Editing Contact geometry (section)

===Relation with symplectic structures===
A consequence of the definition is that the restriction of the 2-form ''ω''&nbsp;=&nbsp;''dα'' to a hyperplane in ''ξ'' is a nondegenerate 2-form. This construction provides any contact manifold ''M'' with a natural [[symplectic bundle]] of rank one smaller than the dimension of ''M''. Note that a symplectic vector space is always even-dimensional, while contact manifolds need to be odd-dimensional.

The [[cotangent bundle]] ''T''*''N'' of any ''n''-dimensional manifold  ''N'' is itself a manifold (of dimension 2''n'') and supports naturally an exact symplectic structure ω = ''dλ''. (This 1-form ''λ'' is sometimes called the [[Liouville form]]). There are several ways to construct an associated contact manifold, some of dimension 2''n''&nbsp;−&nbsp;1, some of dimension 2''n''&nbsp;+&nbsp;1.

;Projectivization
Let ''M'' be the [[projective space|projectivization]] of the cotangent bundle of ''N'': thus ''M'' is fiber bundle over ''N'' whose fiber at a point ''x'' is the space of lines in T*''N'', or, equivalently, the space of hyperplanes in T''N''. The 1-form ''λ'' does not descend to a genuine 1-form on ''M''. However, it is homogeneous of degree 1, and so it defines a 1-form with values in the line bundle O(1), which is the dual of the fibrewise tautological line bundle of ''M''. The kernel of this 1-form defines a contact distribution.

;Energy surfaces
Suppose that ''H'' is a smooth function on T*''N'',   that ''E'' is a regular value for ''H'', so that the level set <math>L=\{(q,p)\in T^*N\mid H(q,p)=E\}</math> is a smooth submanifold of codimension 1.  A vector field ''Y'' is called an Euler (or Liouville) vector field if it is transverse to ''L'' and conformally symplectic,  meaning that the Lie derivative of ''dλ'' with respect to ''Y'' is a multiple of ''dλ'' in a neighborhood of ''L''.

Then the restriction of <math>i_Y \,d\lambda</math> to ''L'' is a contact form on ''L''.

This construction originates in [[Hamiltonian mechanics]], where ''H'' is a Hamiltonian of a mechanical system with the configuration space ''N'' and the phase space ''T''*''N'', and ''E'' is the value of the energy.

;The unit cotangent bundle
Choose a [[Riemannian metric]] on the manifold ''N'' and let ''H'' be the associated kinetic energy.
Then the level set   ''H'' = 1/2  is the  ''unit cotangent bundle'' of ''N'',  a smooth manifold of dimension 2''n''&nbsp;−&nbsp;1 fibering over ''N'' with fibers being spheres. Then the Liouville form restricted to the unit cotangent bundle is a contact structure.  This corresponds to a special case of the second construction, where the flow of the Euler vector field ''Y'' corresponds to linear scaling of momenta p''s'', leaving the ''q''s fixed. The [[vector field]] ''R'', defined by the equalities
: ''λ''(''R'') = 1 and ''dλ''(''R'',&nbsp;''A'') = 0 for all vector fields ''A'',
is called the '''[[#Reeb vector field|Reeb vector field]]''', and it generates the [[geodesic flow]] of the Riemannian metric. More precisely, using the Riemannian metric, one can identify each point of the cotangent bundle of ''N'' with a point of the tangent bundle of ''N'', and then the value of ''R'' at that point of the (unit) cotangent bundle is the corresponding (unit) vector parallel to ''N''.

;First jet bundle
On the other hand, one can build a contact manifold ''M'' of dimension 2''n''&nbsp;+&nbsp;1 by considering the first [[jet bundle]] of the real valued functions on ''N''. This bundle is isomorphic to ''T''*''N''×'''R''' using the [[exterior derivative]] of a function. With coordinates (''x'',&nbsp;''t''), ''M'' has a contact structure
:α = ''dt'' + ''λ''.

Conversely, given any contact manifold ''M'', the product ''M''×'''R''' has a natural structure of a symplectic manifold. If α is a contact form on ''M'', then

:''ω'' = ''d''(''e''<sup>''t''</sup>α)

is a symplectic form on ''M''×'''R''', where ''t'' denotes the variable in the '''R'''-direction.  This new manifold is called the [[symplectization]] (sometimes [[symplectification]] in the literature) of the contact manifold ''M''.