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)

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)