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759045.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759046.png" /> is the subgroup of the holonomy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759047.png" /> generated by the closed curves that are homotopic to zero. It coincides with the pathwise-connected component of the unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759048.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759049.png" /> is at most countable.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759050.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759051.png" /> be a principal fibre bundle satisfying the second axiom of countability; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759052.png" /> be the holonomy group of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759053.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759054.png" />. Then the structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759055.png" /> is reducible to its subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759056.png" />, and the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759057.png" /> is reducible to a connection on the reduced fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759058.png" />, the holonomy group of which coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759059.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759060.png" /> generated by all vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759062.png" /> is the curvature form at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759064.png" /> runs through the set of points which may be connected with the beginning point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759065.png" /> by a horizontal path, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047590/h04759067.png" /> are arbitrary horizontal vectors.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.L. Bishop,  R.J. Crittenden,  "Geometry of manifolds" , Acad. Press  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.L. Bishop,  R.J. Crittenden,  "Geometry of manifolds" , Acad. Press  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR></table>

Latest revision as of 14:59, 8 August 2014

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)
How to Cite This Entry:
Holonomy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holonomy_group&oldid=12095
This article was adapted from an original article by G.F. Laptev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article