## Geometry & Topology

### Geodesible contact structures on $3$–manifolds

Patrick Massot

#### Abstract

In this paper, we study and almost completely classify contact structures on closed 3–manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complete classification of contact structures with negative (maximal) twisting number (which includes the transverse ones) on Seifert manifolds whose base is not a sphere, as well as partial results in the spherical case.

#### Article information

Source
Geom. Topol., Volume 12, Number 3 (2008), 1729-1776.

Dates
Revised: 21 May 2008
Accepted: 25 April 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800108

Digital Object Identifier
doi:10.2140/gt.2008.12.1729

Mathematical Reviews number (MathSciNet)
MR2421139

Zentralblatt MATH identifier
1152.57017

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57R17: Symplectic and contact topology

#### Citation

Massot, Patrick. Geodesible contact structures on $3$–manifolds. Geom. Topol. 12 (2008), no. 3, 1729--1776. doi:10.2140/gt.2008.12.1729. https://projecteuclid.org/euclid.gt/1513800108

#### References

• M Aigner, G M Ziegler, Proofs from The Book, third edition, Springer, Berlin (2004) Including illustrations by K H Hofmann
• D Bennequin, Entrelacements et équations de Pfaff, from: “Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)”, Astérisque 107, Soc. Math. France, Paris (1983) 87–161
• Y Carrière, Flots riemanniens, Astérisque (1984) 31–52 Transversal structure of foliations (Toulouse, 1982)
• V Colin, Chirurgies d'indice un et isotopies de sphères dans les variétés de contact tendues, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 659–663
• F Ding, H Geiges, Symplectic fillability of tight contact structures on torus bundles, Algebr. Geom. Topol. 1 (2001) 153–172
• D Eisenbud, U Hirsch, W Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv. 56 (1981) 638–660
• Y Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990) 29–46
• Y Eliashberg, Contact $3$–manifolds twenty years since J Martinet's work, Ann. Inst. Fourier $($Grenoble$)$ 42 (1992) 165–192
• Y M Eliashberg, W P Thurston, Confoliations, University Lecture Series 13, Amer. Math. Soc. (1998)
• J B Etnyre, K Honda, Knots and contact geometry. I. Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001) 63–120
• P Ghiggini, On tight contact structures with negative maximal twisting number on small Seifert manifolds
• P Ghiggini, Strongly fillable contact $3$–manifolds without Stein fillings, Geom. Topol. 9 (2005) 1677–1687
• P Ghiggini, Tight contact structures on Seifert manifolds over $T\sp 2$ with one singular fibre, Algebr. Geom. Topol. 5 (2005) 785–833
• P Ghiggini, Linear Legendrian curves in $T\sp 3$, Math. Proc. Cambridge Philos. Soc. 140 (2006) 451–473
• E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637–677
• E Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000) 615–689
• E Giroux, Structures de contact sur les variétés fibrées en cercles au-dessus d'une surface, Comment. Math. Helv. 76 (2001) 218–262
• E Giroux, Sur les transformations de contact au-dessus des surfaces, from: “Essays on geometry and related topics, Vol. 1, 2”, Monogr. Enseign. Math. 38, Enseignement Math., Geneva (2001) 329–350
• R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. $(2)$ 148 (1998) 619–693
• T Hangan, R Lutz, Champs d'hyperplans totalement géodésiques sur les sphères, from: “Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)”, Astérisque 107, Soc. Math. France, Paris (1983) 189–200
• K Honda, Confoliations transverse to vector fields, preprint available at http://almaak.usc.edu/\char'176khonda/research.html (1998)
• K Honda, On the classification of tight contact structures. I, Geom. Topol. 4 (2000) 309–368
• K Honda, On the classification of tight contact structures. II, J. Differential Geom. 55 (2000) 83–143
• K Honda, Gluing tight contact structures, Duke Math. J. 115 (2002) 435–478
• M Jankins, W Neumann, Homomorphisms of Fuchsian groups to ${\rm PSL}(2,{\rm R})$, Comment. Math. Helv. 60 (1985) 480–495
• M Jankins, W Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann. 271 (1985) 381–400
• J Kim, Tight contact structures of certain Seifert fibered $3$–manifolds with $e\sb 0=-1$, Pacific J. Math. 221 (2005) 109–122
• P Lisca, G Matić, Stein $4$–manifolds with boundary and contact structures, from: “Symplectic, contact and low-dimensional topology (Athens, GA, 1996)”, Topology Appl. 88 (1998) 55–66
• P Lisca, G Matić, Transverse contact structures on Seifert $3$–manifolds, Algebr. Geom. Topol. 4 (2004) 1125–1144
• R Lutz, Structures de contact et systèmes de Pfaff à pivot, from: “Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)”, Astérisque 107, Soc. Math. France, Paris (1983) 175–187
• R Lutz, Quelques remarques sur la géométrie métrique des structures de contact, from: “South Rhone seminar on geometry, I (Lyon, 1983)”, Travaux en Cours, Hermann, Paris (1984) 75–113
• J D McCarthy, J G Wolfson, Symplectic gluing along hypersurfaces and resolution of isolated orbifold singularities, Invent. Math. 119 (1995) 129–154
• D McDuff, D Salamon, Introduction to symplectic topology, second edition, Oxford Mathematical Monographs, The Clarendon Oxford University Press, New York (1998)
• R Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv. 69 (1994) 155–162
• K Niederkrüger, F Pasquotto, Resolution of symplectic cyclic orbifold singularities
• G Pick, Geometrisches zur Zahlenlehre, Sitzungberichte Lotos Naturwissen Zeitschrift 19 (1899) 311–319
• B L Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. $(2)$ 69 (1959) 119–132
• D Rolfsen, Knots and links, AMS Chelsea Publishing (2003)
• A Sato, T Tsuboi, Contact structures on closed manifolds fibered by the circle, Mem. Inst. Sci. Tech. Meiji Univ. 33 (1994) 41–46
• S Schwartzman, Asymptotic cycles, Ann. of Math. $(2)$ 66 (1957) 270–284
• P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401–487
• W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at http://msri.org/publications/books/gt3m/
• H Wu, Tight contact small Seifert spaces with $e_0$ not $0,-1$ or $-2$
• H Wu, Legendrian vertical circles in small Seifert spaces, Commun. Contemp. Math. 8 (2006) 219–246