## 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

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53B05 Linear and affine connections
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### References:

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