Connections on a manifold

Differential-geometric structures (cf. Differential-geometric structure) on a smooth manifold $ M $ that are connections (cf. Connection) on smooth fibre bundles $ E $ with homogeneous spaces $ G / H $ of the same dimension as $ M $ as typical fibres over the base $ M $. Depending on the choice of the homogeneous space $ G / H $ one obtains, for example, affine, projective, conformal, etc., connections on $ M $( cf. Affine connection; Conformal connection; Projective connection). The general notion of a connection on a manifold was introduced by E. Cartan [1], who called a manifold $ M $ with a connection defined on it a "non-holonomic space with a fundamental groupnon-holonomic space with a fundamental group" .

The modern definition of a connection on a manifold $ M $ is based on the concept of a smooth fibre bundle over the base $ M $. Let $ F = G / H $ be a homogeneous space of the same dimension as $ M $( for example, an affine space, a projective space, etc.). Let $ p : E \rightarrow M $ be a smooth locally trivial fibration with typical fibre $ F $ and suppose that in this fibration there is fixed a smooth section $ s $, that is, a smooth mapping $ s : M \rightarrow E $ such that $ p ( s ( x) ) = x $ for every $ x \in M $. The last condition ensures that $ s $ is a diffeomorphism of $ M $ onto $ s ( M) $, and therefore $ M $ and $ s ( M) $ can be identified, if desired. In other words, to each point $ x \in M $ there is associated a copy $ F _ {x} $ of the homogeneous space $ F $ of the same dimension as $ M $( that is, the fibre of $ p : E \rightarrow M $ over $ x $) with a fixed point $ s ( x) $ that can be identified with $ x $.

A connection on a manifold is a special case of the more general concept of a connection; it can be defined independently as follows. Suppose that for each piecewise-smooth curve $ L ( x _ {0} , x _ {1} ) $ on a manifold $ M $ there is an isomorphism $ \Gamma L : F _ {x _ {1} } \rightarrow F _ {x _ {0} } $ of the tangent homogeneous spaces at the end points of the curve (for example, if $ F $ is an affine or projective space, then $ \Gamma L $ is, respectively, an affine or projective mapping). In addition, suppose that

1) for $ L ( x _ {0} , x _ {1} ) $, $ L ^ \prime ( x _ {1} , x _ {2} ) $, $ L ^ {-1} ( x _ {1} , x _ {0} ) $, and $ L L ^ \prime ( x _ {0} , x _ {2} ) $ one has $ \Gamma L ^ {-1} = ( \Gamma L ) ^ {-1} $, $ \Gamma ( L L ^ \prime ) = ( \Gamma L) ( \Gamma L ^ \prime ) $;

2) for each point $ x \in M $ and for each tangent vector $ X _ {x} \in T _ {x} ( M) $ the isomorphism $ \Gamma L _ {t} : F _ {x _ {t} } \rightarrow F _ {x} $, where $ L _ {t} $ denotes the image of $ [ 0 , t ] $ under the parametrization $ \lambda : [ 0 , 1 ] \rightarrow L ( x , x _ {1} ) $ of $ L $ with tangent vector $ X $, tends to the identity isomorphism as $ t \rightarrow 0 $, and its deviation from the latter depends in its principal part only on $ x $ and $ X $, and this dependence is smooth.

In this case it is said that a connection $ \Gamma $ of type $ F $ is defined on $ M $; the isomorphism $ \Gamma L $ is called the parallel displacement along $ L $. For each curve $ L ( x , x _ {1} ) \in M $ its evolute is defined, that is, the curve in $ F _ {x} $ that consists of the image of the points $ x _ {t} $ of $ L $ under parallel displacement along $ L _ {t} $. It follows from 2) that curves with common tangent vector $ X $ at a point $ x $ have evolutes with common tangent vector $ Y $ that depends smoothly on $ x $ and $ X $. A consequence of this is that for each point $ x $ there is a mapping

$$ f _ {x} : T _ {x} ( M) \rightarrow T _ {s ( x) } ( F _ {x} ) . $$

The connections on a manifold that have been studied most are linear connections, which have the following additional property:

3) the element $ \omega ( x) $ in the Lie algebra $ \mathfrak g $ of the structure group $ G $ that defines the principal part of the deviation of the isomorphism $ \Gamma L _ {t} $ from the identity isomorphism as $ t \rightarrow 0 $ relative to a certain field of frames, depends linearly on $ X $.

In this case $ f _ {x} $ is a linear mapping. If $ f _ {x} $ is an isomorphism for any point $ x $, then one speaks about a non-degenerate connection on a manifold, or about a Cartan connection; in this case the isomorphism $ f _ {x} ^ {-1} $ is also treated as a glueing of the fibration $ p : E \rightarrow M $ to the base $ M $( along a given section $ s $). A Cartan connection on $ M $ is called complete if for each point $ x $, any smooth curve in $ F _ {x} $ that begins at $ x $ is the evolute of a curve on $ M $.

There is another point of view of the general theory of connections, where a linear connection in the fibration $ p : E \rightarrow M $ is defined by using a horizontal distribution $ \Delta $ on $ E $. Then the mapping $ f _ {x} $ is the composite of an isomorphism $ s ^ {*} $ that maps $ X $ into the corresponding tangent vector to $ s ( M) $, followed by a projection of the space $ T _ {s ( x) } ( E) = \Delta _ {s ( x) } \oplus T _ {s ( x) } ( F _ {x} ) $ onto the second direct summand. Hence it follows that a connection is non-degenerate if and only if $ \Delta _ {s ( x) } \cap T _ {s ( x) } ( s ( M) ) = \{ 0 \} $ for any $ x \in M $. To $ M $ all concepts and results developed in the general theory of connections can be applied. Such are, e.g., the holonomy group, the curvature form, the holonomy theorem, etc. The additional structure of a fibre bundle over the manifold $ M $ enables one, however, to introduce certain more special concepts. Apart from evolutes, the most most important of these is the concept of the torsion form of a connection on $ M $ at $ x $.

The Cartan connections in the case when $ F = G / H $ is a homogeneous reductive space (that is, when there is a direct decomposition $ \mathfrak g = \mathfrak k + \mathfrak m $ with the property $ [ \mathfrak h \mathfrak m ] \subset \mathfrak m $) occupy a special position in the theory of connections on a manifold. In this case the curvature form $ \Omega $ splits into two independent objects: its component in $ \mathfrak m $ generates the torsion form, and the component in $ \mathfrak h $ generates the curvature form. The best-known example here is an affine connection on $ M $ for which $ F $ is an affine space of the same dimension as $ M $.

A reductive space $ F $ has an invariant affine connection. More generally, if there is an invariant affine or projective connection on $ F $, then the geodesic lines (cf. Geodesic line) of a connection of type $ F $ are defined on $ M $ as those lines possessing evolutes which are geodesic lines of the given invariant connection.

References

[1]	E. Cartan, "Espaces à connexion affine, projective et conforme" Acta Math. , 48 (1926) pp. 1–42
[2]	G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian)
[3]	Ch. Ehresmann, "Les connexions infinitésimal dans une espace fibré différentiable" , Colloq. de Topologie Bruxelles, 1950 , G. Thone & Masson (1951) pp. 29–55
[4]	S. Kobayashi, "On connections of Cartan" Canad. J. Math. , 8 : 2 (1956) pp. 145–156
[5]	Y.H. Clifton, "On the completeness of Cartan connections" J. Math. Mech. , 16 : 6 (1966) pp. 569–576

Comments

Let $ E = U \times F $ be a trivial vector bundle. The principal part of an element $ e = ( u , f ) \in E $ is the component $ f $. Similarly, if $ \phi : E \rightarrow E $ is a bundle homomorphism ( $ ( u , f ) \mapsto ( u , g( u ) f ) $), then $ g( u ) $, or $ u \mapsto g( u ) $, is its principal part. See also the editorial comments to Connection.