Equi-affine connection

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An affine connection on a smooth manifold of dimension for which there is a non-zero -form on that is covariantly constant with respect to it. The form can be interpreted as the volume function of the parallelepiped constructed from the vectors of the fields ; this condition implies the existence of a volume that is preserved by parallel displacement of vectors. If the affine connection on is given by means of a matrix of local connection forms

and , then the above condition on has the form

Equivalently, an affine connection on 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 , that is, . In the presence of an equi-affine connection the frame bundle of can be reduced to a subbundle with respect to which .


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



[a1] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)
How to Cite This Entry:
Equi-affine connection. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article