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.

- 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.

- 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.

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.

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.

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.

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

- 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.

Last modified: Tue May 23 09:47:53 PDT 2000