×

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] [A] Arnold, V.: Les méthodes mathématiques de la mécanique classique. Moscow: Edition MIR 1976
[2] [Be] Benzecri, J.P.: Variétés localement affines et projectives. Bull. Soc. Math. France88, 229-332 (1960)
[3] [Bu] Buser, P.: A geometric proof of Bieberbach’s theorems on crystallographic groups. L’Enseignement Math.31, 137-145 (1985) · Zbl 0582.20033
[4] [CG] Conze, J.P., Guivarch, Y.: Remarques sur la distalité dans les espaces vectoriels. C. R. Acad. Sci. Paris278, 1083-1086 (1974) · Zbl 0275.54028
[5] [D’A] D’Ambra, G.: Isometry groups of Lorentz manifolds. Invent. Math.92, 555-565 (1988) · Zbl 0647.53046
[6] [F1] Fried, D.: Closed similarity manifolds. Comment. Math. Helv.55, 576-582 (1980) · Zbl 0455.57005
[7] [F2] Fried, D.: Distality, completeness and affine structures. J. Diff. Geom.24, 265-273 (1986) · Zbl 0608.53026
[8] [F3] Fried, D.: Flat spacetimes. J. Diff. Geom.26, 385-396 (1987) · Zbl 0643.53047
[9] [F4] Fried, D.: Polynomials on affine manifolds. Trans. Am. Math. Soc.274, 709-719 (1982) · Zbl 0526.53040
[10] [FG] Fried, D., Goldman, W.: Three-dimensional affine crystallographic groups. Adv. Math.47, 1-49 (1983) · Zbl 0571.57030
[11] [FGH] Fried, D., Goldman, W., Hirsch, M.: Affine manifolds with nilpotent holonomy. Comment. Math. Helv.56, 487-523 (1981) · Zbl 0516.57014
[12] [G1] Goldman, W.: Two examples of affine manifolds. Pac. J. Math.94, 327-330 (1981) · Zbl 0461.53041
[13] [G2] Goldman, W.: Projective structures with fuchsian holonomy. J. Diff. Geom.25, 297-326 (1987) · Zbl 0595.57012
[14] [GH1] Goldman, W., Hirsch, M.: The radiance obstruction and parallel forms on affine manifolds. Trans. Am. Math. Soc.286, 629-649 (1984) · Zbl 0561.57014
[15] [GH2] Goldman, W., Hirsch, M.: Affine manifolds and orbits of algebraic groups. Trans. Am. Math. Soc.295, 175-190 (1986) · Zbl 0591.57013
[16] [GK] Goldman, W., Kamishima, Y.: The fondamental group of a compact flat Lorentz space form is virtually polycyclic. J. Diff. Geom.19, 233-240 (1984) · Zbl 0546.53039
[17] [GP] Guillemin, V., Pollack, A.: Differential topology. New York: Prentice Hall 1974 · Zbl 0361.57001
[18] [Kob] Kobayashi, S.: Projectively invariant distances for affine and projective structures in Differential Geometry, Vol. 2, pp. 127-152, Banach Center Publications, Polish Scientific Publishers, Warsaw, 1984
[19] [Kos] Koszul, J.L.: Variétés localement plates et convexité. Osaka J. Math.2, 285-290 (1965) · Zbl 0173.50001
[20] [Mar] Markus, L.: Cosmological models in differential geometry, mimeographed notes. University of Minnesota, 1962
[21] [Mil] Milnor, J.: Topology from the Differentiable viewpoint. Univ. Press of Virginia, 1965 · Zbl 0136.20402
[22] [P] Palmeira, C.F.B.: Open manifolds foliated by planes. Ann. Math.107, 109-131 (1978) · Zbl 0382.57010
[23] [Sm] Smillie, J.: Affinely flat manifolds, Doctoral Dissertation, University of Chicago, 1977
[24] [ST] Sullivan, D., Thurston, W.: Manifolds with canonical coordinates: some examples. Enseign. Math.29, 15-25 (1983) · Zbl 0529.53025
[25] [T] Thurston, W.: The geometry and topology of 3-manifolds, chapter 3. Princeton University, 1978
[26] [W] Wolf, J.: Spaces of constant curvature. New York: McGraw-Hill 1967 · Zbl 0162.53304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.