# Are flat manifolds affine?

I try to understand an article where it is stated that some results regarding affine manifolds apply to the case of the manifold being a flat, compact Lorentzian manifold.

The definition of affine, in this context, is that the manifold has a maximal atlas of charts whose transitions maps extend to affine mappings on ${R}^n$.

My questions:

Is a flat manifold affine? Especially, is a flat Lorentzian manifold affine? Is a compact, flat Lorentzian manifold affine?

• Hope the edits will do! Oct 27 '15 at 14:47

• In my text an affine manifold is defined as a manifold whose transition maps extend to affine mappings on ${R}^n$. By 'flat' is meant that the sectional curvature vanishes everywhere. Is it an easy thing to show that flat in this sense implies affine? Jan 11 '16 at 4:08