Scientiae Mathematicae Japonicae Online, Vol. 4, (2001), ASSOCIATED WITH CONTACT FORMS ON 3-MANIFOLDS. Atsuhide Mori

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1 Scientiae Mathematicae Japonicae Online, Vol. 4, (200), REMARKS ON THE SPACES OF RIEMANNIAN METRICS ASSOCIATED WITH CONTACT FORMS ON 3-MANIFOLDS Atsuhide Mori Received November 28, 2000; revised March 26, 200 Abstract. We recall and improve the correspondence theorem of Etnyre-Ghrist [] between a positively rescaled Reeb field for a contact -form and a rotational Beltrami field for a Riemannian metric on a closed oriented 3-manifold. Given a contact form, we associate it with the space of Riemannian metrics for which the Reeb field is a Beltrami field with certain additional properties. We obtain a product structure on this space of metrics and then, by applying it, we characterize certain geometric structures on 3-manifolds. Introduction. Recently, J.Etnyre and R.Ghrist [] found a correspondence between a Reeb-like field in contact topology and a rotational Beltrami field in topological hydrodynamics. Let M 3 be a closed oriented 3-manifold. A Reeb-like field on M 3 is a positively rescaled Reeb field for a given positive contact form on M 3. On the other hand, a rotational Beltrami field on M 3 is a non-singular vector field X satisfying r X = fx for some function f > 0. Here r X denotes the curl of X with respect to a given Riemannian metric g and a fixed positive volume form ν on M 3. It is easy to see that the definition of a rotational Beltrami field is independent of the choice of ν. The correspondence theorem says that the set of all Reeb-like fields on M 3 can be regarded as the set of all rotational Beltrami fields on M 3 if we don't fix a contact form nor a metric. We consider the case where the above f can be taken as f with respect to g and the g-induced volume form ν g. Then we call X a normal Beltrami field for g if moreover g(x; X) holds. Fix a positive volume form ν on M 3. Then we can improve the above correspondence theorem as follows. () A Reeb field X ff for some contact form ff with ff ^ dff = ν corresponds to a normal Beltrami field for some ν-inducing metric g (Theorem 5). (2) A rotational Beltrami field for a Riemannian metric g is a positively rescaled normal Beltrami field for some Riemannian metric g 0 (Remark 6). Every positive contact form ff on M 3, therefore, can be associated with the subspace F ff of the space R(M 3 ) of all Riemannian metrics on M 3, where F ff consists of any element g inducing the volume S form ff ^ dff and satisfying g(x ff ; ) =ff. Set B ff = ffij X fi = X ff g(ρ T Λ M 3 )andc ff = fi2b ff F fi (ρr(m 3 )). In this paper, we study these spaces. Our results are the following theorems. Theorem A(Theorem 0). For any positive contact form ff on a closed oriented 3- manifold M 3, the above C ff, F ff and B ff are connected and contractible. Moreover, C ff is fibred trivially by ff fi g fi2bff over B ff. We obtain the following theorem by using Theorem A and a result of Geiges and Gonzalo [2] on the characterization of closed 3-manifolds admitting Cartan structures Mathematics Subject Classification. 57M50.

2 930 A. Mori Theorem B(Theorem 5). Let M 3 be a closed orientable 3-manifold. Then there aretwo contact forms ff and fi on M 3 satisfying ) ff ^ dff = hfi ^ dfi for some function h>0 and 2) X ff?x fi with respect to some g 2C ff C fi if and only if M 3 is diffeomorphic to a SU(2)-, g SL2 -orf E2 -manifold. The class of closed oriented 3-manifolds admitting positive contact forms ff and fi with X fi 6= ±X ff and C ff C fi 6= ; seems very small since such a manifold has to admit another normal Beltrami field than ±X ff for the same metric in C ff.itiseven likely that this class coincides with one stated in Theorem B. 2 The correspondence theorem. We will work in the smooth category throughout this paper. First we prepare some definitions. Definition. Avector field X on a Riemannian 3-manifold (M 3 ;g) is called a Beltrami field if it is everywhere colinear with its curl, that is, r X = fx for some function f on M 3. Here the curl r X is the vector field determined by r X ν = d(g(x; )) for a fixed volume form ν ( denotes the interior product). A non-singular Beltrami field is called a rotational Beltrami field if the above f satisfies f > 0. A rotational Beltrami field for the particularly g-induced volume form ν g is called a normal Beltrami field if f and g(x; X) hold. Remark 2. Beltrami fields form an important and still mysterious class of steady (i.e., time-independent) solutions of the following Euler's equation for a perfect incompressible fluid with a volume form ν (see []). Let fx t g t2r be a family of vector fields. X t can be considered as the velocity field of an perfect incompressible fluid if it satisfies the Euler's equation _X t + r Xt X t = rp t ; L Xt ν =0 for some family p t of functions, called the pressure term. Here r Xt denotes the covariant derivative with respect to g along X t. Then the curl of X t with respect to ν and g satisfies ( Xt r Xt μ)(y ) = Xt d(g(x t ; ))(Y ) = X t g(x t ;Y) Yg(X t ;X t ) g(x t ; r Xt Y r Y X t ) = g(r Xt X t ;Y) g(r Y X t ;X t ) = g(r Xt X t ;Y) 2 Yg(X t;x t ): From now on, we assume that X t is time-independent, that is, X t = X holds for any t 2 R. Put p = p t and P = p + g(x; X). Then the first portion of the Euler's equation yields 2 X r X μ = dp: Then we see that a Beltrami field corresponds to a P -free steady fluid while a normal Beltrami field corresponds to a special kind of pressure-free fluid. Note also that a normal Beltrami field generates a divergence-free geodesical flow. Definition 3. A -form ff on an oriented 3-manifold M 3 is called a positive contact form if ff ^ dff is a positive volume form. Then a vector field X on M 3 is called a Reeb field for ff if Xdff =0and Xff = hold. Such anx always exists and is determined uniquely by

3 METRICS AND CONTACT FORMS 93 ff, sowe denote it by X ff. A Reeb-like field Y is a positively rescaled Reeb-field, that is, Y = fx ff for some function f>0. The following is the Etnyre-Ghrist's correspondence theorem. Theorem 4 ([]). Let M 3 be an oriented 3-manifold. Given a Riemannian metric g on M 3, any rotational Beltrami field for g, if it exists, is a Reeb-like field for some positive contact form on M 3. Conversely, given a positive contact form ff on M 3, any Reeb-like field for ff is a rotational Beltrami field for some Riemannian metric on M 3. Note that the definition of rotational Beltrami fields are independent of the choice of the fixed volume form ν in Definition. The above theorem says that a vector field X is a Reeb-like field for some positive contact form if and only if X is a rotational Beltrami field for some Riemannian metric. It may be difficult, however, to associate the set of all such contact forms with the set of all such metrics in a general way. Sowe state a more detailed correspondence theorem as follows. Theorem 5. Let M 3 be an oriented 3-manifold equipped with a positive volume form ν. Given a ν-inducing Riemannian metric g, any normal Beltrami field with respect to g, if it exists, is a Reeb field for some contact form ff with ff ^ dff = ν on M 3. Conversely, for any contact form ff on M 3 with ff ^ dff = ν, the Reeb field X ff is a normal Beltrami field for some ν-inducing metric. Thus a vector field X is a Reeb field X ff for some contact form ff with ff ^ dff = ν if and only if X is a normal Beltrami field for some ν-inducing metric. Proof. Suppose that r X = X with respect to ν and a ν-inducing metric g and g(x; X) = hold. Putting ff = g(x; ), we have Since g is ν-inducing, we have ff ^ dff = g(x; ) ^ r X ν: ν = g(x; ) ^ g(e 2 ; ) ^ g(e 3 ; ) =g(x; ) ^ Xν for a local orthonormal framing (X; e 2 ;e 3 ). Thus the condition r X = X implies ff^dff = ν. ThenX = X ff holds since Xdff = X Xν = 0 and Xff = g(x; X) =. Conversely, suppose that X = X ff for ff with ff ^ dff = ν. Then choose a local frame (X; e 2 ;e 3 )suchthat (e 2 ;e 3 ) forms a symplectic basis for dff on ker ff. Since there is a global complex structure J on ker ff compatible with dff, wemay assume e 3 = Je 2. Let g be the metric for which (X; e 2 ;e 3 ) is orthonormal. Note that g is globally defined since each transformation map between charts preserves the orthonormality of(e 2 ;e 3 )asanelement of SU() with respect to J. ThenX is a normal Beltrami field for g since g(x; X) =and d(g(x; )) = dff = Xν. This completes the proof. Remark 6. Our normal condition may seem much too strong at a glance. Note that, however, a given rotational Beltrami field for a metric g can be rescaled to be a normal Beltrami field for another metric g 0. This fact follows immediately from the theorems 4 and 5. Moreover we can see, from Moser's theorem, that even when we fix arbitrary volume form ν the Weinstein conjecture translates to whether any normal Beltrami field for any ν-inducing Riemannian metric generates a flow with a closed orbit. 3 The Spaces of metrics. Given a positive volume form ν on a closed oriented 3- manifold M 3, weset Cont(M 3 ) = fall positive contact forms on Mg(ρ T Λ M 3 )and Cont(M 3 ;ν)=fffj ff ^ dff = νg(ρ Cont(M 3 )). Let R(M 3 ) be the space of all Riemannian

4 932 A. Mori metrics on M 3 and R(M 3 ;ν) its subset consisting of all ν-inducing metrics. For any ff 2 Cont(M 3 ), put F ff = fg 2R(M 3 ;ff^ dff)j g(x ff ; ) =ffg; and ef ff = fg 2R(M 3 )j g(x ff ; ) =ffg; B ff = ffi 2 Cont(M 3 )j X fi = X ff g C ff = [ fi2b ff F fi (ρr(m 3 )): Note that we can also define the above C ff by C ff = fg 2R(M 3 )j fi := g(x ff ; ) 2B ff and g 2R(M 3 ;fi^ dfi)g. Then the following lemmas hold. Lemma 7. R(M 3 ) and e Fff are connected andcontractible. Proof. For fixed g 0 2R(M 3 )andany g 2R(M 3 ), the family ftg 0 +( t)gg t2[0;] defines a contraction from R(M 3 )tofg 0 g. If g and g 0 are in Fff e then so are tg 0 +( t)g since (tg 0 +( t)g)(x ff ; ) =tff +( t)ff = ff. This ends the proof. Lemma 8. F ff is connected andcontractible. Proof. Any metric g(2 F ff ) has the following form on each Darboux coordinate (x; y; z) with ff = xdy + dz. g = a b 0 b c + x 2 x 0 x A (a>0;b >0;ac b2 ) where a; b and c are some local functions. Fix the orthonormal framing (X ff ;e 2 ;e ; @z on each Darboux coordinate, whose dual is p b ff(= xdy + dz); adx + p dy; a p dy : a Put h ff = ff Ω ff. Then we have h ff = x 2 x 0 x A on each Darboux coordinate. Note that h ff satisfies h ff (X ff ; ) =ff and h ff (e; ) = 0 for any e 2 ker ff. Then for any g 2 e Fff inducing a volume form fν (f >0), the family ( t)+t f (g h ff )+h ff (t 2 [0; ]) defines a retraction from e Fff to F ff. Note that this retraction fixes any element off ff since f holds in this case. Thus the lemma follows from Lemma 7.

5 METRICS AND CONTACT FORMS 933 Lemma 9. Put ff t =( t)ff+tfi for fi 2B ff.let fy t g t2[0;] be the family of vector fields determined byff ^ fi = Yt (ff t ^ dff t ) and fffi t g t2[0;] the family of diffeomorphisms on M 3 obtained byintegrating Y t under the initial condition ffi 0 = Id M 3. Then F fft =(ffi t ) Λ (F ff ) (t 2 [0; ]) holds. Proof. On each Darboux coordinate (x; y; z) for ff with ff = xdy + dz, wehave fi ff = pdx + qdy for some functions p and q with p z = q z = 0 and q x p y >. This yields Y t = +t(q x p y Then we have L Yt ff t + d dt ff t = pdx qdy + pdx + qdy =0: Thus F fft =(ffi t ) Λ (F ff ) holds for M is closed. This ends the proof. These lemmas imply the following theorem. Theorem 0. For any contact form ff on a closed oriented 3-manifold M 3, the above C ff, F ff and B ff are connected and contractible. Moreover, C ff is fibred trivially by ff fi g fi2bff over the space B ff with projection g 7! g(x ff ; ). Remark. R R We see that fi(2 B ff ) satisfies M 3 fi ^ dfi = M 3 ff^dff from Lemma 9. Note that, however, this does not mean fi ^ dfi = ff ^ dff in general. It may beinteresting to compare Lemma 9 and Theorem 0 with the Gray's stability theorem [3]. Note also that if ±fi is in B ff then B fi = B ff holds. C ff C fi may be non-empty even when ±fi 62 B ff. Here is an example. Example 2. Put M 3 = T 3 = R 3 =(2ßZ) 3, ν = dx ^ dy ^ dz, ff = sin zdx + cos zdy, fi = cos zdx + sin zdy and fl = sinxdy +cosxdz. Then we can easily see that the Eucridean metric dx 2 + dy 2 + dz 2 is in the intersection C ff C fi C fl. Note that all these contact forms are, then, equivalent up to isometries. 4 Cartan structures. We recall a definition and one of the results in Geiges and Gonzalo [2]. Definition 3. We say that a pair (ff; fi) ofcontact forms on a closed oriented 3-manifold M 3 is a Cartan structure on M 3 if ff ^ dff = fi ^ dfi and ff ^ dfi = fi ^ dff = 0 hold. Theorem 4 ([2]). Let M 3 be a closed orientable 3-manifold. Then M 3 admits a Cartan structure if and only if M 3 is diffeomorphic to a quotient of the Lie group G under a discrete and cocompact subgroup acting from the left, where G is one of the following: ) SU(2), 2) g SL2, the universal cover of PSL(2; R) or 3) f E2, the universal cover of the orientation-preserving isometry group of the Euclidean R 2. The manifolds satisfying the above condition are called SU(2)-manifolds, g SL2 -manifolds or f E2 -manifolds respectively. As an application of Theorem 0 and this characterization theorem, we can prove the following theorem.

6 934 A. Mori Theorem 5. Let M 3 be a closed orientable 3-manifold. Then there are two contact forms ff and fi on M 3 satisfying ) ff ^ dff = hfi ^ dfi for some function h>0 and 2) X ff?x fi with respect to some g 2C ff C fi if and only if M 3 is diffeomorphic to a SU(2)-, g SL2 -orf E2 -manifold. Proof. First, we prove the `if' part. The Lie algebra of G = SU(2); g SL2 or f E2 admits a basis (e ;e 2 ;e 3 ) with [e ;e 2 ]=ffie 3 ; [e 2 ;e 3 ]=e ; [e 3 ;e ]=e 2 where ffi = +; or 0 respectively (see [2]). We regard these as left-invariant vector fields on G. Then take the dual frame ( ; 2 ; 3 ) and set ff = and fi = 2. Then the pair (ff; fi) is a Cartan structure. Take the metric g and the orientation of M by means of the oriented orthonormal basis (e ;e 3 ;e 2 ). Then we have for example dff(e 3 ;e 2 )=ff([e 2 ;e 3 ]) = = 3 ^ 2 (e 3 ;e 2 ) and conclude the `if' part. Next, we prove the `only if' part. Theorem 0 implies that there is the (unique) fibre F ff 0(ff 0 2B ff )containing the metric g. Thus by changing ff and fi if necessary, wemay assume g 2F ff F fi. Then we have ker ff?x ff and ker fi?x fi.thus dffj ker fi = dfij ker ff =0 holds. This yields fi ^ dff = ff ^ dfi = 0. The pair (ff; fi), therefore, forms a Cartan structure on M 3.Thus the theorem follows from Theorem 4. References [] J.Etnyre and R.Ghrist: Contact topology and hydrodynamics I: Beltrami fields and the Seifert conjecture, Nonlinearity 3 (2000), [2] H.Geiges and J.Gonzalo: Contact geometry and complex surfaces, Invent. Math. 2(995), [3] J.W.Gray: Some global properties of contact structures, Ann. of Math. 69(959) Department of Mathematics, Kyoto Sangyo University, Kyoto, JAPAN. rimo@cc.kyoto-su.ac.jp