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An isomorphism of fibres over the end-points  $  x _ {0} $
and  $  x _ {1} $
of a piecewise-smooth curve  $  L( x _ {0} , x _ {1} ) $
in the base  $  M $
of a smooth fibre space  $  E $
defined by some [[Connection|connection]] given in  $  E $; 
in particular, a linear isomorphism between the tangent spaces  $  T _ {x _ {0}  } ( M) $
and  $  T _ {x _ {1}  } ( M) $
defined along a curve  $  L \in M $
of some [[Affine connection|affine connection]] given on  $  M $. 
The development of the concept of a parallel displacement began with the ordinary parallelism on the Euclidean plane  $  E  ^ {2} $, 
for which F. Minding (1837) indicated a way of generalizing it to the case of a surface  $  M $
in  $  E  ^ {3} $
by means of the development of a curve  $  L \in M $
onto the plane  $  E  ^ {2} $, 
a notion he introduced. This served as the starting point for T. Levi-Civita [[#References|[1]]], who, by forming analytically a parallel displacement of the tangent vector to a surface, discovered that it depends only on the metric of the surface and on this basis generalized it at once to the case of an  $  n $-
dimensional Riemannian space (see [[Levi-Civita connection|Levi-Civita connection]]). H. Weyl [[#References|[2]]] placed the concept of parallel displacement of a tangent vector at the base of the definition of an affine connection on a smooth manifold  $  M $. 
Further generalizations of the concept are linked with the development of a general theory of connections.

Suppose that on a smooth manifold  $  M $
an affine connection is given by means of the matrix of local connection forms:

$$ 
\omega  ^ {i}  =  \Gamma _ {k}  ^ {l} ( x)  dx  ^ {k} ,\ \ 
\omega _ {j}  ^ {i}  =  \Gamma _ {jn}  ^ {i} ( x) \omega  ^ {k} ,\ \ 
 \mathop{\rm det}  | \Gamma _ {k}  ^ {i} |  \neq  0.
$$

One says that a vector  $  X _ {0} \in T _ {x _ {0}  } ( M) $
is obtained by parallel displacement from a vector  $  X _ {1} \in T _ {x _ {1}  } ( M) $
along a smooth curve  $  L( x _ {0} , x _ {1} ) \in M $
if on  $  L $
there is a smooth vector field  $  X $
joining  $  X _ {0} $
and  $  X _ {1} $
and such that  $  \nabla _ {Y} X = 0 $. 
Here  $  Y $
is the field of the tangent vector of  $  L $
and  $  \nabla _ {Y} X $
is the covariant derivative of  $  X $
relative to  $  Y $, 
which is defined by the formula

$$ 
\omega  ^ {i} ( \nabla _ {Y} X)  =  Y \omega  ^ {i} ( X) + \omega _ {k}  ^ {i} ( Y)
\omega  ^ {k} ( X).
$$

Thus, the coordinates  $  \zeta  ^ {i} = \omega  ^ {i} ( X) $
of  $  X $
must satisfy along  $  L $
the system of differential equations

$$ 
d \zeta  ^ {i} + \zeta  ^ {k} \omega _ {k}  ^ {i}  =  0.
$$

From the linearity of this system it follows that a parallel displacement along  $  L $
determines a certain isomorphism between  $  T _ {x _ {0}  } ( M) $
and  $  T _ {x _ {1}  } ( M) $. 
A parallel displacement along a piecewise-smooth curve is defined as the composition of the parallel displacements along its smooth pieces.

The automorphisms of the space  $  T _ {x} ( M) $
defined by parallel displacements along closed piecewise-smooth curves  $  L( x, x ) $
form the linear [[Holonomy group|holonomy group]]  $  \Phi _ {x} $; 
here  $  \Phi _ {x} $
and  $  \Phi _ {x  ^  \prime   } $
are always conjugate to each other. If  $  \Phi _ {x} $
is discrete, that is, if its component of the identity is a singleton, then one talks of an affine connection with a (local) absolute parallelism of vectors, or of a (locally) flat connection. Then the parallel displacement for any  $  x _ {0} $
and  $  x _ {1} $
does not depend on the choice of  $  L( x _ {0} , x _ {1} ) $
from one homotopy class; for this it is necessary and sufficient that the curvature tensor of the connection vanishes.

On the basis of the parallel displacement of a vector one defines the parallel displacement of a covector and, more generally, of a tensor. One says that the field of a covector  $  \theta $
on  $  L $
accomplishes a parallel displacement if for any vector field  $  X $
on  $  L $
accomplishing the parallel displacement the function  $  \theta ( X) $
is constant along  $  L $. 
More generally, one says that a tensor field  $  T $
of type  $  ( 2, 1) $, 
say, accomplishes a parallel displacement along  $  L $
if for any  $  X $, 
$  Y $
and  $  \theta $
accomplishing a parallel displacement the function  $  T( X, Y, \theta ) $
is constant along  $  L $. 
For this it is necessary and sufficient that the components  $  T _ {jk}  ^ {i} $
satisfy along  $  L $
the system of differential equations

$$ 
dT _ {jk}  ^ {i}  =  T _ {lk}  ^ {i} \omega _ {j}  ^ {l} + T _ {jl}  ^ {i} \omega _ {k}  ^ {l} - T _ {jk}  ^ {l} \omega _ {l}  ^ {i} .
$$

After E. Cartan introduced in the 1920's [[#References|[3]]] a space of projective or conformal connection and the general concept of a connection on a manifold, the notion of parallel displacement obtained a more general content. In its most general meaning it is considered nowadays as the analysis of connections in principal fibre spaces or fibre spaces associated to them. There is a way of defining the very concept of a connection by means of that of parallel displacement, which is then defined axiomatically. However, a connection can be given by a [[Horizontal distribution|horizontal distribution]] or some other equivalent manner, for example, a [[Connection form|connection form]]. Then for every curve  $  L( x _ {0} , x _ {1} ) $
in the base  $  M $
its horizontal liftings are defined as integral curves of the horizontal distribution over  $  L $. 
A parallel displacement is then the name for a mapping that puts the end-points of these liftings in the fibre over  $  x _ {1} $
into correspondence with their other end-points in the fibre over  $  x _ {0} $. 
The concepts of the holonomy group and of a (locally) flat connection are defined similarly; the latter are also characterized by the vanishing of the [[Curvature form|curvature form]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  T. Levi-Civita,   "Nozione di parallelismo in una varietá qualunque e consequente specificazione geometrica della curvatura riemanniana"  ''Rend. Circ. Mat. Padova'' , '''42'''  (1917)  pp. 173–205</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Weyl,   "Raum, Zeit, Materie" , Springer  (1923)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Cartan,   "Les groupes d'holonomie des espaces généralisés"  ''Acta Math.'' , '''48'''  (1926)  pp. 1–42</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Nomizu,   "Lie groups and differential geometry" , Math. Soc. Japan  (1956)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. II</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Lichnerowicz,   "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR>
</table>