# 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)