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A contact form on a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k1100102.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k1100103.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k1100104.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k1100105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k1100106.png" /> is everywhere non-zero. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k1100107.png" /> is called a contact manifold. See also [[Contact structure|Contact structure]].
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A contact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k1100108.png" /> carries a distinguished [[Vector field|vector field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k1100109.png" />, called the characteristic vector field or Reeb field, which is uniquely determined by the following equations: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001011.png" /> for all vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001012.png" />. The flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001013.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001014.png" /> (when it is complete) is called the contact flow. Sometimes the name "contact flow" is used for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001015.png" />-dimensional foliation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001016.png" /> consisting of the unparametrized orbits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001017.png" />, [[#References|[a5]]].
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If the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001018.png" /> is a Riemannian foliation in the sense of Reinhart–Molino [[#References|[a7]]], i.e., if there is a holonomy-invariant transverse metric for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001020.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001021.png" />-contact flow, and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001022.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001024.png" />-contact manifold. This definition is equivalent to requiring that the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001026.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001027.png" />-parameter group of isometries for some contact metric (a [[Riemannian metric|Riemannian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001028.png" /> such that there exists an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001029.png" /> of the [[Tangent bundle|tangent bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001033.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001034.png" /> for all vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001037.png" />). If one has in addition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001039.png" /> is the [[Levi-Civita connection|Levi-Civita connection]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001040.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001041.png" /> is a Sasakian manifold, [[#References|[a4]]], [[#References|[a12]]].
+
A contact form on a smooth  $  ( 2n + 1 ) $-
 +
dimensional manifold $  M $
 +
is a $  1 $-
 +
form  $  \alpha $
 +
such that $  \alpha \wedge ( d \alpha ) ^ {n} $
 +
is everywhere non-zero. The pair  $  ( M, \alpha ) $
 +
is called a contact manifold. See also [[Contact structure|Contact structure]].
  
As a consequence of the Meyer–Steenrod theorem [[#References|[a6]]], a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001042.png" />-contact flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001043.png" /> on a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001044.png" />-dimensional manifold is almost periodic: the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001046.png" /> in the isometry group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001047.png" /> (of the associated contact metric) is a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001048.png" />, of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001049.png" /> in between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001051.png" />, which acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001052.png" /> while preserving the contact form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001053.png" />, [[#References|[a3]]]. The "completely integrable" case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001054.png" /> has been studied in [[#References|[a2]]]: these structures are determined by the image of their contact moment mapping.
+
A contact manifold  $  ( M, \alpha ) $
 +
carries a distinguished [[Vector field|vector field]] $  Z $,
 +
called the characteristic vector field or Reeb field, which is uniquely determined by the following equations: $  \alpha ( Z ) = 1 $
 +
and $  d \alpha ( Z,X ) = 0 $
 +
for all vector fields  $  X $.  
 +
The flow  $  \phi _ {t} $
 +
generated by  $  Z $(
 +
when it is complete) is called the contact flow. Sometimes the name "contact flow" is used for the  $  1 $-
 +
dimensional foliation  $  {\mathcal F} $
 +
consisting of the unparametrized orbits of  $  Z $,  
 +
[[#References|[a5]]].
  
The existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001055.png" />-contact flows poses restrictions on the topology of the manifold. For instance, since a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001056.png" />-contact flow can be approximated by a periodic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001057.png" />-contact flow, only Seifert fibred compact manifolds can carry a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001058.png" />-contact flow. Another example of a restriction is the Tachibana theorem, asserting that the first [[Betti number|Betti number]] of a compact Sasakian manifold is either zero or even, [[#References|[a9]]]. This shows that no torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001059.png" /> can carry a Sasakian structure. Actually, P. Rukimbira [[#References|[a8]]] showed that no torus can carry a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001060.png" />-contact flow.
+
If the flow  $  {\mathcal F} $
 +
is a Riemannian foliation in the sense of Reinhart–Molino [[#References|[a7]]], i.e., if there is a holonomy-invariant transverse metric for  $  {\mathcal F} $,
 +
then  $  {\mathcal F} $
 +
is called a  $  K $-
 +
contact flow, and the pair  $  ( M, \alpha ) $
 +
is called a  $  K $-
 +
contact manifold. This definition is equivalent to requiring that the flow  $  \phi _ {t} $
 +
of  $  Z $
 +
is a $  1 $-
 +
parameter group of isometries for some contact metric (a [[Riemannian metric|Riemannian metric]]  $  g $
 +
such that there exists an endomorphism  $  J $
 +
of the [[Tangent bundle|tangent bundle]]  $  TM $
 +
such that  $  JZ = 0 $,
 +
$  J  ^ {2} X = - X + \alpha ( X ) Z $,
 +
$  d \alpha ( X,Y ) = g ( X,JY ) $,
 +
and  $  g ( X,Y ) = g ( JX,JY ) + \alpha ( X ) \alpha ( Y ) $
 +
for all vector fields  $  X $
 +
and  $  Y $
 +
on  $  M $).
 +
If one has in addition  $  ( \nabla _ {X} J ) Y = g ( X,Y ) Z - \alpha ( Y ) X $,
 +
where  $  \nabla $
 +
is the [[Levi-Civita connection|Levi-Civita connection]] of $  g $,
 +
then one says that  $  ( M, \alpha ) $
 +
is a Sasakian manifold, [[#References|[a4]]], [[#References|[a12]]].
  
A. Weinstein [[#References|[a11]]] has conjectured that the contact flow of a compact contact manifold has at least one periodic orbit. Despite important breakthroughs (including [[#References|[a10]]]), this conjecture is not quite settled at present (1996). However, it is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001061.png" />-contact flows on compact manifolds have at least two periodic orbits [[#References|[a3]]].
+
As a consequence of the Meyer–Steenrod theorem [[#References|[a6]]], a  $  K $-
 +
contact flow $  \phi _ {t} $
 +
on a compact $  ( 2n + 1 ) $-
 +
dimensional manifold is almost periodic: the closure of  $  \phi _ {t} $
 +
in the isometry group of  $  M $(
 +
of the associated contact metric) is a torus  $  T  ^ {k} $,
 +
of dimension  $  k $
 +
in between  $  1 $
 +
and  $  n + 1 $,
 +
which acts on  $  M $
 +
while preserving the contact form  $  \alpha $,
 +
[[#References|[a3]]]. The "completely integrable" case  $  k = n + 1 $
 +
has been studied in [[#References|[a2]]]: these structures are determined by the image of their contact moment mapping.
  
Examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001062.png" />-contact manifolds include the contact manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001063.png" /> with a periodic contact flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001064.png" /> (these include the regular contact manifolds), such as the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001065.png" /> equipped with the contact form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001066.png" /> that is the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001067.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001068.png" />-form
+
The existence of $  K $-
 +
contact flows poses restrictions on the topology of the manifold. For instance, since a  $  K $-
 +
contact flow can be approximated by a periodic  $  K $-
 +
contact flow, only Seifert fibred compact manifolds can carry a $  K $-
 +
contact flow. Another example of a restriction is the Tachibana theorem, asserting that the first [[Betti number|Betti number]] of a compact Sasakian manifold is either zero or even, [[#References|[a9]]]. This shows that no torus  $  T ^ {2n + 1 } $
 +
can carry a Sasakian structure. Actually, P. Rukimbira [[#References|[a8]]] showed that no torus can carry a  $  K $-
 +
contact flow.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001069.png" /></td> </tr></table>
+
A. Weinstein [[#References|[a11]]] has conjectured that the contact flow of a compact contact manifold has at least one periodic orbit. Despite important breakthroughs (including [[#References|[a10]]]), this conjecture is not quite settled at present (1996). However, it is known that  $  K $-
 +
contact flows on compact manifolds have at least two periodic orbits [[#References|[a3]]].
  
on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001070.png" />. More generally, compact contact hypersurfaces (in the sense of M. Okumura) [[#References|[a1]]] in Kähler manifolds of constant positive holomorphic sectional curvature carry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001071.png" />-contact flows. A large set of examples is provided by the Brieskorn manifolds: In [[#References|[a12]]] it is shown that every Brieskorn manifold admits many Sasakian structures, hence carries many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001072.png" />-contact flows.
+
Examples of  $  K $-
 +
contact manifolds include the contact manifolds  $  ( M, \alpha ) $
 +
with a periodic contact flow  $  \phi _ {t} $(
 +
these include the regular contact manifolds), such as the sphere  $  S ^ {2n + 1 } $
 +
equipped with the contact form  $  \alpha $
 +
that is the restriction to  $  S ^ {2n + 1 } $
 +
of the  $  1 $-
 +
form
 +
 
 +
$$
 +
\sum _ {i = 1 } ^ { {n }  + 1 } x _ {i}  dy _ {i} - y _ {i}  dx _ {i}  $$
 +
 
 +
on $  \mathbf R ^ {2n + 2 } $.  
 +
More generally, compact contact hypersurfaces (in the sense of M. Okumura) [[#References|[a1]]] in Kähler manifolds of constant positive holomorphic sectional curvature carry $  K $-
 +
contact flows. A large set of examples is provided by the Brieskorn manifolds: In [[#References|[a12]]] it is shown that every Brieskorn manifold admits many Sasakian structures, hence carries many $  K $-
 +
contact flows.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Banyaga, "On characteristics of hypersurfaces in symplectic manifolds" , ''Proc. Symp. Pure Math.'' , '''54''' , Amer. Math. Soc. (1993) pp. 9–17 {{MR|1216525}} {{ZBL|0792.58015}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Banyaga, P. Molino, "Complete integrability in contact geometry" , ''Memoirs'' , Amer. Math. Soc. (submitted)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Banyaga, P. Rukimbira, "On characteristics of circle invariant presymplectic forms" ''Proc. Amer. Math. Soc.'' , '''123''' (1995) pp. 3901–3906 {{MR|1307491}} {{ZBL|0849.58025}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D.E. Blair, "Contact manifolds in Riemannian geometry" , ''Lecture Notes in Mathematics'' , '''509''' , Springer (1976) {{MR|0467588}} {{ZBL|0319.53026}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> Y. Carrière, "Flots riemanniens" ''Astérisque'' , '''116''' (1982) pp. 31–52 {{MR|1046241}} {{MR|0755161}} {{MR|0744829}} {{ZBL|0996.37500}} {{ZBL|0548.58033}} {{ZBL|0524.57018}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S.B. Meyer, N.E. Steenrod, "The group of isometries of a Riemannian manifold" ''Ann. of Math.'' , '''40''' (1939) pp. 400–416 {{MR|1503467}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P. Molino, "Riemannian foliations" , ''Progress in Math.'' , Birkhäuser (1984) {{MR|0761580}} {{MR|0755169}} {{ZBL|0576.57022}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> P. Rukimbira, "Some remarks on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001073.png" />-contact flows" ''Ann. Global Anal. and Geom.'' , '''11''' (1993) pp. 165–171 {{MR|1225436}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> S. Tachibana, "On harmonic tensors in compact sasakian spaces" ''Tohoku Math. J.'' , '''17''' (1965) pp. 271–284 {{MR|0190878}} {{ZBL|0132.16203}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> C. Viterbo, "A proof of the Weinstein conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001074.png" />" ''Ann. Inst. H. Poincaré. Anal. Non-Lin.'' , '''4''' (1987) pp. 337–356 {{MR|917741}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Weinstein, "On the hypothesis of Rabinowicz' periodic orbit theorem" ''J. Diff. Geom.'' , '''33''' (1978) pp. 353–358</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> K. Yano, M. Kon, "Structures on manifolds" , World Sci. (1984) {{MR|0794310}} {{ZBL|0557.53001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Banyaga, "On characteristics of hypersurfaces in symplectic manifolds" , ''Proc. Symp. Pure Math.'' , '''54''' , Amer. Math. Soc. (1993) pp. 9–17 {{MR|1216525}} {{ZBL|0792.58015}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Banyaga, P. Molino, "Complete integrability in contact geometry" , ''Memoirs'' , Amer. Math. Soc. (submitted)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Banyaga, P. Rukimbira, "On characteristics of circle invariant presymplectic forms" ''Proc. Amer. Math. Soc.'' , '''123''' (1995) pp. 3901–3906 {{MR|1307491}} {{ZBL|0849.58025}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D.E. Blair, "Contact manifolds in Riemannian geometry" , ''Lecture Notes in Mathematics'' , '''509''' , Springer (1976) {{MR|0467588}} {{ZBL|0319.53026}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> Y. Carrière, "Flots riemanniens" ''Astérisque'' , '''116''' (1982) pp. 31–52 {{MR|1046241}} {{MR|0755161}} {{MR|0744829}} {{ZBL|0996.37500}} {{ZBL|0548.58033}} {{ZBL|0524.57018}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S.B. Meyer, N.E. Steenrod, "The group of isometries of a Riemannian manifold" ''Ann. of Math.'' , '''40''' (1939) pp. 400–416 {{MR|1503467}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P. Molino, "Riemannian foliations" , ''Progress in Math.'' , Birkhäuser (1984) {{MR|0761580}} {{MR|0755169}} {{ZBL|0576.57022}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> P. Rukimbira, "Some remarks on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001073.png" />-contact flows" ''Ann. Global Anal. and Geom.'' , '''11''' (1993) pp. 165–171 {{MR|1225436}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> S. Tachibana, "On harmonic tensors in compact sasakian spaces" ''Tohoku Math. J.'' , '''17''' (1965) pp. 271–284 {{MR|0190878}} {{ZBL|0132.16203}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> C. Viterbo, "A proof of the Weinstein conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110010/k11001074.png" />" ''Ann. Inst. H. Poincaré. Anal. Non-Lin.'' , '''4''' (1987) pp. 337–356 {{MR|917741}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Weinstein, "On the hypothesis of Rabinowicz' periodic orbit theorem" ''J. Diff. Geom.'' , '''33''' (1978) pp. 353–358</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> K. Yano, M. Kon, "Structures on manifolds" , World Sci. (1984) {{MR|0794310}} {{ZBL|0557.53001}} </TD></TR></table>

Revision as of 22:14, 5 June 2020


A contact form on a smooth $ ( 2n + 1 ) $- dimensional manifold $ M $ is a $ 1 $- form $ \alpha $ such that $ \alpha \wedge ( d \alpha ) ^ {n} $ is everywhere non-zero. The pair $ ( M, \alpha ) $ is called a contact manifold. See also Contact structure.

A contact manifold $ ( M, \alpha ) $ carries a distinguished vector field $ Z $, called the characteristic vector field or Reeb field, which is uniquely determined by the following equations: $ \alpha ( Z ) = 1 $ and $ d \alpha ( Z,X ) = 0 $ for all vector fields $ X $. The flow $ \phi _ {t} $ generated by $ Z $( when it is complete) is called the contact flow. Sometimes the name "contact flow" is used for the $ 1 $- dimensional foliation $ {\mathcal F} $ consisting of the unparametrized orbits of $ Z $, [a5].

If the flow $ {\mathcal F} $ is a Riemannian foliation in the sense of Reinhart–Molino [a7], i.e., if there is a holonomy-invariant transverse metric for $ {\mathcal F} $, then $ {\mathcal F} $ is called a $ K $- contact flow, and the pair $ ( M, \alpha ) $ is called a $ K $- contact manifold. This definition is equivalent to requiring that the flow $ \phi _ {t} $ of $ Z $ is a $ 1 $- parameter group of isometries for some contact metric (a Riemannian metric $ g $ such that there exists an endomorphism $ J $ of the tangent bundle $ TM $ such that $ JZ = 0 $, $ J ^ {2} X = - X + \alpha ( X ) Z $, $ d \alpha ( X,Y ) = g ( X,JY ) $, and $ g ( X,Y ) = g ( JX,JY ) + \alpha ( X ) \alpha ( Y ) $ for all vector fields $ X $ and $ Y $ on $ M $). If one has in addition $ ( \nabla _ {X} J ) Y = g ( X,Y ) Z - \alpha ( Y ) X $, where $ \nabla $ is the Levi-Civita connection of $ g $, then one says that $ ( M, \alpha ) $ is a Sasakian manifold, [a4], [a12].

As a consequence of the Meyer–Steenrod theorem [a6], a $ K $- contact flow $ \phi _ {t} $ on a compact $ ( 2n + 1 ) $- dimensional manifold is almost periodic: the closure of $ \phi _ {t} $ in the isometry group of $ M $( of the associated contact metric) is a torus $ T ^ {k} $, of dimension $ k $ in between $ 1 $ and $ n + 1 $, which acts on $ M $ while preserving the contact form $ \alpha $, [a3]. The "completely integrable" case $ k = n + 1 $ has been studied in [a2]: these structures are determined by the image of their contact moment mapping.

The existence of $ K $- contact flows poses restrictions on the topology of the manifold. For instance, since a $ K $- contact flow can be approximated by a periodic $ K $- contact flow, only Seifert fibred compact manifolds can carry a $ K $- contact flow. Another example of a restriction is the Tachibana theorem, asserting that the first Betti number of a compact Sasakian manifold is either zero or even, [a9]. This shows that no torus $ T ^ {2n + 1 } $ can carry a Sasakian structure. Actually, P. Rukimbira [a8] showed that no torus can carry a $ K $- contact flow.

A. Weinstein [a11] has conjectured that the contact flow of a compact contact manifold has at least one periodic orbit. Despite important breakthroughs (including [a10]), this conjecture is not quite settled at present (1996). However, it is known that $ K $- contact flows on compact manifolds have at least two periodic orbits [a3].

Examples of $ K $- contact manifolds include the contact manifolds $ ( M, \alpha ) $ with a periodic contact flow $ \phi _ {t} $( these include the regular contact manifolds), such as the sphere $ S ^ {2n + 1 } $ equipped with the contact form $ \alpha $ that is the restriction to $ S ^ {2n + 1 } $ of the $ 1 $- form

$$ \sum _ {i = 1 } ^ { {n } + 1 } x _ {i} dy _ {i} - y _ {i} dx _ {i} $$

on $ \mathbf R ^ {2n + 2 } $. More generally, compact contact hypersurfaces (in the sense of M. Okumura) [a1] in Kähler manifolds of constant positive holomorphic sectional curvature carry $ K $- contact flows. A large set of examples is provided by the Brieskorn manifolds: In [a12] it is shown that every Brieskorn manifold admits many Sasakian structures, hence carries many $ K $- contact flows.

References

[a1] A. Banyaga, "On characteristics of hypersurfaces in symplectic manifolds" , Proc. Symp. Pure Math. , 54 , Amer. Math. Soc. (1993) pp. 9–17 MR1216525 Zbl 0792.58015
[a2] A. Banyaga, P. Molino, "Complete integrability in contact geometry" , Memoirs , Amer. Math. Soc. (submitted)
[a3] A. Banyaga, P. Rukimbira, "On characteristics of circle invariant presymplectic forms" Proc. Amer. Math. Soc. , 123 (1995) pp. 3901–3906 MR1307491 Zbl 0849.58025
[a4] D.E. Blair, "Contact manifolds in Riemannian geometry" , Lecture Notes in Mathematics , 509 , Springer (1976) MR0467588 Zbl 0319.53026
[a5] Y. Carrière, "Flots riemanniens" Astérisque , 116 (1982) pp. 31–52 MR1046241 MR0755161 MR0744829 Zbl 0996.37500 Zbl 0548.58033 Zbl 0524.57018
[a6] S.B. Meyer, N.E. Steenrod, "The group of isometries of a Riemannian manifold" Ann. of Math. , 40 (1939) pp. 400–416 MR1503467
[a7] P. Molino, "Riemannian foliations" , Progress in Math. , Birkhäuser (1984) MR0761580 MR0755169 Zbl 0576.57022
[a8] P. Rukimbira, "Some remarks on -contact flows" Ann. Global Anal. and Geom. , 11 (1993) pp. 165–171 MR1225436
[a9] S. Tachibana, "On harmonic tensors in compact sasakian spaces" Tohoku Math. J. , 17 (1965) pp. 271–284 MR0190878 Zbl 0132.16203
[a10] C. Viterbo, "A proof of the Weinstein conjecture for " Ann. Inst. H. Poincaré. Anal. Non-Lin. , 4 (1987) pp. 337–356 MR917741
[a11] A. Weinstein, "On the hypothesis of Rabinowicz' periodic orbit theorem" J. Diff. Geom. , 33 (1978) pp. 353–358
[a12] K. Yano, M. Kon, "Structures on manifolds" , World Sci. (1984) MR0794310 Zbl 0557.53001
How to Cite This Entry:
K-contact-flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K-contact-flow&oldid=24485
This article was adapted from an original article by A. Banyaga (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article