# K-contact-flow

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 $R$-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 $\mathbf{R} ^ { 2 n }$" 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=50268
This article was adapted from an original article by A. Banyaga (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article