42 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICSObserve that being tight or overtwisted is a property of 6. rather than ,.
It is a non trivial result of D. Bennequin that the contact form >-o = dz+ xdy
on JR^3 is tight, see [4]
Theorem 1.17 (Bennequin). The standard contact form dz+ xdy on JR^3 is tight.Exercise 1.18. Find a precise formula for an overtwisted contact form on JR^3.
Show that the Reeb vector field associated to the structure you defined has a peri-
odic orbit.
We need the following definition.
Definition 1.19. A Reeb-like vector field X on M is said to be overtwisted if
ixD. = d>.
ix>.> 0
for some volume form D. and some one-form,\ which is an overtwisted contact form.If similarly ,\ is a tight contact form we say that X is tight.
Figure 2. The characteristic foliation on an overtwisted disk is the foliation
associated to the one-dimensional part of En TV. Generically there are only
a finite number of points with a tangency between E and V. Our example is
in fact the best possible.I.. ··
' .. ··.. ····
.. ····/~( ~3:,;r~:~~ ,..
.. ·········· i
.·Figure 3. This shows an overtwisted contact structure on JR^3