View source for Equi-affine connection

An [[Affine connection|affine connection]] on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e0359401.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e0359402.png" /> for which there is a non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e0359403.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e0359404.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e0359405.png" /> that is covariantly constant with respect to it. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e0359406.png" /> can be interpreted as the volume function of the parallelepiped constructed from the vectors of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e0359407.png" />; this condition implies the existence of a volume that is preserved by [[Parallel displacement(2)|parallel displacement]] of vectors. If the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e0359408.png" /> is given by means of a matrix of local connection 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/e/e035/e035940/e0359409.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/e/e035/e035940/e03594010.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e03594011.png" />, then the above condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e03594012.png" /> has 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/e/e035/e035940/e03594013.png" /></td> </tr></table>

Equivalently, an affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e03594014.png" /> is equi-affine if and only if its [[Holonomy group|holonomy group]] is the affine unimodular group. In the case of a torsion-free connection this condition is equivalent to the symmetry of the [[Ricci tensor|Ricci tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e03594015.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e03594016.png" />. In the presence of an equi-affine connection the frame bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e03594017.png" /> can be reduced to a subbundle with respect to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035940/e03594018.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Norden,   "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.A. Schouten,   "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  (Translated from German)</TD></TR></table>