LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 41Assume that Xis Reeb-like. Then, by definition, ixD = d>.. and ix>..> 0 for a
suitable choice of volume form D and one-form>... Observe thatixd>.. = ixixD = 0.
Hence, in view of ix>.. > 0 we see that >..Ad>.. is a volume form. Moreover X = f X 0 ,
where Xo is the vector field defined byix 0 d>.. = 0 and ix 0 >.. = 1.
Clearly Xo = y · X where f =ix>...
A one-form>.. such that >..Ad>.. is a volume form is called a contact form. Thevector field X 0 is called the associated Reeb vector field. By defining ~ = kern(>..)
we obtain a plane-field distribution. called the associated contact structure. Notethat d>..I(~ EB 0 defines a symplectic product on every fibre. Hence ((, d>..) is a
symplectic vector bundle over M. Denote, if there is an ambiguity, by X>. the Reeb
vector field associated to>... Moreover write L>. for the line bundle over M spannedby X>. and 6 for the associated contact structure.
Hence, given a contact form >.. on a closed three-manifold M, we obtain a
canonical decomposition of the tangent space of M:into a line bundle with a preferred section and a symplectic vector bundle.
The standard example on ffi.^3 is >.. 0 = dz + xdy. Then
X>, 0 (x,y,z) = (0,0,1)
60 (x, y, z) = span{(l, 0, 0), (0, 1, -x)}
d>.. 0 (x, y, z)((l, 0, 0), (0, 1, -x)) = 1.
That this is the most general local example follows from the following exercise.Exercise.1.15. Given a contact form Ton Mand m E M there exists for every
point a E ffi.^3 a local diffeomorphism <p: U(a) ----t V(m) such that
<p*T =(dz+ xdy)IU(a).
We need the following definition due to Eliashberg.Definition 1.16. A contact form >.. or its associated contact structure ~ is said to
be overtwisted if there exists an embedded disk D such that
T(8D) c ~l8D
T zD </.. ~z for all z E 8D.A contact form >.. is said to be tight if it is not overtwisted.
~·
Figure 1. The standard contact structure on JR^3.