Autour de la conjecture de L. Markus sur les variétés affines. (Around the conjecture of L. Markus on affine manifolds). (French) Zbl 0682.53051

Let M be a compact affine manifold of dimension M with holonomy representation \(h: \pi_ 1(M)\to \Gamma \subset Aff({\mathbb{R}}^ n)\) (and holonomy group \(\Gamma)\). Then the kernel of h defines a covering \(\hat M\to M\) and M is called complete iff \(\hat M\) is affine-diffeomorphic to \({\mathbb{R}}^ n\). The main result of the present paper states that if the linear holomy group, \(L(\Gamma)\subset GL({\mathbb{R}}^ n),\) (the linear part of \(\Gamma)\) satisfies \((i)\quad disc(L(\Gamma))\leq 1\) and \((ii)\quad L(\Gamma)\subset SL({\mathbb{R}}^ n)\) then M is complete. Here disc(L(\(\Gamma)\)) is the discompacity of the group L(\(\Gamma)\) (an integer \(\leq n)\) introduced in the present paper and measuring the non- compacity of L(\(\Gamma)\). This gives a partial answer to the conjecture of Markus that an affine compact unimodular manifold is complete. Note that \(disc(SO(1,n-1))=1\) and \(disc(SL({\mathbb{R}}^ n))=n-1\). Also a series of other interesting corollaries of the main result are given.
Reviewer: A.Juhl


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B05 Linear and affine connections
Full Text: DOI EuDML


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