Chern’s conjecture for special affine manifolds

  title={Chern’s conjecture for special affine manifolds},
  author={Bruno Klingler},
  journal={Annals of Mathematics},
  • B. Klingler
  • Published 1 July 2017
  • Mathematics
  • Annals of Mathematics
An affine manifold X in the sense of differential geometry is a differentiable manifold admitting an atlas of charts with value in an affine space, with locally constant affine change of coordinates. Equivalently, it is a manifold whose tangent bundle admits a flat torsion free connection. Around 1955 Chern conjectured that the Euler characteristic of any compact affine manifold has to vanish. I will explain a proof of this conjecture in the case where X moreover admits a parallel volume form. 
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