Connection object
A differential-geometric object on a smooth principal fibre bundle that is used to define a horizontal distribution
of a connection in
. Let
be the bundle of all tangent frames to
such that the first
vectors
are tangent to the corresponding fibre, and are generated by
basis elements in the Lie algebra of the structure group
of
,
. A connection object then consists of functions
on
such that the subspace of
is spanned by the vectors
. Furthermore, the
must satisfy the following conditions on
:
![]() | (1) |
They are expressed by using the -forms on
that occur in the structure equations for the forms
given by the co-basis dual to
;
![]() | (2) |
A connection object also defines a corresponding connection form , given by the relation
, and its curvature form
, given by the formulas:
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For example, let be the space of affine tangent frames of an
-dimensional smooth manifold
. Then the second equation in (2) has the form
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and (1) reduces to
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Under parallel displacement one must have . If a local chart is chosen in
, and if in its domain one makes the transition to the natural frame of the chart, i.e.
, then the parallel displacement is defined by
. The classical definition of a connection object of an affine connection on
is given by the set of functions
defined on the domains of the charts such that under transition to the coordinates of another chart these functions are transformed according to the formulas
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Here this follows from the condition of invariance under displacement.
Connection object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection_object&oldid=46477