The systolic constant of orientable Bieberbach 3-manifolds

Chady El Mir[1]; Jacques Lafontaine[2]

  • [1] Department of Mathematics and computer science, Beirut Arab University, P.O.Box 11 - 50 - 20 Riad El Solh 11072809, Beirut, Lebanon
  • [2] Institut de Mathématiques et Modélisation de Montpellier, CNRS, UMR 5149, Université Montpellier 2, CC 0051, Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 3, page 623-648
  • ISSN: 0240-2963

Abstract

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A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact 3 -dimensional orientable Bieberbach manifolds which are not tori, and prove that it cannot be realized by a flat metric. We also highlight a metric that we construct on one type of such manifolds ( C 2 ) which has interesting geometric properties: it is extremal in its conformal class and the systole is realized by “very many” geodesics.

How to cite

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El Mir, Chady, and Lafontaine, Jacques. "The systolic constant of orientable Bieberbach 3-manifolds." Annales de la faculté des sciences de Toulouse Mathématiques 22.3 (2013): 623-648. <http://eudml.org/doc/275394>.

@article{ElMir2013,
abstract = {A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact $3$-dimensional orientable Bieberbach manifolds which are not tori, and prove that it cannot be realized by a flat metric. We also highlight a metric that we construct on one type of such manifolds ($C_2$) which has interesting geometric properties: it is extremal in its conformal class and the systole is realized by “very many” geodesics.},
affiliation = {Department of Mathematics and computer science, Beirut Arab University, P.O.Box 11 - 50 - 20 Riad El Solh 11072809, Beirut, Lebanon; Institut de Mathématiques et Modélisation de Montpellier, CNRS, UMR 5149, Université Montpellier 2, CC 0051, Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France},
author = {El Mir, Chady, Lafontaine, Jacques},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {aspherical; systolic ratio; flat metric},
language = {eng},
month = {6},
number = {3},
pages = {623-648},
publisher = {Université Paul Sabatier, Toulouse},
title = {The systolic constant of orientable Bieberbach 3-manifolds},
url = {http://eudml.org/doc/275394},
volume = {22},
year = {2013},
}

TY - JOUR
AU - El Mir, Chady
AU - Lafontaine, Jacques
TI - The systolic constant of orientable Bieberbach 3-manifolds
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 3
SP - 623
EP - 648
AB - A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact $3$-dimensional orientable Bieberbach manifolds which are not tori, and prove that it cannot be realized by a flat metric. We also highlight a metric that we construct on one type of such manifolds ($C_2$) which has interesting geometric properties: it is extremal in its conformal class and the systole is realized by “very many” geodesics.
LA - eng
KW - aspherical; systolic ratio; flat metric
UR - http://eudml.org/doc/275394
ER -

References

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  1. Babenko (I.).— Souplesse intersystolique forte des variétés fermées et des polyèdres, Ann. Inst. Fourier 52 no. 4, p. 1259-1284 (2002). Zbl1023.53025MR1927080
  2. Bavard (C.).— Inégalité isosystolique pour la bouteille de Klein, Math. Ann. 274, p. 439-441 (1986) Zbl0578.53032MR842624
  3. Bavard (C.).— Inégalités isosystoliques conformes pour la bouteille de Klein, Geom. Dedicata 27, p. 349-355 (1988). Zbl0667.53033MR960206
  4. Bavard (C.).— Inégalités isosystoliques conformes, Comment. Math. Helv 67, p. 146-166 (1992). Zbl0783.53041MR1144618
  5. Bavard (C.).— Une remarque sur la géométrie systolique de la bouteille de Klein, Arch. Math. (Basel) 87 (2006), No 1, p. 72-74 (1993). Zbl1109.53037MR2246408
  6. Berger (M.).— Systoles et applications selon Gromov, Séminaire N. Bourbaki, exposé 771, Astérisque 216, p. 279-310 (1993). Zbl0789.53040MR1246401
  7. Burago (D.), Burago (Y.D.), Ivanov (S.).— A course in metric geometry, Graduate studies in Mathematics (33), Amer. Math. Soc., Providence, R.I. (2001). Zbl0981.51016MR1835418
  8. Calabi (E.).— Extremal isosystolic metrics for compact surfaces, Actes de la table ronde de géométrie différentielle, Semin. Congr 1, Soc.Math.France p. 146-166 (1996). 3). Zbl0884.58029MR1427758
  9. Charlap (L.S.).— Bieberbach Groups and Flat Manifolds, Springer Universitext, Berlin (1986). Zbl0608.53001MR862114
  10. Cheeger (J.), Ebin (D.).— Comparison Theorems in Riemannian Geometry, North-Holland Publishing Co., Amsterdam (1975). Zbl1142.53003MR458335
  11. Elmir (C.), Lafontaine (J.).— Sur la géométrie systolique des variétés de Bieberbach, Geom. Dedicata. 136, p. 95-110 (2008) Zbl1190.53037MR2443345
  12. Gallot (S.), Hulin (D.), Lafontaine (J.).— Riemannian Geometry, 3rd edition, Springer, Berlin Heidelberg (2004). Zbl0636.53001MR2088027
  13. Gromov (M.).— Filling Riemannian manifolds, J. Diff. Geom. 18, p. 1-147 (1983) Zbl0515.53037MR697984
  14. Jenkins (J.A.).— On the existence of certain general extremal metrics, Ann. of Math. 66, p. 440-453 (1957). Zbl0082.06301MR90648
  15. Katz (M.G.).— Systolic Geometry and Topology, Math. Surveys and Monographs 137, Amer. Math. Soc., Providence, R.I. (2007). Zbl1149.53003MR2292367
  16. Pu (P.M.).— Some inequalities in certain non-orientable riemannian manifolds. Pacific J. Math. 2, p. 55-71 (1952). Zbl0046.39902MR48886
  17. Thurston (W.P.).— Three-Dimensional Geometry and Topology, edited by S. Levy, Princeton University Press, Princeton (1997). Zbl0873.57001MR1435975
  18. Wolf (J.A.).— Spaces of constant curvature, Publish or Perish, Boston (1974). Zbl0281.53034MR343214

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