Connection on a principal bundle
The user responsible for this page is Jost Eschenburg. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Definition
Let be a Lie group with Lie algebra
and
a principal bundle for
over a
smooth manifold
. A connection on
is a distribution
(a subbundle of the tangent bundle)
on
, called the "horizontal distribution", which is
-invariant and
complementary to the vertical distribution
on
.
The decomposition can be given by the projection
onto the vertical distribution. Since each vertical
space
can be identified with
(see Principal bundle),
this map
can be viewed as a
-valued 1-form on
,
a linear map
; this is called the connection form.
The -valued 2-form
is called curvature form
and measures the non-integrability of the distribution
, see the theory page Connections for details.
A connection on a
-principal bundles
induces a distribution on any associated
bundle
(see Principal bundle) since
passes
trivially to
and by
-invariance to
. The induced distribution is called a connection on
. If
is a vector bundle (the action of
on
is linear), the connection
on
is closely related to a covariant derivative (see Connections).