View source for Parallel displacement(2)

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.

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:

<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>

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

<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>

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

<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>

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.

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.

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

<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>

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]].

====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>



====Comments====


====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>