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Closed ray affine manifolds

Abstract : We consider closed manifolds that possess a so called rank one ray structure. That is a (flat) affine structure such that the linear part is given by the products of a diagonal transformation and a commuting rotation. We show that closed manifolds with a rank one ray structure are either complete or their developing map is a cover onto the complement of an affine subspace. This result extends the geometric picture given by Fried on closed similarity manifolds. We prove, in the line of Markus conjecture, that if the rank one ray geometry has parallel volume, then closed manifolds are necessarily complete. Finally, we show that the automorphism group of a closed manifold acts non properly when the manifold is complete.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03358563
Contributor : Raphaël Alexandre Connect in order to contact the contributor
Submitted on : Wednesday, September 29, 2021 - 2:29:28 PM
Last modification on : Thursday, September 30, 2021 - 11:56:53 AM

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  • HAL Id : hal-03358563, version 1
  • ARXIV : 2109.14417

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Raphaël Alexandre. Closed ray affine manifolds. 2021. ⟨hal-03358563⟩

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