Affine connection

A differential-geometric structure on a smooth manifold , a special kind of connection on a manifold (cf. Connections on a manifold), when the smooth fibre bundle attached to has the affine space of dimension as its typical fibre. The structure of such an involves the assignment to each point of a copy of the affine space , which is identified with the tangent centro-affine space . In an affine connection each smooth curve with origin and each one of its points is thus provided with an affine mapping which satisfies the condition formulated below. Let be covered with coordinate domains, each provided with a smooth field of affine frames in . The origin of these frames coincides with (i.e. smooth vector fields, linearly independent at each point of the domain, are given). The requirement is that, as , when moves along towards , the mapping tends to become the identity mapping, and that the principal part of its deviation from the identity mapping be defined, with respect to some frame, by the system of linear differential forms

(1)

Thus, for , the image of the frame at is the system consisting of the point in with position vector and vectors , where is the tangent vector to at , and

A manifold with an affine connection defined on it is called a space with an affine connection. During the transformation of a frame of the field at an arbitrary point according to the formulas , , i.e. when passing to an arbitrary element of the principal fibre bundle of frames in the tangent spaces with origins at the point , the forms (1) are replaced by the following -forms on :

(2)

while the -forms

(3)

are transformed as follows:

where and are composed from the forms (2) according to (3). The equations (3) are called the structure equations of the affine connection on . Here the left-hand sides — the so-called torsion forms and curvature forms — are semi-basic (cf. Torsion form; Curvature form), i.e. they are linear combinations of the :

(4)

All -forms and , defined on and satisfying equations (3) with left-hand sides of type (4), define a certain affine connection on . The mapping for a curve is obtained as follows. A smooth field of frames is chosen in a coordinate neighbourhood of the origin of the curve , and the image of the frame at point is defined as the solution of the system

(5)

for the initial conditions , where are the defining equations of the curve . The curve which is described in by the point with position vector with respect to is known as the development of . The field of frames in the coordinate neighbourhood may be so chosen that ; then . In the intersection of the coordinate neighbourhoods, , i.e. and

(6)

(7)

Here and are, respectively, the torsion tensor and the curvature tensor of the affine connection on . An affine connection on may be defined by a system of functions on each coordinate neighbourhood which transforms in the intersection of two neighbourhoods according to formula (5). The system is called the object of the affine connection. The mapping is obtained with the aid of (5) into which

is to be substituted.

If, in some neighbourhood of the point , a vector field is given, then, when , the vector is mapped into the vector (where is the solution of system (5)). The differential of this in at :

is called the covariant differential of the field with respect to the given affine connection. Here

form a tensor field, called the covariant derivative of the field . If a second vector field is given, the covariant derivative of the field in the direction of is defined as

which may also be defined with respect to an arbitrary field of frames by the formula

An affine connection on may also be defined as a bilinear operator which assigns a vector field to each two vector fields and , and which possesses the properties:

where is a smooth function on . The relation between these definitions is established by the formula where is the field of frames. The fields of the torsion tensor and curvature tensor

are defined by the formulas:

A vector field is said to be parallel along the curve if holds identically with respect to , i.e. if, along ,

Parallel vector fields are used to effect parallel displacement of vectors (and, generally, of tensors) in an affine connection, representing a linear mapping of the tangent vector spaces , defined by the mapping . In this sense any affine connection generates a linear connection on .

A curve is called a geodesic line in a given affine connection if its development is a straight line; in other words, if, by a suitable parametrization, its tangent vector field is parallel to it. Geodesic lines are defined with respect to a local coordinate system by the system

Through each point, in each direction passes one geodesic line.

There is a one-to-one correspondence between affine connections on and connections in principal fibre bundles of free affine frames in , generated by them. To closed curves with origin and end at there correspond affine transformations , which form the non-homogeneous holonomy group of the given affine connection. The corresponding linear automorphisms form the homogeneous holonomy group. In accordance with the holonomy theorem, the Lie algebras of these groups are defined by the -forms of torsion and curvature . The Bianchi identities apply to the latter:

In particular, for torsion-free affine connections, when , these identities reduce to the following:

The concept of an affine connection arose in 1917 in Riemannian geometry (in the form of the Levi-Civita connection); it found an independent meaning in 1918–1924 owing to work of H. Weyl [1] and E. Cartan .

References

[1]	H. Weyl, "Raum, Zeit, Materie" , Springer (1923)
[2a]	E. Cartan, "Sur les variétés a connexion affine et la théorie de la relativité généralisée (première partie)" Ann. Sci. École Norm. Sup. , 40 (1923) pp. 325–412
[2b]	E. Cartan, "Sur les variétés a connexion affine et la théorie de la relativité généralisée (première partie suite)" Ann. Sci. École Norm. Sup. , 41 (1924) pp. 1–25
[2c]	E. Cartan, "Sur les variétés a connexion affine et la théorie de la relativité généralisée (deuxième partie)" Ann. Sci. École Norm. Sup. , 42 (1925) pp. 17–88
[3a]	E. Cartan, "Sur les variétés à connexion projective" Bull. Soc. Math. France , 52 (1924) pp. 205–241
[3b]	E. Cartan, "Sur les espaces à connexion conforme" Ann. Soc. Polon. Math. , 2 (1923) pp. 171–221
[4]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[5]	M.M. Postnikov, "The variational theory of geodesics" , Saunders (1967) (Translated from Russian)

Comments

Instead of the articles [3a], [3b], one may consult [a1]. Useful additional up-to-date references in English are [a2] and [a3].

References

[a1]	A. Lichnerowicz, "Théorie globale des connexions et des groupes d'holonomie" , Cremonese (1955)
[a2]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
[a3]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)