Linear connection

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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)