# Parallel displacement(2)

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) |

[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) |

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Parallel displacement(2).

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