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A differential-geometric object on a smooth principal fibre bundle  $  P $
that is used to define a [[Horizontal distribution|horizontal distribution]]  $  \Delta $
of a connection in  $  P $. 
Let  $  R _ {0} ( P) $
be the bundle of all tangent frames to  $  P $
such that the first  $  r $
vectors  $  e _ {1} \dots e _ {r} $
are tangent to the corresponding fibre, and are generated by  $  r $
basis elements in the Lie algebra of the structure group  $  G $
of  $  P $, 
$  r =  \mathop{\rm dim}  G $. 
A connection object then consists of functions  $  \Gamma _ {i}  ^  \rho  $
on  $  R _ {0} ( P) $
such that the subspace of  $  \Delta $
is spanned by the vectors  $  e _ {i} + \Gamma _ {i}  ^  \rho  e _  \rho  $
$  ( \rho , \sigma = 1 \dots r;  i , j , \dots = r + 1 \dots r+ n ) $. 
Furthermore, the  $  \Gamma _ {i}  ^  \rho  $
must satisfy the following conditions on  $  R _ {0} ( P) $:

$$ \tag{1 }
d \Gamma _ {i}  ^  \rho  - \Gamma _ {j}  ^  \rho  \omega _ {i}  ^ {j} +
\Gamma _ {i}  ^  \sigma  \omega _  \sigma   ^  \rho  + \omega _ {i}  ^  \rho 
 =  \Gamma _ {ij}  ^  \rho  \omega  ^ {j} .
$$

They are expressed by using the  $  1 $-
forms on  $  R _ {0} ( P) $
that occur in the structure equations for the forms  $  \omega  ^ {i} , \omega  ^  \rho  $
given by the co-basis dual to  $  \{ e _ {i} , e _  \rho  \} $;

$$ \tag{2 }
\left .

\begin{array}{c}
d \omega  ^ {i}  =  \omega  ^ {j} \wedge \omega _ {j}  ^ {i} ,  \\
d \omega  ^  \rho   =  - 
\frac{1}{2}
 C _ {\sigma \tau }   ^  \rho 
\omega  ^  \sigma  \wedge \omega  ^  \tau  + \omega  ^ {i} \wedge
\omega _ {i}  ^  \rho  ,  \\
\omega _  \sigma   ^  \rho   =  - C _ {\sigma \tau }   ^  \rho 
\omega  ^  \tau  .  \\
\end{array}
 \right \}
$$

A connection object also defines a corresponding [[Connection form|connection form]]  $  \theta $, 
given by the relation  $  \theta  ^  \rho  = \omega  ^  \rho  - \Gamma _ {i}  ^  \rho  \omega  ^ {i} $, 
and its [[Curvature form|curvature form]]  $  \Omega $, 
given by the formulas:

$$ 
\Omega  ^  \rho   =  - 
\frac{1}{2}

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

$$ 
R _ {ij}  ^  \rho   =  - 2 ( \Gamma _ {[ ij ] } 
 ^  \rho  + C _ {\sigma \tau }   ^  \rho  \Gamma _ {i}  ^  \sigma  \Gamma _ {j}  ^  \tau  ) .
$$

For example, let  $  P $
be the space of affine tangent frames of an  $  n $-
dimensional smooth manifold  $  M $. 
Then the second equation in (2) has the form

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

and (1) reduces to

$$ 
d \Gamma _ {ik}  ^ {j} - \Gamma _ {lk}  ^ {j} \omega _ {i}  ^ {l} -
\Gamma _ {il}  ^ {j} \omega _ {k}  ^ {l} + \Gamma _ {ik}  ^ {l}
\omega _ {l}  ^ {j} + \omega _ {ik}  ^ {j}  = \ 
\Gamma _ {jkl}  ^ {i} \omega  ^ {l} .
$$

Under [[Parallel displacement(2)|parallel displacement]] one must have  $  \omega _ {j}  ^ {i} - \Gamma _ {jk}  ^ {i} \omega  ^ {k} = 0 $. 
If a local chart is chosen in  $  M $, 
and if in its domain one makes the transition to the natural frame of the chart, i.e.  $  \omega  ^ {k} = dx  ^ {k} $, 
then the parallel displacement is defined by  $  \omega _ {j}  ^ {i} = \Gamma _ {jk}  ^ {i}  dx  ^ {k} $. 
The classical definition of a connection object of an affine connection on  $  M $
is given by the set of functions  $  \Gamma _ {jk}  ^ {i} $
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

$$ 
\Gamma _ {st} ^ { \prime r }  = \ 

\frac{\partial  x ^ {\prime r } }{\partial  x  ^ {i} }

\frac{\partial  x  ^ {j} }{\partial  x ^ {\prime s } }

\frac{\partial  x  ^ {k} }{\partial  x ^ {\prime t } }

\Gamma _ {jk}  ^ {i} +

\frac{\partial   ^ {2} x  ^ {i} }{\partial  x ^ {\prime s } \partial  x ^ {\prime t } }

\frac{\partial  x ^ {\prime r } }{\partial  x  ^ {i} }
 .
$$

Here this follows from the condition of invariance under displacement.