Difference between revisions of "K-contact-flow"
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− | + | 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]]. | ||
− | + | 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]]]. | ||
− | + | 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]]]. | ||
− | + | 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. | ||
− | + | 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. | ||
− | + | 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 | + | 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 ![]() |
[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=24485