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A linear differential form  $  \theta $
on a principal fibre bundle  $  P $
that takes values in the Lie algebra  $  \mathfrak g $
of the structure group  $  G $
of  $  P $. 
It is defined by a certain [[Linear connection|linear connection]]  $  \Gamma $
on  $  P $, 
and it determines this connection uniquely. The values of the connection form  $  \theta _ {y} ( Y) $
in terms of  $  \Gamma $, 
where  $  y \in P $
and  $  Y \in T _ {y} ( P) $, 
are defined as the elements of  $  \mathfrak g $
which, under the action of  $  G $
on  $  P $, 
generate the second component of  $  Y $
relative to the direct sum  $  T _ {y} ( F) = \Delta _ {y} \oplus T _ {y} ( G _ {y} ) $. 
Here  $  G _ {y} $
is the fibre of  $  P $
that contains  $  y $
and  $  \Delta $
is the horizontal distribution of  $  \Gamma $. 
The horizontal distribution  $  \Delta $, 
and so the connection  $  \Gamma $, 
can be recovered from the connection form  $  \theta $
in the following way.

The Cartan–Laptev theorem. For a form  $  \theta $
on  $  P $
with values in  $  \mathfrak g $
to be a connection form it is necessary and sufficient that: 1)  $  \theta _ {y} ( Y) $, 
for  $  Y \in T _ {y} ( G _ {y} ) $, 
is the element of  $  \mathfrak g $
that generates  $  Y $
under the action of  $  G $
on  $  P $; 
and 2) the  $  \mathfrak g $-valued  $  2 $-form

$$ 
\Omega  =  d \theta +

\frac{1}{2}
 [ \theta , \theta ] ,
$$

formed from  $  \theta $, 
is semi-basic, or horizontal, that is,  $  \Omega _ {y} ( Y , Y _ {1} ) = 0 $
if at least one of the vectors  $  Y , Y _ {1} $
belongs to  $  T _ {y} ( G _ {y} ) $. 
The  $  2 $-form  $  \Omega $
is called the [[Curvature form|curvature form]] of the connection. If a basis  $  \{ e _ {1} \dots e _ {r} \} $
is defined in  $  \mathfrak g $, 
then condition 2) can locally be expressed by the equalities:

$$ 
d \theta  ^  \rho  +

\frac{1}{2}
 C _ {\sigma \tau }   ^  \rho 
\theta  ^  \sigma  \wedge
\theta  ^  \tau   = \ 

\frac{1}{2}
 R _ {ij}  ^  \rho 
\omega  ^ {i} \wedge \omega  ^ {j} ,
$$

where  $  \omega  ^ {1} \dots \omega  ^ {n} $
are certain linearly independent semi-basic  $  1 $-forms. The necessity of condition 2) was established in this form by E. Cartan [[#References|[1]]]; its sufficiency under the additional assumption of 1) was proved by G.F. Laptev [[#References|[2]]]. The equations

for the components of the connection form are called the structure equations for the connection in  $  P $, 
the  $  R _ {ij}  ^  \rho  $
define the curvature object.

As an example, let  $  P $
be the space of affine frames in the tangent bundle of an  $  n $-dimensional smooth manifold  $  M $. 
Then  $  G $
and  $  \mathfrak g $
are, respectively, the group and the Lie algebra of matrices of the form

$$ 
\left \| 
\begin{array}{cc}
 1  &a  ^ {i}  \\
 0  &A _ {j}  ^ {i}  \\
\end{array}
 \right \| ,\ \ 
 \mathop{\rm det}  | A _ {j}  ^ {i} |
 \neq  0 ,
$$

and

$$ 
\left \| 
\begin{array}{cc}
 0  &\mathfrak g  ^ {i}  \\
 0  &\mathfrak g _ {j}  ^ {i}  \\
\end{array}
 \right \| \ \ 
( i , j = 1 \dots n ) .
$$

By the Cartan–Laptev theorem, the  $  \mathfrak g $-valued  $  1 $-form

$$ 
\theta  = \ 
\left \| 
\begin{array}{cc}
 0  &\omega  ^ {i}  \\
 0  &\omega _ {j}  ^ {i}  \\
\end{array}
 \right \|
$$

on  $  P $
is the connection form of a certain [[Affine connection|affine connection]] on  $  M $
if and only if

$$ 
d \omega  ^ {i} +
\omega _ {j}  ^ {i} \wedge
\omega  ^ {j}  = \ 

\frac{1}{2}
 T _ {jk} ^ { i }
\omega  ^ {j} \wedge \omega  ^ {k} ,
$$

$$ 
d \omega _ {j}  ^ {i}  =  \omega _ {k}  ^ {i} \wedge \omega _ {j}  ^ {k}  =  
\frac{1}{2}
 R _ {jkl}  ^ {i} \omega  ^ {k} \wedge \omega  ^ {l} .
$$

Here  $  T _ {jk} ^ { i } $
and  $  R _ {jkl}  ^ {i} $
form, respectively, the torsion and curvature tensors of the affine connection on  $  M $. 
The last two equations for the components of the connection form are called the structure equations for the affine connection on  $  M $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,   "Espaces à connexion affine, projective et conforme"  ''Acta Math.'' , '''48'''  (1926)  pp. 1–42</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR></table>