Background Readings in Contact Geometry

There is no introductory text for contact geometry. Below we have listed some books and papers that might help one begin studying contact geometry. This might be a somewhat biased list but we hope it is helpful.

Basics

• McDuff and Salamon "Introduction to Symplectic Topology"
  Section 3.4 - Short simple well written account of classical stuff.
• B. Aebischer, et al, "Symplectic Geometry: An introduction based on the seminar in Bern, 1992," Birkhauser.
  Chapter 8 - Does more than the above reference. Probably skip sections 8.4.2 and 8.4.3
• E. Giroux, "Topologie de contact en dimension $3$ (autour des travaux de Yakov Eliashberg)" (French) ["Contact topology in dimension $3$ (about the works of Yakov Eliashberg)"] Seminaire Bourbaki, Vol. 1992/93. Asterisque No. 216, (1993), Exp. No. 760, 3, 7--33.
  Very thorough survey of contact geometry circa 1992.

Characteristic Foliations and Contact Convexity

• Y. Eliashberg, "Contact $3$-manifolds twenty years since J. Martinet's work" Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165--192.
  Discusses manipulations of the characteristic foliations of surfaces, a lot of fundamental ideas here. Definitely should read sections 1-3. The other sections culminate in the classification of contact structures on the three ball.

Other papers discussing useful manipulations of the characteristic foliation are:

• Y. Eliashberg, "Legendrian and transversal knots in tight contact $3$-manifolds," Topological methods in modern mathematics (Stony Brook, NY, 1991), 171--193, Publish or Perish, Houston, TX, 1993
• Y. Eliashberg; M. Fraser, "Classification of topologically trivial Legendrian knots'" Geometry, topology, and dynamics (Montreal, PQ, 1995), 17--51, CRM Proc. Lecture Notes, 15, Amer. Math. Soc., Providence, RI, 1998.
• J. Etnyre, "Transversal torus knots," Geometry & Topology 3 (1999), 253-268.
  

A powerful method for manipulating characteristic foliations is developed in:

• E. Giroux, "Convexity en topologie de contact" (French) ["Convexity in contact topology"] Comment. Math. Helv. 66 (1991), no. 4, 637--677.
  First paper that really discusses convex surfaces. A great discussion of all the basic theorems.

Symplectic Convexity

See the following paper and the references therein:

• J. Etnyre "Symplectic convexity in low-dimensional topology" Symplectic, contact and low-dimensional topology (Athens, GA, 1996). Topology Appl. 88 (1998), no. 1-2, 3--25
  Symplectic convexity is very useful in constructing tight contact manifold. It also provides a bridge between contact and symplectic geometry.
  

For the construction of many contact manifolds using convexity see:

• R. Gompf, "Handlebody construction of Stein surfaces," Ann. of Math. (2) 148 (1998), no. 2, 619--693.
• A. Weinstein, "Contact surgery and symplectic handlebodies," Hokkaido Math. J. 20 (1991), no. 2, 241--251.

Other Constructions

Many constructions of contact structures may be found in:

• H. Geiges, "Constructions of contact manifolds," Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 455--464.

Some of these constructions can be shown to produce tight contact structures (sometimes):

• J. Etnyre and R. Ghrist, "Tight contact 3-manifolds and dynamics," Proceedings of the AMS 127 (1999), 3697-3706.

Holomorphic Curves in Contact Geometry

There are many useful books on holomorphic curves in symplectic geometry. These would be useful too look at, particularly the proofs of Gromov compactness. See for example:

• Aebischer op. cit. chapters 6 and 7.
• D. McDuff and D. Salamon, "J-holomorphic Curves and Quantum Cohomology," University Lecture Series 6, AMS.
• C. Hummel, "Gromov's compactness theorem for pseudoholomorphic curves" Birkhauser.
  

For a more direct application to contact structures see:

• H. Hofer, "Holomorphic curves and dynamics in dimension three," Park City/IAS institute lecture notes, AMS
• H. Hofer and M Kriener, "Holomorphic curves in contact dynamics," to appear in the proceedings of a conference on the occasion of the 70th birthday of P. Lax and L. Nirenberg

For contact homology see:

• Y. Eliashberg, "Invariants in contact topology," Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 327--338 (electronic)
• Y. Chekanov, "Differential algebra of Legendrian links," preprint, 1999.

Foliations and Contact Geometry

There is a very promising connection between foliation theory and contact geometry that has only begun to be pursued. See

• Y. Eliashberg; WIlliam P. Thurston, "Confoliations'" University Lecture Series, 13, AMS

Contact Geometry and Seiberg-Witten Theory

• P. Kronheimer; T. Mrowka, "Monopoles and contact structures," Invent. Math. 130 (1997), no. 2, 209--255.
• P. Lisca, "Symplectic fillings and positive scalar curvature," Geom. Topol. 2 (1998), 103--116.