Linear connection

A linear connection on a differentiable manifold $ M $ is a differential-geometric structure on $ M $ associated with an affine connection on $ M $. For every affine connection a parallel displacement of vectors is defined, which makes it possible to define for every curve $ L ( x _ {0} , x _ {1} ) $ in $ M $ a linear mapping of tangent spaces $ T _ {x _ {1} } ( M) \rightarrow T _ {x _ {0} } ( M) $. In this sense an affine connection determines a linear connection on $ M $, to which all concepts and constructions can be transferred which only depend on the displacement of vectors and, more generally, of tensors. A linear connection on $ M $ is a connection in the principal bundle $ B ( M) $ of frames in the tangent spaces $ T _ {x} ( M) $, $ x \in M $, and is defined in one of the following three equivalent ways:

1) by a connection object $ \Gamma _ {jk} ^ {i} $, satisfying the following transformation law on intersections of domains of local charts:

$$ \overline \Gamma \; {} _ {jk} ^ {i} = \ \frac{\partial \overline{x}\; {} ^ {i} }{\partial x ^ {r} } \frac{\partial x ^ {s} }{\partial \overline{x}\; {} ^ {j} } \frac{\partial x ^ {t} }{\partial \overline{x}\; {} ^ {k} } \Gamma _ {st} ^ {r} + \frac{\partial ^ {2} x ^ {r} }{\partial \overline{x}\; {} ^ {j} \partial \overline{x}\; {} ^ {k} } \frac{\partial \overline{x}\; {} ^ {i} }{\partial x ^ {r} } ; $$

2) by a matrix of $ 1 $- forms $ \omega _ {j} ^ {i} $ on the principal frame bundle $ B ( M) $, such that the $ 2 $- forms

$$ d \omega _ {j} ^ {i} + \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} = \Omega _ {j} ^ {i} $$

in each local coordinate system can be expressed in the form

$$ \Omega _ {j} ^ {i} = \frac{1}{2} R _ {jkl} ^ {i} d x ^ {k} \wedge d x ^ {l} ; $$

3) by the bilinear operator $ \nabla $ of covariant differentiation, which associates with two vector fields $ X , Y $ on $ M $ a third vector field $ \nabla _ {Y} X $ and has 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 $.

Every linear connection on $ M $ uniquely determines an affine connection on $ M $ canonically associated with it. It is determined by the involute of any curve $ L ( x _ {0} , x _ {1} ) $ in $ M $. To obtain this involute one must first define $ n = \mathop{\rm dim} M $ linearly independent parallel vector fields $ X _ {1} \dots X _ {n} $ along $ L $, then expand the tangent vector field to $ L $ in terms of them,

$$ \dot{x} ( t) = \mu ^ {i} ( t) X _ {i} ( t), $$

and finally find in $ T _ {x _ {0} } ( M) $ the solution $ x ( t) $ of the differential equation

$$ \dot{x} ( t) = \mu ^ {i} ( t) X _ {i} ( 0) $$

with initial value $ x ( 0) = 0 $. At an arbitrary point $ x _ {t} $ of $ L $ an affine mapping of tangent affine spaces

$$ ( A _ {n} ) _ {x _ {t} } \rightarrow \ ( A _ {n} ) _ {x _ {0} } $$

is now defined by a mapping of frames

$$ \{ x _ {t} , X _ {i} ( t) \} \rightarrow \ \{ y _ {t} , X _ {i} ( 0) \} , $$

where $ {x _ {0} y _ {t} } vec = x ( t) $.

A linear connection is often identified with the affine connection canonically associated with it, by using the one-to-one correspondence between them.

A linear connection on a vector bundle is a differential-geometric structure on a differentiable vector bundle $ \pi : X \rightarrow B $ which associates with every piecewise-smooth curve $ L $ in $ B $ beginning at $ x _ {0} $ and ending at $ x _ {1} $ a linear isomorphism of the fibres $ \pi ^ {-} 1 ( x _ {0} ) $ and $ \pi ^ {-} 1 ( x _ {1} ) $ as vector spaces, called parallel displacement along $ L $. A linear connection is determined by a horizontal distribution on the principal bundle $ P $ of frames in the fibres of the given vector bundle. Analytically, a linear connection is specified by a matrix of $ 1 $- forms $ \omega _ \alpha ^ \beta $ on $ P $, where $ \alpha , \beta = 1 \dots k $, where $ k $ denotes the dimension of the fibres, such that the $ 2 $- forms

$$ d \omega _ \alpha ^ \beta + \omega _ \alpha ^ \gamma \wedge \omega _ \gamma ^ \beta = \Omega _ \alpha ^ \beta $$

are semi-basic, that is, in every local coordinate system $ ( x ^ {i} ) $ on $ B $ they can be expressed in the form

$$ \Omega _ \alpha ^ \beta = \frac{1}{2} R _ {\alpha i j } ^ \beta \ d x ^ {i} \wedge d x ^ {j} . $$

The horizontal distribution is determined, moreover, by the differential system $ \omega _ \alpha ^ \beta = 0 $ on $ P $. The $ 2 $- forms $ \Omega _ \alpha ^ \beta $ are called curvature forms. According to the holonomy theorem they determine the holonomy group of the linear connection.

A linear connection in a fibre bundle $ E $ is a connection under which the tangent vectors of horizontal curves beginning at a given point $ y $ of $ E $ form a vector subspace $ \Delta _ {y} $ of $ T _ {y} ( E) $; the linear connection is determined by the horizontal distribution $ \Delta $: $ y \mapsto \Delta _ {y} $.

References

[1]	A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[2]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)

References

[a1]	S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III