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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?

Vertex
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1 Answers1

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An affine manifold is defined a connection whose curvature and torsion forms vanish. If you define by a flat manifold a manifold endowed with a differentiable metric whose curvature vanishes, then such a manifold is affine since the torsion form of a differentiable metric vanishes. But the fact that only the curvature vanishes does not imply the existence of an affine structure.

Tsemo Aristide
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  • Is there a simple counterexample to that?

    Can compactness help?

     – Vertex Oct 26 '15 at 22:23
  • 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?

     – Vertex Jan 11 '16 at 04:08