K-contact-flow
A contact form on a smooth -dimensional manifold
is a
-form
such that
is everywhere non-zero. The pair
is called a contact manifold. See also Contact structure.
A contact manifold carries a distinguished vector field
, called the characteristic vector field or Reeb field, which is uniquely determined by the following equations:
and
for all vector fields
. The flow
generated by
(when it is complete) is called the contact flow. Sometimes the name "contact flow" is used for the
-dimensional foliation
consisting of the unparametrized orbits of
, [a5].
If the flow is a Riemannian foliation in the sense of Reinhart–Molino [a7], i.e., if there is a holonomy-invariant transverse metric for
, then
is called a
-contact flow, and the pair
is called a
-contact manifold. This definition is equivalent to requiring that the flow
of
is a
-parameter group of isometries for some contact metric (a Riemannian metric
such that there exists an endomorphism
of the tangent bundle
such that
,
,
, and
for all vector fields
and
on
). If one has in addition
, where
is the Levi-Civita connection of
, then one says that
is a Sasakian manifold, [a4], [a12].
As a consequence of the Meyer–Steenrod theorem [a6], a -contact flow
on a compact
-dimensional manifold is almost periodic: the closure of
in the isometry group of
(of the associated contact metric) is a torus
, of dimension
in between
and
, which acts on
while preserving the contact form
, [a3]. The "completely integrable" case
has been studied in [a2]: these structures are determined by the image of their contact moment mapping.
The existence of -contact flows poses restrictions on the topology of the manifold. For instance, since a
-contact flow can be approximated by a periodic
-contact flow, only Seifert fibred compact manifolds can carry a
-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
can carry a Sasakian structure. Actually, P. Rukimbira [a8] showed that no torus can carry a
-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 -contact flows on compact manifolds have at least two periodic orbits [a3].
Examples of -contact manifolds include the contact manifolds
with a periodic contact flow
(these include the regular contact manifolds), such as the sphere
equipped with the contact form
that is the restriction to
of the
-form
![]() |
on . More generally, compact contact hypersurfaces (in the sense of M. Okumura) [a1] in Kähler manifolds of constant positive holomorphic sectional curvature carry
-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
-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 ![]() |
[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 ![]() |
[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 |
K-contact-flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K-contact-flow&oldid=47473