View source for K-contact-flow

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]].

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]]].

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]]].

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.

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.

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]]].

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

<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>

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.

====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</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</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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Y. Carrière,   "Flots riemanniens"  ''Astérisque'' , '''116'''  (1982)  pp. 31–52</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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P. Molino,   "Riemannian foliations" , ''Progress in Math.'' , Birkhäuser  (1984)</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</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</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</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)</TD></TR></table>