Equi-affine connection
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
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and , then the above condition on
has the form
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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
.
References
[1] | A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian) |
Comments
References
[a1] | J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German) |
Equi-affine connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equi-affine_connection&oldid=18185