A bilinear form associated to contact sub-conformal manifolds

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1 Dfferental Geometry and ts Applcatons ) A blnear form assocated to contact sub-conformal manfolds E. Falbel a,,j.m.veloso b a Insttut de Mathématques, Unversté Perre et Mare Cure, Analyse Algébrque, 4, place Jusseu, F Pars, France b Departamento de Matemátca CCEN, Unversdade Federal do Pará, 66059, Belém PA, Brazl Receved 20 August 2004; receved n revsed form 28 September 2005 Avalable onlne 6 June 2006 Communcated by O. Kowalsk Abstract We defne a blnear form assocated to a sub-remannan contact manfold. It transforms by scalar multples under subconformal transformatons and wth further hypothess t s naturally defned on certan torus bundles over the contact manfold Elsever B.V. All rghts reserved. MSC: 32F25; 53B25 Keywords: Contact manfolds; Fefferman metrc; Sub-Remannan manfolds Lorentz form 1. Introducton A pseudohermtan structure on a contact manfold M gves rse to a Lorentz conformal metrc on a crcle bundle over M see [4] for a survey). That conformal structure s the same for all pseudohermtan structures defnng the same CR structure. Ths was the method gven n [10] generalzng the constructon of the Lorentz form assocated to a real strctly pseudoconvex hypersurfaces n C n n [8]. Another constructon of the Lorentz structure for abstract CR manfolds s gven n [1]. In fact, that was the frst constructon for an abstract CR manfold but ts constructon nvolves the constructon of a Cartan connecton [2,3,9,11] on a fber bundle canoncally assocated to a CR manfold. A natural generalzaton of pseudohermtan geometry on a contact manfold s sub-remannan geometry whch s a metrc structure only defned on the contact dstrbuton. The sub-conformal structure assocated to the sub- Remannan manfold has an assocated fber bundle but n general t s not a prncpal bundle so that the approach n [1] becomes mpossble. The goal of ths paper s to construct a form whch changes by a scalar factor under sub-conformal transformatons followng the constructon n [10] see also [7] for a related constructon). Ths constructon s very general and holds even when there exsts no natural crcle bundle assocated to the manfold. Instead, under approprate hypothess, there exsts a torus bundle where the blnear form s naturally defned see Theorem 3.1) and eventually, under more * Correspondng author. E-mal addresses: falbel@math.jusseu.fr E. Falbel), veloso@ufpa.br J.M. Veloso) /$ see front matter 2006 Elsever B.V. All rghts reserved. do: /j.dfgeo

2 36 E. Falbel, J.M. Veloso / Dfferental Geometry and ts Applcatons ) restrctve condtons, s well defned on a crcle bundle over the sub-remannan manfold see Theorem 3.2). Ths generalzes the Fefferman s metrc. 2. Sub-Remannan and sub-conformal manfolds In ths secton we defne the geometrcal structures we wll use, namely the sub-remannan structures and subconformal structures. Let D be a contact dstrbuton on a manfold M. Defnton M,D,g)s a sub-remannan structure f g s a metrc on D. 2. M, D, g) s a sub-conformal structure f g s a conformal class of sub-remannan metrcs. Let π : TM TM/Dbe the quotent map. Defnton 2.2. The Lev form α : D D TM/Ds the skew-symmetrc form defned as αx,y) = π[x, Y ]). Fxng a base v of TM/D defnes the Lev form α v as a real valued form. Let θ v be the contact form of ths dstrbuton such that θ v π 1 v) = 1, then the Lev form s gven by dθ v X, Y ) = α v X, Y ). If we have a metrc on D, defne a skew-symmetrc operator H v on the dstrbuton by α v X, Y ) = gh v X, Y ). As α v s non-degenerate, we can always choose v such that det H v = 1 and ths determnes a unque v gnorng orentaton effects. Observe that f we let tg be a new metrc and choose v as above, then α 1 t v X, Y ) = tgh vx, Y ) so the defnton of H v does not depend on a metrc nsde a conformal class of metrcs. Fxng a metrc on D, denote by H ths operator. Its normal form s gven n the followng lemma. Lemma 2.1. Let V be a 2n dmensonal real vector space wth a scalar product. If H s a nondegenerated skewsymmetrc operator, then there exsts an orthonormal bass of V such that the matrx of H s λ λ Λ = n λ λ n wth λ > 0, = 1,...,n. We also need the followng smple lemma whch characterzes the subgroup of SO2n) commutng wth Λ. Lemma 2.2. Suppose that Λ s such that λ d1 + +d k 1 +1 = =λ d1 + +d k = ν k for 1 k r, wth d 1 + +d r = n, and ν 1 <ν 2 < <ν r, where the ν k are real numbers, and that A SO2n) satsfes AΛA T = Λ. Then A Ud 1 ) Ud r ).

3 E. Falbel, J.M. Veloso / Dfferental Geometry and ts Applcatons ) Sub-Remannan geometry Let M,D,g)be a sub-remannan manfold of dmenson 2n + 1, where D s a contact dstrbuton on M, and g a postve quadratc form on D. We wll further suppose that TM/Ds orented. We consder the SO2n) bundle E of coframes θ, θ ) on M such that the θ restrcted to D are orthonormal, θ s the postve contact form defned such that dθ = h j θ θ j, wth h j = h j and det h j = 1. Observe that a change n coframes gven by θ = aj θ j, aj ) SO2n) mples that h j = ak h kla j l.one we consder the tautologcal forms defned by θ,θ whch we denote by the same letters. E s a prncpal fber bundle wth a rght acton by SO2n). We wll consder the coframes as lne vectors where SO2n) acts by matrx multplcaton from the rght. Proposton 3.1. See [6,12]) There exst unque connecton forms ω j and torson forms τ on E satsfyng 1) dθ = θ j ω j + θ τ, 2) wth ω j = ωj and τ θ = 0. 3) Proof. Let ω j and τ be any forms satsfyng the frst equaton. If ω j and τ also satsfy the equaton, then θ j ωj ) ω j + θ τ τ ) = 0. From Cartan s lemma we have ωj ω j = a jk θ k + bj θ, τ τ = bk θ k wth ajk = a kj. We wll choose a jk,b j such that the condtons n the theorem be satsfed for ω j, τ.toverfy condton 3) we must have 0 = τ θ = τ θ + b k θ k θ. If we wrte τ = τ k θ k, then τ k + bk ) θ k θ = 0 and usng Cartan s lemma agan τ k + b k = a k wth a k = ak. On the other hand f ω j = ωj s satsfed, and wrtng ω j = ω jk θ k + w j θ we obtan ω jk + ω j k + a jk + ) aj k θ k + w j + wj + b j + ) bj θ = 0. We get two equatons w j + wj + a j + aj τ j τ j = 0, ω jk + ωj k + a jk + aj k = 0. The frst equaton, recallng that aj s symmetrc, has soluton a j = τ j + τ j 2 w j + wj 2 therefore bj second equaton can be solved usng the permutaton trck, as n Remannan geometry. s determned. The Eq. 3) s equvalent to τ = τj θ j, wth τj = τ j. If we dfferentate 1), and apply 2) we get dhj h kj ω k + h kωj k ) θ θ j + 2h j τ θ j θ = 0.

4 38 E. Falbel, J.M. Veloso / Dfferental Geometry and ts Applcatons ) A smple computaton shows that there exst unque functons b j k and b j so that we can wrte and dh j h kj ω k + h kω k j = b j kθ k + b j θ 4) 2h j τ θ j = b j θ θ j, where b j k = b jk,b j k + b jk + b kj = 0,b j = b j. We wll use the followng notatons: 5) Ω = ω j ), H = h j ) and H 1 = h j ). Defnton 3.1. On E we defne, ς = Tr H 1 Ω ) = h j ω j Conformal change n the sub-remannan metrc In ths secton we wll study the transformaton n the connecton forms when the metrc on D undergoes a conformal change of the form g = e 2f g where f s a real functon on M. For ths we need to compare the structure equatons for the metrcs g and g. Let s frst ntroduce some notaton a study of the nvarants of sub-conformal structures can be found n [5]). Defne f 0 and f usng the formula and wrte df = f 0 θ + f θ, 6) f = h j f j. If we dfferentate 6), we get df0 f τ ) θ + df j f ω j + f 0h j θ ) θ j = 0. Applyng Cartan s lemma we obtan df j f ω j + f 0h j θ = f jk θ k + f j0 θ, df 0 f τ = f 0j θ j + f 00 θ, wth f jk = f kj and f 0j = f j0. It s a drect verfcaton, applyng 4), that Then where dh j = h jk ω k hk ω j k + hk b klm θ m + b kl θ ) h jl. df + f j ω j f 0θ = f j θ j + f 0 θ, f j = hk f kj b klj f l) and f 0 = hk f k0 b kl f l ). The contact form assocated to g s 7) θ = e 2f θ.

5 E. Falbel, J.M. Veloso / Dfferental Geometry and ts Applcatons ) Verfyng that dθ = h j θ θ j, we obtan that the new coframes are gven by θ = e f θ + f θ ) 8) 9) wth h j = h j.lete be the bundle of coframes θ,θ ) assocated to the sub-remannan manfold M,D,g ). Proposton 3.2. The applcaton gven by F : E E Fθ,θ ) = θ,θ ) = e 2f θ,e f θ + f θ )) s a somorphsm of SO2n)-bundles. Proof. Let θ, θ ) be a new coframe of E, wth θ = aj θ j, aj ) SO2n). Fromdθ = h j θ θ j we obtan h j = ak hkl a j l. Then Fθ, θ ) = θ, θ ), where θ = e f θ + f θ), df = f j θ j + f 0 θ, and f = h j f j. That mples f = aj f j, f = ak f k and θ = e f aj θ j + aj f j θ)= aj θ j. We proved that F θ,a j θ j ) = θ,a j θ j ), what ends the proof. We consder from now on the tautologcal forms on E usng, by abuse of notaton, the same letters θ j and θ.as before there exst unque forms ω j and τ such that dθ = θ j ω j + θ τ, ω j = ω j, and τ θ = 0. In what follows, we wll omt the functon F when comparng the bundles E and E. That s, we wll wrte, by abuse of notaton, α = F α ) for a form α defned on E. Dfferentatng 9) and applyng 2) and 10), we obtan 10) Proposton 3.3. The connecton and torson forms of E and E satsfy the followng formulae: ω j = ω j + e f f j δk f δ j k + f h jk f j h k f k ) h j θ k + e 2f f j f f f j f j 1 ) 2 f j θ 11) and τ = e 2f τj + f 0δj + f f j + f j f 1 2 f j 1 ) 2 f j θ j. It follows from 11) the followng Proposton 3.4. ς = TrH 1 Ω) and ς = TrH 1 Ω ) are related by ς = ς + 2n + 4)f θ + 2n + 2)f f + h j f j ) θ. Applyng 7) we get d f θ ) = f j θ θ j + θ f 0 θ + f τ ).

6 40 E. Falbel, J.M. Veloso / Dfferental Geometry and ts Applcatons ) Then dς = dς + 2n + 4)f j + 2n + 2)f k f k + h kl fl k ) ) hj θ θ j mod θ. Our goal s to construct a conformally nvarant blnear form on a certan bundle over M usng ς, dς and the tautologcal forms θ and θ. In order to do so we frst need to reduce the structure group of E. Eventually we wll mpose that H s constant on the reduced bundle, but we wll carry our computaton n a more general settng wth some regularty assumptons on H Reducton of the structure group We suppose now that the canoncal form Λ of H = h j ) as n Lemma 2.2 has n every pont of Mrvalues ν k, each one wth multplcty d k. More explctly, the ν k are real functons on M. Defnton 3.2. E 1 s the subset of ponts of E such that H = Λ. From Lemma 2.2 we have that E 1 s a Ud 1 ) Ud r ) subbundle of E. We denote a coframe n E 1 by θ k where 1 k r and d 1 + +d k k d 1 + +d k wth d 1 + +d r = n. If Y = Y 1 + +Y r wth Y k sud k ), then Λ 1 Y = 2ν 1 1 J 1Y νr 1 J r Y r, where J k s a lnear operator such that Jk θ l = δk l θ l+n and Jk θ l+n = δk l θ l. Observe that Jk θ 0 only for d 1 + +d k d 1 + +d k The torus bundle over M 12) The subgroup SUd 1 ) SUd r ) s normal n Ud 1 ) Ud r ). We defne T as the quotent bundle of E 1 by SUd 1 ) SUd r ). T s a U1) U1) bundle,.e. T s a r-torus prncpal bundle over M: Proposton 3.5. T = E 1 /SUd 1 ) SUd r ) s a U1) U1)-prncpal bundle. Parts of the sub-remannan connecton descend to forms defned on the torus bundle. We use the followng crteron: A dfferental form ϕ on E projects on T f and only f 1. R g ϕ = ϕ for every g SUd 1) SUd r ), 2. ϕx ) = 0 for every X sud 1 ) + +sud r ). Restrcted to E 1 we have ς = TrΛ 1 Ω) = k k 2 ν k ω k k +n, where d k k d k and 1 k r. Also, Defnton 3.3. On E 1 we defne, ω k = 2 k ω k k +n where d k k d k. Proposton 3.6. The forms ς and ω k project to T = E 1 /SUd 1 ) SUd r ). Proof. We prove frst the result for the form ς. In order to verfy R a ς = ς for a Ud 1) Ud r ), t suffce to verfy the formula on vertcal vectors, because ς vanshes on horzontal vectors. Suppose X ud 1 ) + +ud r ),

7 E. Falbel, J.M. Veloso / Dfferental Geometry and ts Applcatons ) a Ud 1 ) Ud r ) and let X be the vertcal vector feld on E 1 nduced by X. FromR a X = a 1 Xa) and observng that aλ = Λa,wehave R a ςx ) = ςr a X ) = Tr Λ 1 a 1 Xa ) = Tr Λ 1 X ) = ςx ). To end the proof t s enough to verfy that f Y sud 1 ) + +sud r ) Tr Λ 1 Y = 0. Recall that f Y = Y Y r wth Y k sud k ), then Λ 1 Y = 2ν 1 1 J 1Y νr 1 J r Y r and Tr Λ 1 Y = 2νk 1 TrJ k Y k ) = 0. The same argument shows that the forms ω k = 2 k ω k k +n project to T. Proposton 3.7. The projected forms ω 1,...,ω r ) defne a connecton on T wth values n u1) + +u1). Proposton 3.8. The form dς descends to a form on M f and only f H s constant. Proof. As Ra ς = ς for every a Ud 1) Ud r ), then L X ς = 0 for every X ud 1 ) + +ud r ). We wrte dx )ς) = d k ν 1 k ω k X )) and observe that ω k X ) s constant for every X. Therefore dx )ς) = 0 f and only f ν k are constant. It follows from the formula L X = dx ) + X )d that X )dς = 0. We conclude that dς s projectable on M. The same argument shows that the forms dω k can always be projected. In the rest of that secton we suppose that H s constant. Gven a two form ϖ = V j θ θ j + V θ θ on M, defne Tr ϖ = h j V j. We can now defne on T the form Defnton 3.4. σ = 1 n + 2 ς ) 1 4n + 1) Trdς)θ. We defne now a blnear form on T of type 2n + 1, 1,r 1), that s, wth 2n + 1 postve egenvalues, 1 negatve egenvalue and wth a r 1)-dmensonal kernel. Defnton 3.5. Let b be the blnear symmetrc form of type 2n + 1, 1,r 1) b = θ θ + θσ. Observe that for any 2-form ϖ on M, Tr ϖ ) = Trϖ )e 2f.From12) we obtan Lemma 3.1. so Tr dς = Tr dς + 4n + 4) f j hj + nf f )) e 2f Proposton 3.9. σ = σ 2f θ f f θ.

8 42 E. Falbel, J.M. Veloso / Dfferental Geometry and ts Applcatons ) Puttng together the formulas above we fnally obtan the conformal nvarance of the blnear form. Theorem 3.1. Let M,D,g) be a sub-remannan manfold such that H s constant on E 1. Then there exsts a torus bundle T over M and the blnear form b = θ θ + θσ on T s a conformal nvarant,.e., b = e 2f b, f g = e 2f g on D. Proof. From the defnton we get b = e 2f θ + f θ ) θ + f θ ) + e 2f θ σ 2f θ f f θ ) = e 2f b. Corollary 3.1. When H 2 = Id on E 1 the torus bundle T s a U1) bundle and the blnear form b s a conformally nvarant Lorentz metrc 3.4. Crcle bundles In that secton we suppose that H s constant on T. Consder now g = { X u1) + +u1): σx ) = 0 }. k ω kx ) As σx ) = n+2 1 ν k = 0, G = exp g s a closed subgroup of U1) U1) f and only f there exsts relatvely prme postve ntegers m 1,...,m r such that m 1 ν 1 = m 2 ν 2 = =m r ν r. We may now defne the U1) bundle N = T/G. We have proved Theorem 3.2. The blnear form b descends to a Lorentz metrc on N defned by L = θ θ + θσ whch s conformally nvarant, that s, f g = e 2f g on D then L = e 2f L on N. Acknowledgements The authors thank the Unversty of Pars VI and the Unversty of Pará UFPA) for generous support whle preparng ths work. References [1] D. Burns, K. Dederch, S. Shnder, Dstngushed curves n pseudoconvex boundares, Duke Math. J ) [2] E. Cartan, Sur la géométre pseudo-conforme des hypersurfaces de deux varables complexes, I, Ann. Math. Pura Appl. 4) ) or Ouevres II, 2, ); II, Ann. Scuola Norm. Sup. Psa 2) ) or Ouevres III, 2, ). [3] S.S. Chern, J. Moser, Real hypersurfaces n complex manfolds, Acta Math ) [4] S. Dragomr, Pseudohermtan geometry, Bull. Math. Soc. Sc. Math. Roumane 4393) 3) 2000) [5] E. Falbel, J.M. Veloso, A parallelsm for contact conformal sub-remannan geometry, Forum Mathematcum ) [6] E. Falbel, J.M. Veloso, J.A. Verderes, Constant curvature models n sub-remannan geometry, Matematca Contemp ) [7] F. Farrs, An ntrnsc constructon of Fefferman s CR metrc, Pacfc J. Math ) 1986) [8] C. Fefferman, Monge-Ampère equatons, the Bergman kernel, and the geometry of pseudo-convex domans, Ann. of Math. 2) ) Correcton, Ann. of Math ) [9] M. Kuransh, CR geometry and Cartan geometry, Forum Math. 9 2) 1997)

9 E. Falbel, J.M. Veloso / Dfferental Geometry and ts Applcatons ) [10] J.M. Lee, The Fefferman metrc and pseudohermtan nvarants, Trans. Amer. Math. Soc ) [11] N. Tanaka, On the equvalence problems assocated wth smple graded Le algebras. Graded Le algebras and geometrcal structures, Hokkado Math. J ) [12] S. Webster, Pseudo-hermtan structures on a real hypersurface, J. Dfferental Geom )