Connection on a principal bundle
From Diffgeom
Contents
Definition
Setup
Let be a differential manifold,
a Lie group acting on
, and
a principal
-bundle.
Definition part
A principal -connection on this principal
-bundle is a differential 1-form on
with values in the Lie algebra
of
which is
-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on
.
In other words, it is an element of
such that:
-
where
denotes right multiplication by
. This condition is
-equivariance
- If
and
is the fundamental vector field corresponding to
, then
identically on
.
Related notions
- Connection on a vector bundle
- Ehresmann connection
- Linear connection
Facts
Viewing a connection on a vector bundle as a principal connection
Fill this in later
Transport using principal connections
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