View source for Horizontal distribution

<!--
h0480201.png
$#A+1 = 47 n = 0
$#C+1 = 47 : ~/encyclopedia/old_files/data/H048/H.0408020 Horizontal distribution
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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====
<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>