Parallel displacement(2)
An isomorphism of fibres over the end-points and
of a piecewise-smooth curve
in the base
of a smooth fibre space
defined by some connection given in
; in particular, a linear isomorphism between the tangent spaces
and
defined along a curve
of some affine connection given on
. The development of the concept of a parallel displacement began with the ordinary parallelism on the Euclidean plane
, for which F. Minding (1837) indicated a way of generalizing it to the case of a surface
in
by means of the development of a curve
onto the plane
, 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
-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
. Further generalizations of the concept are linked with the development of a general theory of connections.
Suppose that on a smooth manifold an affine connection is given by means of the matrix of local connection forms:
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One says that a vector is obtained by parallel displacement from a vector
along a smooth curve
if on
there is a smooth vector field
joining
and
and such that
. Here
is the field of the tangent vector of
and
is the covariant derivative of
relative to
, which is defined by the formula
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Thus, the coordinates of
must satisfy along
the system of differential equations
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From the linearity of this system it follows that a parallel displacement along determines a certain isomorphism between
and
. 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 defined by parallel displacements along closed piecewise-smooth curves
form the linear holonomy group
; here
and
are always conjugate to each other. If
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
and
does not depend on the choice of
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 on
accomplishes a parallel displacement if for any vector field
on
accomplishing the parallel displacement the function
is constant along
. More generally, one says that a tensor field
of type
, say, accomplishes a parallel displacement along
if for any
,
and
accomplishing a parallel displacement the function
is constant along
. For this it is necessary and sufficient that the components
satisfy along
the system of differential equations
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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 in the base
its horizontal liftings are defined as integral curves of the horizontal distribution over
. A parallel displacement is then the name for a mapping that puts the end-points of these liftings in the fibre over
into correspondence with their other end-points in the fibre over
. 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) |
Parallel displacement(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallel_displacement(2)&oldid=48115