Abstract
A similarity structure on a connected manifold M is a Riemannian metric on its universal cover \({\tilde{M}}\) such that the fundamental group of M acts on \({\tilde{M}}\) by similarities. If the manifold M is compact, we show that the universal cover admits a de Rham decomposition with at most two factors, one of which is Euclidean. Very recently, after Belgun and Moroianu conjectured that the number of factors was at most one, Matveev and Nikolayevsky found an example with two factors. When the non-flat factor has dimension 2, we give a complete classification of the examples with two factors. In greater dimensions, we make the first steps towards such a classification by showing that M is a fibration (with singularities) by flat Riemannian manifolds. Up to a finite covering of M, we may assume that these manifolds are flat tori. We also prove a version of the de Rham decomposition theorem for the universal covers of manifolds endowed with locally metric connections. During the proof, we define a notion of transverse (not necessarily flat) similarity structure on foliations, and show that foliations endowed with such a structure are either transversally flat or transversally Riemannian. None of these results assumes analyticity.
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References
Alperin, R.C.: An elementary account of Selberg’s lemma. Enseign. Math. (2) 33(3–4), 269–273 (1987)
Asuke, T.: Classification of Riemannian flows with transverse similarity structures. Ann. Fac. Sci. Toulouse Math. (6) 6(2), 203–227 (1997)
Auslander, L.: Bieberbach’s theorem on space groups and discrete uniform subgroups of Lie groups. II. Am. J. Math. 83, 276–280 (1961)
Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume. Math. Ann. 70(3), 297–336 (1911)
Belgun, F., Moroianu, A.: On the irreducibility of locally metric connections. J. Reine Angew. Math. 714, 123–150 (2016)
Carrière, Y.: Flots riemanniens. Astérisque, (116), 31–52 (1984). Transversal structure of foliations (Toulouse, 1982)
de Rham, G.: Sur la reductibilité d’un espace de Riemann. Comment. Math. Helv. 26, 328–344 (1952)
Fried, D.: Closed similarity manifolds. Comment. Math. Helv. 55(4), 576–582 (1980)
Frances, C., Tarquini, C.: Autour du théorème de Ferrand–Obata. Ann. Glob. Anal. Geom. 21(1), 51–62 (2002)
Gallot, S.: Équations différentielles caractéristiques de la sphère. Ann. Sci. École Norm. Sup. (4) 12(2), 235–267 (1979)
Ghys, É.: Flots transversalement affines et tissus feuilletés. Mém. Soc. Math. Fr. (N.S.) (46), 123–150 (1991). Analyse globale et physique mathématique (Lyon, 1989)
Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. I. Interscience Publishers, A division of Wiley, New York (1963)
Kobayashi, T.: Proper action on a homogeneous space of reductive type. Math. Ann. 285(2), 249–263 (1989)
Madani, F., Moroianu, A., Pilca, M.: On Weyl-reducible locally conformally Kähler structures. arXiv:1705.10397
Matveev, V.S., Nikolayevsky, Y.: A counterexample to Belgun–Moroianu conjecture. C. R. Math. Acad. Sci. Paris 353(5), 455–457 (2015)
Matveev, V.S., Nikolayevsky, Y: Locally conformally berwald manifolds and compact quotients of reducible manifolds by homotheties (2015). arXiv:1506.08935
Molino, P.: Riemannian Foliations. Progress in Mathematics, vol. 73. Birkhäuser Boston Inc, Boston (1988)
Nishimori, T.: A note on the classification of nonsingular flows with transverse similarity structures. Hokkaido Math. J. 21(3), 381–393 (1992)
Ponge, R., Reckziegel, H.: Twisted products in pseudo-Riemannian geometry. Geom. Dedic 48(1), 15–25 (1993)
Raghunathan, M.S.: Discrete subgroups of Lie groups. Springer, New York (1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68
Starkov, A.N.: Ergodic decomposition of flows on homogeneous spaces of finite volume. Mat. Sb., 180(12), 1614–1633, 1727 (1989)
Tarquini, C.: Feuilletages conformes. Ann. Inst. Fourier (Grenoble) 54(2), 453–480 (2004)
Tarquini, C.: Feuilletages de type fini compact. C. R. Math. Acad. Sci. Paris 339(3), 209–214 (2004)
Acknowledgements
This paper contains results which were obtained during my PhD thesis: I would like to thank my advisor Abdelghani Zeghib for his help. This work was also supported by the ERC Avanced Grant 320939, Geometry and Topology of Open Manifolds (GETOM). I would like to thank the referees for helping me improve the paper.
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Communicated by F. C. Marques.
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Kourganoff, M. Similarity structures and de Rham decomposition. Math. Ann. 373, 1075–1101 (2019). https://doi.org/10.1007/s00208-018-1748-y
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DOI: https://doi.org/10.1007/s00208-018-1748-y