Summary
For any subgroupG of (ℝn), we introduce some integer discG≦n called thediscompacity ofG. This number measures to what extent the closure ofG is not compact. The Markus' conjecture says that a compact affinely flat unimodular manifold is complete. Our main result (called the ≪discompact theorem≫) is that this conjecture is true under the assumption that the linear holonomy i.e. the parallel transport has discompacity ≦1. Because discSO(n−1, 1)=1, this ensures that a compact flat Lorentz manifoldM is geodesically complete. Hence, by a previous result of W. Goldman and Y. Kamishima [GK], such aM is, up to finite covering, a solvmanifold. This achieves the proof of a Bieberbach's theorem for compact Lorentz flat manifolds.
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Références
[A] Arnold, V.: Les méthodes mathématiques de la mécanique classique. Moscow: Edition MIR 1976
[Be] Benzecri, J.P.: Variétés localement affines et projectives. Bull. Soc. Math. France88, 229–332 (1960)
[Bu] Buser, P.: A geometric proof of Bieberbach's theorems on crystallographic groups. L'Enseignement Math.31, 137–145 (1985)
[CG] Conze, J.P., Guivarch, Y.: Remarques sur la distalité dans les espaces vectoriels. C. R. Acad. Sci. Paris278, 1083–1086 (1974)
[D'A] D'Ambra, G.: Isometry groups of Lorentz manifolds. Invent. Math.92, 555–565 (1988)
[F1] Fried, D.: Closed similarity manifolds. Comment. Math. Helv.55, 576–582 (1980)
[F2] Fried, D.: Distality, completeness and affine structures. J. Diff. Geom.24, 265–273 (1986)
[F3] Fried, D.: Flat spacetimes. J. Diff. Geom.26, 385–396 (1987)
[F4] Fried, D.: Polynomials on affine manifolds. Trans. Am. Math. Soc.274, 709–719 (1982)
[FG] Fried, D., Goldman, W.: Three-dimensional affine crystallographic groups. Adv. Math.47, 1–49 (1983)
[FGH] Fried, D., Goldman, W., Hirsch, M.: Affine manifolds with nilpotent holonomy. Comment. Math. Helv.56, 487–523 (1981)
[G1] Goldman, W.: Two examples of affine manifolds. Pac. J. Math.94, 327–330 (1981)
[G2] Goldman, W.: Projective structures with fuchsian holonomy. J. Diff. Geom.25, 297–326 (1987)
[GH1] Goldman, W., Hirsch, M.: The radiance obstruction and parallel forms on affine manifolds. Trans. Am. Math. Soc.286, 629–649 (1984)
[GH2] Goldman, W., Hirsch, M.: Affine manifolds and orbits of algebraic groups. Trans. Am. Math. Soc.295, 175–190 (1986)
[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)
[GP] Guillemin, V., Pollack, A.: Differential topology. New York: Prentice Hall 1974
[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
[Kos] Koszul, J.L.: Variétés localement plates et convexité. Osaka J. Math.2, 285–290 (1965)
[Mar] Markus, L.: Cosmological models in differential geometry, mimeographed notes. University of Minnesota, 1962
[Mil] Milnor, J.: Topology from the Differentiable viewpoint. Univ. Press of Virginia, 1965
[P] Palmeira, C.F.B.: Open manifolds foliated by planes. Ann. Math.107, 109–131 (1978)
[Sm] Smillie, J.: Affinely flat manifolds, Doctoral Dissertation, University of Chicago, 1977
[ST] Sullivan, D., Thurston, W.: Manifolds with canonical coordinates: some examples. Enseign. Math.29, 15–25 (1983)
[T] Thurston, W.: The geometry and topology of 3-manifolds, chapter 3. Princeton University, 1978
[W] Wolf, J.: Spaces of constant curvature. New York: McGraw-Hill 1967
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Carrière, Y. Autour de la conjecture de L. Markus sur les variétés affines. Invent Math 95, 615–628 (1989). https://doi.org/10.1007/BF01393894
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DOI: https://doi.org/10.1007/BF01393894