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An isomorphism of fibres over the end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713302.png" /> of a piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713303.png" /> in the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713304.png" /> of a smooth fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713305.png" /> defined by some [[Connection|connection]] given in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713306.png" />; in particular, a linear isomorphism between the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713308.png" /> defined along a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p0713309.png" /> of some [[Affine connection|affine connection]] given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133010.png" />. The development of the concept of a parallel displacement began with the ordinary parallelism on the Euclidean plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133011.png" />, for which F. Minding (1837) indicated a way of generalizing it to the case of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133013.png" /> by means of the development of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133014.png" /> onto the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133015.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133016.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133017.png" />. Further generalizations of the concept are linked with the development of a general theory of connections.
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Suppose that on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133018.png" /> an affine connection is given by means of the matrix of local connection forms:
+
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 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133019.png" /></td> </tr></table>
+
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.
  
One says that a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133020.png" /> is obtained by parallel displacement from a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133021.png" /> along a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133022.png" /> if on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133023.png" /> there is a smooth vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133024.png" /> joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133026.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133027.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133028.png" /> is the field of the tangent vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133030.png" /> is the covariant derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133031.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133032.png" />, which is defined by the formula
+
Suppose that on a smooth manifold  $  M $
 +
an affine connection is given by means of the matrix of local connection forms:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133033.png" /></td> </tr></table>
+
$$
 +
\omega  ^ {i}  = \Gamma _ {k}  ^ {l} ( x)  dx  ^ {k} ,\ \
 +
\omega _ {j}  ^ {i}  = \Gamma _ {jn}  ^ {i} ( x) \omega  ^ {k} ,\ \
 +
\mathop{\rm det}  | \Gamma _ {k}  ^ {i} |  \neq  0.
 +
$$
  
Thus, the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133035.png" /> must satisfy along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133036.png" /> the system of differential equations
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133037.png" /></td> </tr></table>
+
$$
 +
\omega  ^ {i} ( \nabla _ {Y} X)  = Y \omega  ^ {i} ( X) + \omega _ {k}  ^ {i} ( Y)
 +
\omega  ^ {k} ( X).
 +
$$
  
From the linearity of this system it follows that a parallel displacement along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133038.png" /> determines a certain isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133040.png" />. A parallel displacement along a piecewise-smooth curve is defined as the composition of the parallel displacements along its smooth pieces.
+
Thus, the coordinates  $  \zeta  ^ {i} = \omega  ^ {i} ( X) $
 +
of $  X $
 +
must satisfy along $  L $
 +
the system of differential equations
  
The automorphisms of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133041.png" /> defined by parallel displacements along closed piecewise-smooth curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133042.png" /> form the linear [[Holonomy group|holonomy group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133043.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133045.png" /> are always conjugate to each other. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133046.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133048.png" /> does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133049.png" /> from one homotopy class; for this it is necessary and sufficient that the curvature tensor of the connection vanishes.
+
$$
 +
d \zeta  ^ {i} + \zeta  ^ {k} \omega _ {k}  ^ {i}  = 0.
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133051.png" /> accomplishes a parallel displacement if for any vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133052.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133053.png" /> accomplishing the parallel displacement the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133054.png" /> is constant along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133055.png" />. More generally, one says that a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133056.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133057.png" />, say, accomplishes a parallel displacement along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133058.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133061.png" /> accomplishing a parallel displacement the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133062.png" /> is constant along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133063.png" />. For this it is necessary and sufficient that the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133064.png" /> satisfy along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133065.png" /> the system of differential equations
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133066.png" /></td> </tr></table>
+
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.
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133067.png" /> in the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133068.png" /> its horizontal liftings are defined as integral curves of the horizontal distribution over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133069.png" />. A parallel displacement is then the name for a mapping that puts the end-points of these liftings in the fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133070.png" /> into correspondence with their other end-points in the fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071330/p07133071.png" />. 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]].
+
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====
 
====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></table>
 
<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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><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>
 
<table><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>

Latest revision as of 08:05, 6 June 2020


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 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 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 [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). H. Weyl [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 $ \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 [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 or some other equivalent manner, for example, a 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.

References

[1] 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
[2] H. Weyl, "Raum, Zeit, Materie" , Springer (1923)
[3] E. Cartan, "Les groupes d'holonomie des espaces généralisés" Acta Math. , 48 (1926) pp. 1–42
[4] K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956)
[5] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)

Comments

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. II
[a2] A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
How to Cite This Entry:
Parallel displacement(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallel_displacement(2)&oldid=16862
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article