I would like to study manifolds endowed with a linear connexion $\nabla$ which is torsion free and locally flat i.e. its curvature is $0$ (such a connexion is called flat if in addition, its holonomy is trivial).

I am looking for a good reference:

  • to prove that such a manifold admits an atlas with affine transition maps (restriction of such maps on opens of $\mathbb{R}^n$).

  • to characterize the universal covering and the fundamental group of such manifolds and its link to Bieberbach groups.

Is it true that a compact affine manifold has a trivial Euler characteristic? (Chern conjecture)

Thank you for any help!

  • 2
    Chern conjecture is still open – Misha Verbitsky Aug 11 '14 at 0:37
  • One version of the Chern conjecture (for isoparametric hypersurfaces): "Let $M$ be a closed, minimally immersed hypersurface of the $(n {+} 1)$-dimensional sphere $S^{n+1}$ with constant scalar curvature. Then $M$ is isoparametric."---Mike Scherfner, TU Berlin. – Joseph O'Rourke Aug 11 '14 at 1:07
  • § V.4 of Kobayashi-Nomizu, in particular Theorem 4.2, might be a good start. – abx Aug 11 '14 at 6:18
  • Thanks abx, I will look for the Kobayashi-Nomizu book. Thanks Misha and Joseph. – user56980 Aug 12 '14 at 0:07

Kostant and Sullivan proved that the Euler characteristic of a compact complete affine manifolds must vanish, affirming the Chern conjecture in the complete case (Bull. AMS 81 (1975)). Benzecri proved that a closed surface which admits a (possibly imcomplete) affine structure must have zero Euler characteristic. In general a good reference is the book "Spaces of constant Curvature" by Joseph A. Wolf (in particular for the link to Bieberbach groups). Also, several survey articles by W. Goldman on affine manifolds are a very good reference, e.g. Locally homogeneous manifolds, see page $11$ for recent work on Chern's conjecture.

  • Many thanks Dietrich for your nice answer. I find the Goldman survey article very interesting! – user56980 Aug 12 '14 at 0:10
  • There is now a claimed proof of the full Chern conjecture (arxiv.org/abs/1603.07248). Any comments? – Danu Jul 4 '17 at 10:03
  • Yes, "a proof of Chern conjecture has recently be claimed", see wikipedia. There have been corrections already, though. There is also a recent paper by B. Klingler in the Annals of Mathematics, see here, under the additional assumption that we have a parallel volume form. – Dietrich Burde Jul 4 '17 at 11:14

Here is a sketch of why you get a $(G,X)$ structure. On any sufficiently small open set you can find a parallel frame for the connection, which by the torsion free hypothesis consists of mutually commuting vector fields. Thus, these can be simultaneously integrated to give you coordinates. On competing overlaps, the frames are related by linear maps, and since the connection is flat, these linear maps are locally constant. Since the transition maps between these frames are linear maps, the transition maps between the coordinates are given by affine maps.

  • Many thanks Andy, for your useful response. – user56980 Aug 12 '14 at 0:08

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