View source for Linear connection

A linear connection on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591401.png" /> is a differential-geometric structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591402.png" /> associated with an [[Affine connection|affine connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591403.png" />. For every affine connection a [[Parallel displacement(2)|parallel displacement]] of vectors is defined, which makes it possible to define for every curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591404.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591405.png" /> a linear mapping of tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591406.png" />. In this sense an affine connection determines a linear connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591407.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591408.png" /> is a connection in the principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l0591409.png" /> of frames in the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914011.png" />, and is defined in one of the following three equivalent ways:

1) by a connection object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914012.png" />, satisfying the following transformation law on intersections of domains of local charts:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914013.png" /></td> </tr></table>

2) by a matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914014.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914015.png" /> on the principal frame bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914016.png" />, such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914017.png" />-forms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914018.png" /></td> </tr></table>

in each local coordinate system can be expressed in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914019.png" /></td> </tr></table>

3) by the bilinear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914020.png" /> of [[Covariant differentiation|covariant differentiation]], which associates with two vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914021.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914022.png" /> a third vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914023.png" /> and has the properties:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914024.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914025.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914026.png" /> is a smooth function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914027.png" />.

Every linear connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914028.png" /> uniquely determines an affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914029.png" /> canonically associated with it. It is determined by the involute of any curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914031.png" />. To obtain this involute one must first define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914032.png" /> linearly independent parallel vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914033.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914034.png" />, then expand the tangent vector field to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914035.png" /> in terms of them,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914036.png" /></td> </tr></table>

and finally find in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914037.png" /> the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914038.png" /> of the differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914039.png" /></td> </tr></table>

with initial value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914040.png" />. At an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914042.png" /> an affine mapping of tangent affine spaces

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914043.png" /></td> </tr></table>

is now defined by a mapping of frames

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914044.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914045.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914046.png" /> which associates with every piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914047.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914048.png" /> beginning at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914049.png" /> and ending at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914050.png" /> a linear isomorphism of the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914052.png" /> as vector spaces, called [[Parallel displacement(2)|parallel displacement]] along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914053.png" />. A linear connection is determined by a [[Horizontal distribution|horizontal distribution]] on the principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914054.png" /> of frames in the fibres of the given vector bundle. Analytically, a linear connection is specified by a matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914055.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914056.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914059.png" /> denotes the dimension of the fibres, such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914060.png" />-forms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914061.png" /></td> </tr></table>

are semi-basic, that is, in every local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914062.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914063.png" /> they can be expressed in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914064.png" /></td> </tr></table>

The horizontal distribution is determined, moreover, by the differential system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914065.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914066.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914067.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914068.png" /> are called curvature forms. According to the holonomy theorem they determine the [[Holonomy group|holonomy group]] of the linear connection.

A linear connection in a fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914069.png" /> is a connection under which the tangent vectors of horizontal curves beginning at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914070.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914071.png" /> form a vector subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914072.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914073.png" />; the linear connection is determined by the [[Horizontal distribution|horizontal distribution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914074.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059140/l05914075.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Lichnerowicz,   "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lang,   "Introduction to differentiable manifolds" , Interscience  (1967)  pp. App. III</TD></TR></table>