Affine connection

A differential-geometric structure on a smooth manifold $ M $, a special kind of connection on a manifold (cf. Connections on a manifold), when the smooth fibre bundle $ E $ attached to $ M $ has the affine space $ A _ {n} $ of dimension $ n = { \mathop{\rm dim}\nolimits} \ M $ as its typical fibre. The structure of such an $ E $ involves the assignment to each point $ x \in M $ of a copy of the affine space $ ( A _ {n} ) _ {x} $, which is identified with the tangent centro-affine space $ T _ {x} (M) $. In an affine connection each smooth curve $ L \in M $ with origin $ x _ {0} $ and each one of its points $ x _ {t} $ is thus provided with an affine mapping $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $ which satisfies the condition formulated below. Let $ M $ be covered with coordinate domains, each provided with a smooth field of affine frames in $ (A _ {n} ) _ {x} $. The origin of these frames coincides with $ x $( i.e. $ n $ smooth vector fields, linearly independent at each point $ x $ of the domain, are given). The requirement is that, as $ t \rightarrow 0 $, when $ x _ {t} $ moves along $ L $ towards $ x _ {0} $, the mapping $ (A _ {n} ) _ {x _ t} \rightarrow (A _ {n} ) _ { x _ 0 } $ 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

$$ \tag{1} \left . { {\omega ^ {i} \ = \ \Gamma _ {k} ^ {i} \ dx ^ {k} ,\ \ \mathop{\rm det}\nolimits \ | \Gamma _ {k} ^ {i} | \ \neq \ 0,} \atop {\omega _ {j} ^ {i} \ = \ \Gamma _ {jk} ^ {i} \omega ^ {k} .}} \right \} $$

Thus, for $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $, the image of the frame at $ x _ {t} $ is the system consisting of the point in $ (A _ {n} ) _ {x _ 0} $ with position vector $ e _ {i} [ \omega ^ {i} (X) t + \epsilon ^ {i} (t) ] $ and $ n $ vectors $ e _ {i} [ \delta _ {j} ^ {i} + \omega _ {j} ^ {i} (X) t + \epsilon _ {j} ^ {i} (t) ] $, where $ X $ is the tangent vector to $ L $ at $ x _ {0} $, and

$$ \lim\limits _ {t \rightarrow 0} \ \frac{\epsilon ^ {i} (t)}{t} \ = \ 0 ,\ \ \lim\limits _ {t \rightarrow 0} \ \frac{\epsilon _ {j} ^ {i} (t)}{t} \ = \ 0. $$

A manifold $ M $ 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 $ x \in M $ according to the formulas $ e _ {i ^ \prime} = A _ {i ^ \prime} ^ {j} e _ {j} $, $ e _ {j} = A _ {j} ^ {i ^ \prime} e _ {i ^ \prime} $, i.e. when passing to an arbitrary element of the principal fibre bundle $ P $ of frames in the tangent spaces $ ( A _ {n} ) _ {x} $ with origins at the point $ x $, the forms (1) are replaced by the following $ 1 $- forms on $ P $:

$$ \tag{2} \left . { {\omega ^ {i ^ \prime} \ = \ A _ {j} ^ {i ^ \prime} \omega ^ {j} ,} \atop {\omega _ {j ^ \prime} ^ {i ^ \prime} \ = \ A _ {k} ^ {i ^ \prime} \ d A _ {j ^ \prime} ^ {k} + A _ {k} ^ {i ^ \prime} A _ {j ^ \prime} ^ {l} \omega _ {l} ^ {k} ,} } \right \} $$

while the $ 2 $- forms

$$ \tag{3} \left . { {\Omega ^ {i} \ = \ d \omega ^ {i} + \omega _ {j} ^ {i} \wedge \omega ^ {j} ,} \atop {\Omega _ {j} ^ {i} \ = \ d \omega _ {j} ^ {i} + \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} ,}} \right \} $$

are transformed as follows:

$$ \Omega ^ {i ^ \prime} \ = \ A _ {j} ^ {i ^ \prime} \Omega ^ {j} ,\ \ \Omega _ {j ^ \prime} ^ {i ^ \prime} \ = \ A _ {k} ^ {i ^ \prime} A _ {j ^ \prime} ^ {l} \Omega _ {l} ^ {k} , $$

where $ \Omega ^ {i ^ \prime} $ and $ \Omega _ {j ^ \prime} ^ {i ^ \prime} $ are composed from the forms (2) according to (3). The equations (3) are called the structure equations of the affine connection on $ M $. Here the left-hand sides — the so-called torsion forms $ \Omega ^ {i} $ and curvature forms $ \Omega _ {j} ^ {i} $ — are semi-basic (cf. Torsion form; Curvature form), i.e. they are linear combinations of the $ \omega ^ {k} \wedge \omega ^ {l} $:

$$ \tag{4} \left . { {\Omega ^ {i} \ = \ \frac{1}{2} S _ {jk} ^ {i} \omega ^ {j} \wedge \omega ^ {k} ,} \atop {\Omega _ {j} ^ {i} \ = \ \frac{1}{2} R _ {jkl} ^ {i} \omega ^ {k} \wedge \omega ^ {l} .}} \right \} $$

All $ 1 $- forms $ \omega ^ {i} $ and $ \omega _ {j} ^ {i} $, defined on $ P $ and satisfying equations (3) with left-hand sides of type (4), define a certain affine connection on $ M $. The mapping $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $ for a curve $ L \in M $ is obtained as follows. A smooth field of frames is chosen in a coordinate neighbourhood of the origin $ x _ {0} $ of the curve $ L $, and the image of the frame at point $ x _ {t} $ is defined as the solution $ \{ x(t),\ e _ {i} (t) \} $ of the system

$$ \tag{5} \left . { {du \ = \ ( \omega ^ {i} ) _ {x(t)} ( \dot{x} (t) ) u _ {i} ,} \atop {du _ {j} \ = \ ( \omega _ {j} ^ {i} ) _ {x(t)} ( \dot{x} (t)) u _ {i} ,}} \right \} $$

for the initial conditions $ u(0) = 0,\ u _ {i} (0) = e _ {i} $, where $ x ^ {i} = x ^ {i} (t) $ are the defining equations of the curve $ L $. The curve which is described in $ ( A _ {n} ) _ {x _ 0} $ by the point with position vector $ x(t) $ with respect to $ x _ {0} $ is known as the development of $ L $. The field of frames in the coordinate neighbourhood may be so chosen that $ \omega ^ {i} = d x ^ {i} $; then $ \omega _ {j} ^ {i} = \Gamma _ {jk} ^ {i} \ d x ^ {k} $. In the intersection of the coordinate neighbourhoods, $ dx ^ {i ^ \prime} = ( {\partial x ^ {i ^ \prime}} / \partial x ^ {j} ) \omega ^ {j} $, i.e. $ A _ {j} ^ {i ^ \prime} = {\partial x ^ {i ^ \prime}} / \partial x ^ {j} $ and

$$ \tag{6} \Gamma _ {j ^ \prime k ^ \prime} ^ {i ^ \prime} \ = \ \frac{\partial x ^ {i ^ \prime}}{\partial x ^ i} \frac{\partial x ^ j}{\partial x ^ {j ^ \prime}} \frac{\partial x ^ k}{\partial x ^ {k ^ \prime}} \Gamma _ {jk} ^ {i} + \frac{\partial ^ {2} x ^ i}{\partial x ^ {j ^ \prime} \partial x ^ {k ^ \prime}} \frac{\partial x ^ {i ^ \prime}}{\partial x ^ i} , $$

$$ \tag{7} \left . { {S _ {jk} ^ {i} \ = \ \Gamma _ {jk} ^ {i} - \Gamma _ {kj} ^ {i} ,} \atop {R _ {j k l} ^ {i} \ = \ \frac{\partial \Gamma _ {jl} ^ i}{\partial x ^ k} - \frac{\partial \Gamma _ {jk} ^ i}{\partial x ^ l} + \Gamma _ {pk} ^ {i} \Gamma _ {jl} ^ {p} - \Gamma _ {pl} ^ {i} \Gamma _ {jk} ^ {p} .}} \right \} $$

Here $ S _ {jk} ^ {i} $ and $ R _ {jkl} ^ {i} $ are, respectively, the torsion tensor and the curvature tensor of the affine connection on $ M $. An affine connection on $ M $ may be defined by a system of functions $ \Gamma _ {jk} ^ {i} $ on each coordinate neighbourhood which transforms in the intersection of two neighbourhoods according to formula (5). The system $ \Gamma _ {jk} ^ {i} $ is called the object of the affine connection. The mapping $ ( A _ {n} ) _ {x _ t} $ is obtained with the aid of (5) into which

$$ \omega ^ {i} \ = \ dx ^ {i} ,\ \ \omega _ {j} ^ {i} \ = \ \Gamma _ {jk} ^ {i} \ dx ^ {k} , $$

is to be substituted.

If, in some neighbourhood of the point $ x _ {0} $, a vector field $ X = \xi ^ {i} e _ {i} $ is given, then, when $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $, the vector $ X _ {x(t)} $ is mapped into the vector $ \xi ^ {i} ( x ^ {t} ) e _ {i} (t) $( where $ \{ e _ {i} (t) \} $ is the solution of system (5)). The differential of this in $ ( A _ {n} ) _ {x _ 0} $ at $ t = 0 $:

$$ ( d \xi ^ {i} + \xi ^ {i} \omega _ {j} ^ {i} ) e _ {i} \ = \ \left ( \frac{\partial \xi ^ i}{\partial x ^ k} + \xi ^ {j} \Gamma _ {jk} ^ {i} \right ) \ dx ^ {k} e _ {i} , $$

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

$$ \nabla _ {k} \xi ^ {i} \ = \ \frac{\partial \xi ^ i}{\partial x ^ k} + \xi ^ {j} \Gamma _ {jk} ^ {i} $$

form a tensor field, called the covariant derivative of the field $ X = \xi ^ {i} e _ {i} $. If a second vector field $ Y = \eta ^ {k} e _ {k} $ is given, the covariant derivative of the field $ X $ in the direction of $ Y $ is defined as

$$ \nabla _ {Y} X \ = \ \eta ^ {k} \nabla _ {k} \xi ^ {i} e _ {i} , $$

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

$$ \omega ^ {i} ( \nabla _ {Y} X ) \ = \ Y \omega ^ {i} (X) + \omega _ {k} ^ {i} (Y) \omega ^ {k} (X). $$

An affine connection on $ M $ may also be defined as a bilinear operator $ \nabla $ which assigns a vector field $ \nabla _ {Y} X $ to each two vector fields $ X $ and $ Y $, and which possesses the properties:

$$ \nabla _ {Y} ( f X ) \ = \ ( Y f ) X + f \nabla _ {Y} X , \ \ \nabla _ {fY} X \ = \ f \nabla _ {Y} X , $$

where $ f $ is a smooth function on $ M $. The relation between these definitions is established by the formula $ \nabla _ {e _ k} e _ {j} = \Gamma _ {jk} ^ {i} e _ {i} $ where $ \{ e _ {i} \} $ is the field of frames. The fields of the torsion tensor and curvature tensor

$$ S ( X ,\ Y ) \ = \ S _ {jk} ^ {i} \xi ^ {j} \eta ^ {k} e _ {i} \ = \ \Omega ^ {i} ( X ,\ Y ) e _ {i} , $$

$$ R ( X ,\ Y ) Z \ = \ ( R _ {jkl} ^ {i} \xi ^ {k} \eta ^ {l} ) \xi ^ {j} e _ {i} \ = \ \Omega _ {j} ^ {i} ( X ,\ Y ) \omega ^ {j} ( Z ) e _ {i} $$

are defined by the formulas:

$$ S ( X ,\ Y ) \ = \ \nabla _ {X} Y - \nabla _ {Y} X - [ X ,\ Y ], $$

$$ R ( X ,\ Y ) \ = \ \nabla _ {X} \nabla _ {Y} Z - \nabla _ {Y} \nabla _ {X} Z - \nabla _ {[ X ,\ Y ]} Z . $$

A vector field $ X $ is said to be parallel along the curve $ L $ if $ \nabla _ {\dot{x} (t)} X _ {x(t)} = 0 $ holds identically with respect to $ t $, i.e. if, along $ L $,

$$ d \xi ^ {i} + \xi ^ {j} \omega _ {j} ^ {i} \ = \ 0 . $$

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 $ T _ {x _ t} (M) \rightarrow T _ {x _ 0} (M) $, defined by the mapping $ ( A _ {n} ) _ {x _ t} \rightarrow ( A _ {n} ) _ {x _ 0} $. In this sense any affine connection generates a linear connection on $ M $.

A curve $ L $ 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 $ \dot{x} (t) $ is parallel to it. Geodesic lines are defined with respect to a local coordinate system by the system

$$ \frac{d ^ {2} x ^ i}{d t ^ 2} + \Gamma _ {jk} ^ {i} \frac{d x ^ j}{dt} \frac{d x ^ k}{dt} \ = \ 0. $$

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

There is a one-to-one correspondence between affine connections on $ M $ and connections in principal fibre bundles of free affine frames in $ (A _ {n} ) _ {x} ,\ x \in M $, generated by them. To closed curves with origin and end at $ x $ there correspond affine transformations $ ( A _ {n} ) _ {x} \rightarrow ( A _ {n} ) _ {x} $, which form the non-homogeneous holonomy group of the given affine connection. The corresponding linear automorphisms $ T _ {x} (M) \rightarrow T _ {x} (M) $ form the homogeneous holonomy group. In accordance with the holonomy theorem, the Lie algebras of these groups are defined by the $ 2 $- forms of torsion $ \Omega ^ {i} $ and curvature $ \Omega _ {j} ^ {i} $. The Bianchi identities apply to the latter:

$$ d \Omega ^ {i} \ = \ \Omega _ {j} ^ {i} \wedge \omega ^ {i} - \omega _ {j} ^ {i} \wedge \Omega ^ {j} ,\ \ d \Omega _ {j} ^ {i} \ = \ \Omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} - \omega _ {k} ^ {i} \wedge \Omega _ {j} ^ {k} . $$

In particular, for torsion-free affine connections, when $ \Omega ^ {i} = 0 $, these identities reduce to the following:

$$ R _ {jkl} ^ {i} + R _ {klj} ^ {i} + R _ {ljk} ^ {i} \ = \ 0 ,\ \ \nabla _ {m} R _ {jkl} ^ {i} + \nabla _ {k} R _ {jlm} ^ {i} + \nabla _ {l} R _ {jmk} ^ {i} \ = \ 0 . $$

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 à 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 à 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 à 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) Zbl 0116.39101
[a2]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
[a3]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)