# Parallel displacement(2)

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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:

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

Thus, the coordinates of must satisfy along the system of differential equations

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

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)