Connection form
A linear differential form on a principal fibre bundle
that takes values in the Lie algebra
of the structure group
of
. It is defined by a certain linear connection
on
, and it determines this connection uniquely. The values of the connection form
in terms of
, where
and
, are defined as the elements of
which, under the action of
on
, generate the second component of
relative to the direct sum
. Here
is the fibre of
that contains
and
is the horizontal distribution of
. The horizontal distribution
, and so the connection
, can be recovered from the connection form
in the following way.
The Cartan–Laptev theorem. For a form on
with values in
to be a connection form it is necessary and sufficient that: 1)
, for
, is the element of
that generates
under the action of
on
; and 2) the
-valued
-form
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formed from , is semi-basic, or horizontal, that is,
if at least one of the vectors
belongs to
. The
-form
is called the curvature form of the connection. If a basis
is defined in
, then condition 2) can locally be expressed by the equalities:
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where are certain linearly independent semi-basic
-forms. The necessity of condition 2) was established in this form by E. Cartan [1]; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev [2]. The equations
for the components of the connection form are called the structure equations for the connection in , the
define the curvature object.
As an example, let be the space of affine frames in the tangent bundle of an
-dimensional smooth manifold
. Then
and
are, respectively, the group and the Lie algebra of matrices of the form
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and
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By the Cartan–Laptev theorem, the -valued
-form
![]() |
on is the connection form of a certain affine connection on
if and only if
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Here and
form, respectively, the torsion and curvature tensors of the affine connection on
. The last two equations for the components of the connection form are called the structure equations for the affine connection on
.
References
[1] | E. Cartan, "Espaces à connexion affine, projective et conforme" Acta Math. , 48 (1926) pp. 1–42 |
[2] | G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian) |
[3] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969) |
Connection form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connection_form&oldid=13808