Equi-affine connection

An 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 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 is the affine unimodular group. In the case of a torsion-free connection this condition is equivalent to the symmetry of the 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

[1]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)

References

[a1]	J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)