This is a very good question.

It is observed by William M. Goldman that an affinely flat manifold (or just affine manifold) is a manifold with a distinguished coordinate atlas with locally affine coordinate changes: 

http://www2.math.umd.edu/~wmg/MFO_Goldman.pdf

For an in-depth view of affine manifolds, see

Louis Auslander. The structure of complete locally affine manifolds

http://www.sciencedirect.com/science/article/pii/0040938364900126

For examples of affine manifolds, see

Conditions for an affine manifold with torsion to have a Riemann-Cartan structure

https://www.researchgate.net/publication/243031001_Conditions_for_an_affine_manifold_with_torsion_to_have_a_Riemann-Cartan_structure

See, also:

Dulac Criteria for Autonomous Systems Having an Invariant Affine Manifold

https://www.researchgate.net/publication/2617598_Dulac_Criteria_for_Autonomous_Systems_Having_an_Invariant_Affine_Manifold

Article Conditions for an affine manifold with torsion to have a Rie...

Article Dulac Criteria for Autonomous Systems Having an Invariant Af...

Dear  Prof. Peters

Thank you very much for your answer and very interesting and helpful links, in particular the link on Dulac criteria.

Best Regards

Ali Taghavi