# 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

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 .

#### 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) |

**How to Cite This Entry:**

Equi-affine connection.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Equi-affine_connection&oldid=18185