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A smooth distribution on a smooth fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480201.png" /> with Lie structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480202.png" /> (i.e. a smooth field of linear subspaces of the tangent spaces to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480203.png" />) that defines a [[Connection|connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480204.png" /> in the sense that the horizontal liftings of curves in the base manifold are integral curves of this distribution. A horizontal distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480205.png" /> is transversal to the fibres, i.e. at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480206.png" /> a direct decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480207.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480208.png" /> is the fibre containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h0480209.png" />. The additional conditions that must be imposed on a transversal distribution, sufficient to make it a horizontal distribution in the general case, are quite complex. In the particular case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802010.png" /> being the total space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802011.png" /> of a principal fibre bundle, they must guarantee the invariance of the distribution with respect to the action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802013.png" />. In this case these conditions are formulated using the connection forms that have as annihilator the horizontal distribution, and are expressed in the Cartan–Laptev theorem. It follows from the relevant structure equations that if the smooth vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802016.png" /> are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802017.png" /> at any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802019.png" /> has the component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802022.png" /> is the [[Curvature form|curvature form]]. Thus, a horizontal distribution is involutory if and only if the connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802023.png" /> defined by it is flat.
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A horizontal distribution on a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802024.png" /> associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802025.png" /> is always the image of some horizontal distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802027.png" /> under canonical projections of the factorizations that are used to construct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802028.png" /> starting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802029.png" />. In the general case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802030.png" /> is obtained by factorization from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802031.png" /> with respect to the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802032.png" /> according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802033.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802034.png" /> be the corresponding canonical projection. Each horizontal distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802035.png" /> is obtained as the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802037.png" /> is the natural lifting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802038.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802039.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802040.png" />. In the more special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802041.png" /> is a homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802042.png" />, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802043.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802044.png" /> and each horizontal distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802045.png" /> is obtained as the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802046.png" /> under the canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048020/h04802047.png" />.
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 +
A smooth distribution on a smooth fibre bundle $  E $
 +
with Lie structure group  $  G $
 +
(i.e. a smooth field of linear subspaces of the tangent spaces to  $  E $)
 +
that defines a [[Connection|connection]] on  $  E $
 +
in the sense that the horizontal liftings of curves in the base manifold are integral curves of this distribution. A horizontal distribution  $  \Delta $
 +
is transversal to the fibres, i.e. at any point  $  y \in E $
 +
a direct decomposition  $  T _ {y} ( E) = \Delta _ {y} \oplus T _ {y} ( F _ {y} ) $
 +
holds, where  $  F _ {y} $
 +
is the fibre containing  $  y $.
 +
The additional conditions that must be imposed on a transversal distribution, sufficient to make it a horizontal distribution in the general case, are quite complex. In the particular case of  $  E $
 +
being the total space  $  P $
 +
of a principal fibre bundle, they must guarantee the invariance of the distribution with respect to the action of the group  $  G $
 +
on  $  P $.  
 +
In this case these conditions are formulated using the connection forms that have as annihilator the horizontal distribution, and are expressed in the [[Cartan–Laptev theorem]]. It follows from the relevant structure equations that if the smooth vector fields  $  X $
 +
and  $  Y $
 +
on  $  P $
 +
are such that  $  X _ {y} , Y _ {y} \in \Delta _ {y} $
 +
at any  $  y \in P $,
 +
then  $  [ XY ] _ {y} $
 +
has the component  $  \Omega _ {y} ( X, Y) $
 +
in  $  T _ {y} ( E _ {y} ) $,
 +
where  $  \Omega $
 +
is the [[Curvature form|curvature form]]. Thus, a horizontal distribution is involutory if and only if the connection on  $  P $
 +
defined by it is flat.
 +
 
 +
A horizontal distribution on a bundle  $  E $
 +
associated to  $  P $
 +
is always the image of some horizontal distribution $  \Delta $
 +
on $  P $
 +
under canonical projections of the factorizations that are used to construct $  E $
 +
starting from $  P $.  
 +
In the general case, $  E $
 +
is obtained by factorization from $  P \times F $
 +
with respect to the action of $  G $
 +
according to the formula $  ( y, f  ) \cdot g = ( y \cdot g, g  ^ {-1} \cdot f  ) $.  
 +
Let $  \pi : P \times F \rightarrow E $
 +
be the corresponding canonical projection. Each horizontal distribution on $  E $
 +
is obtained as the image $  \pi  ^ {*} \overline \Delta \; $,  
 +
where $  \overline \Delta \; $
 +
is the natural lifting of $  \Delta $
 +
from $  P $
 +
to $  P \times F $.  
 +
In the more special case when $  F $
 +
is a homogeneous space $  G/H $,  
 +
the space $  E $
 +
is identified with $  P/H $
 +
and each horizontal distribution on $  E $
 +
is obtained as the image $  \pi  ^ {*} \Delta $
 +
under the canonical projection $  \pi : P \rightarrow P/H $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Nomizu,  "Lie groups and differential geometry" , Math. Soc. Japan  (1956)</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">  Ü.G. Lumiste,  "Connections in homogeneous bundles"  ''Transl. Amer. Math. Soc. (2)'' , '''92'''  (1970)  pp. 231–274  ''Mat. Sb.'' , '''69 (111)''' :  3  (1966)  pp. 434–469</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Nomizu,  "Lie groups and differential geometry" , Math. Soc. Japan  (1956)</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">  Ü.G. Lumiste,  "Connections in homogeneous bundles"  ''Transl. Amer. Math. Soc. (2)'' , '''92'''  (1970)  pp. 231–274  ''Mat. Sb.'' , '''69 (111)''' :  3  (1966)  pp. 434–469</TD></TR></table>

Latest revision as of 17:36, 13 January 2021


A smooth distribution on a smooth fibre bundle $ E $ with Lie structure group $ G $ (i.e. a smooth field of linear subspaces of the tangent spaces to $ E $) that defines a connection on $ E $ in the sense that the horizontal liftings of curves in the base manifold are integral curves of this distribution. A horizontal distribution $ \Delta $ is transversal to the fibres, i.e. at any point $ y \in E $ a direct decomposition $ T _ {y} ( E) = \Delta _ {y} \oplus T _ {y} ( F _ {y} ) $ holds, where $ F _ {y} $ is the fibre containing $ y $. The additional conditions that must be imposed on a transversal distribution, sufficient to make it a horizontal distribution in the general case, are quite complex. In the particular case of $ E $ being the total space $ P $ of a principal fibre bundle, they must guarantee the invariance of the distribution with respect to the action of the group $ G $ on $ P $. In this case these conditions are formulated using the connection forms that have as annihilator the horizontal distribution, and are expressed in the Cartan–Laptev theorem. It follows from the relevant structure equations that if the smooth vector fields $ X $ and $ Y $ on $ P $ are such that $ X _ {y} , Y _ {y} \in \Delta _ {y} $ at any $ y \in P $, then $ [ XY ] _ {y} $ has the component $ \Omega _ {y} ( X, Y) $ in $ T _ {y} ( E _ {y} ) $, where $ \Omega $ is the curvature form. Thus, a horizontal distribution is involutory if and only if the connection on $ P $ defined by it is flat.

A horizontal distribution on a bundle $ E $ associated to $ P $ is always the image of some horizontal distribution $ \Delta $ on $ P $ under canonical projections of the factorizations that are used to construct $ E $ starting from $ P $. In the general case, $ E $ is obtained by factorization from $ P \times F $ with respect to the action of $ G $ according to the formula $ ( y, f ) \cdot g = ( y \cdot g, g ^ {-1} \cdot f ) $. Let $ \pi : P \times F \rightarrow E $ be the corresponding canonical projection. Each horizontal distribution on $ E $ is obtained as the image $ \pi ^ {*} \overline \Delta \; $, where $ \overline \Delta \; $ is the natural lifting of $ \Delta $ from $ P $ to $ P \times F $. In the more special case when $ F $ is a homogeneous space $ G/H $, the space $ E $ is identified with $ P/H $ and each horizontal distribution on $ E $ is obtained as the image $ \pi ^ {*} \Delta $ under the canonical projection $ \pi : P \rightarrow P/H $.

References

[1] K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956)
[2] R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964)
[3] Ü.G. Lumiste, "Connections in homogeneous bundles" Transl. Amer. Math. Soc. (2) , 92 (1970) pp. 231–274 Mat. Sb. , 69 (111) : 3 (1966) pp. 434–469
How to Cite This Entry:
Horizontal distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Horizontal_distribution&oldid=11300
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article