View source for Equi-affine connection

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An [[Affine connection|affine connection]] on a smooth manifold  $  M $
of dimension  $  n $
for which there is a non-zero  $  n $-
form  $  \Phi $
on  $  M $
that is covariantly constant with respect to it. The form  $  \Phi ( X _ {1} \dots X _ {n} ) $
can be interpreted as the volume function of the parallelepiped constructed from the vectors of the fields  $  X _ {1} \dots X _ {n} $; 
this condition implies the existence of a volume that is preserved by [[Parallel displacement(2)|parallel displacement]] of vectors. If the affine connection on  $  M $
is given by means of a matrix of local connection forms

$$ 
\omega  ^ {i}  =  \Gamma _ {k}  ^ {i} ( k)  d x  ^ {k} ,\ \ 
 \mathop{\rm det}  | \Gamma _ {k}  ^ {i} |  \neq  0 ,
$$

$$ 
\omega _ {j}  ^ {i}  =  \Gamma _ {jk}  ^ {i} ( k) \omega  ^ {k} ,
$$

and  $  \Phi = \lambda \omega  ^ {1} \wedge \dots \wedge \omega  ^ {n} $, 
then the above condition on  $  \Phi $
has the form

$$ 
d \lambda  =  \lambda \omega _ {i}  ^ {i} .
$$

Equivalently, an affine connection on  $  M $
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]]  $  R _ {kl} = R _ {kli}  ^ {i} $, 
that is,  $  R _ {kl} = R _ {lk} $. 
In the presence of an equi-affine connection the frame bundle of  $  M $
can be reduced to a subbundle with respect to which  $  \omega _ {i}  ^ {i} = 0 $.

====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>