ON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS

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1 Mem. Fac. Inegaed As and Sci., Hioshima Univ., Se. IV, Vol , Dec. 00 ON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS YOSHIO AGAOKA *, BYUNG HAK KIM ** AND JIN HYUK CHOI ** *Depamen of Mahemaics, Faculy of Inegaed As and Sciences, Hioshima Univesiy, Higashi-Hioshima , Japan **Depamen of Mahemaics and Insiue of Naual Sciences, Kyung Hee Univesiy, Suwon , Koea Absac Le M be a 3-dimensional almos conac meic manifold saisfying (*)-condiion. We denoe such a manifold by M *. We pove ha if M * is η-einsein, hen M * is eihe Sasaian o cosymplecic manifold, and is a space of consan cuvaue. Consequenly M * is eihe fla o isomeic o he 3-dimensional uni sphee if M * is complee and simply conneced. 1. Inoducion The confomal cuvaue enso C is invaian unde confomal ansfomaions and vanishes idenically fo 3-dimensional manifolds. Using his fac many auhos [1, 3, 4, 6] sudied 3-dimensional almos conac manifolds. In [5], hey inoduced a new class of almos conac manifold M * conaining quasi-sasaian and ans-sasaian sucue. Moeove hey consuced non-ivial examples. In his pape, we sudy a 3- dimensional η-einsein manifold M * by use of he fac ha C vanishes idenically and he special fom of Ricci cuvaue. Consequenly, we pove ha he 3-dimensional η-einsein manifold M * becomes eihe Sasaian o cosymplecic manifold, and is a space of consan cuvaue. In he cosymplecic case, M * is fla, and if M * is Sasaian, complee and simply conneced, hen M * is isomeic o he 3-dimensional uni sphee, ha is M * is eihe fla o isomeic o S 3 (1) unde his opological condiion.. Almos conac meic sucue Le M be an m-dimensional eal diffeeniable manifold of class C coveed by a sysem of coodinae neighbohoods { Ux ; h, in which hee ae given a enso field φ of ype (1,1), a veco field ξ and a 1-fom η saisfying (.1) φ X = X η( X) ξ, φ ξ = 0, η( φx)= 0, η( ξ)= 1 fo any veco field X on M. Such a se of φξη,, is called an almos conac sucue and we call a manifold wih an almos conac sucue an almos conac manifold. In an almos conac manifold, if hee is given a Riemannian meic g such ha Key wods: Confomal cuvaue enso, almos conac meic manifold, space of consan cuvaue. ** This wo was suppoed by ABRL Gan Poj. No. R fom KOSEF and Engineeing Foundaion. Received Ocobe 1 00; Acceped Novembe 1 00

2 30 YOSHIO AGAOKA, BYUNG HAK KIM AND JIN HYUK CHOI fo all veco fields X and Y on M, we say M has an almos conac meic sucue and g is called a compaible meic. Seing Y = ξ, we have immediaely η( X)= g( X, ξ). The fundamenal -fom Φ is defined by Φ( XY, )= g( φxy, ). I is nown ha he almos conac sucue φξη,, is nomal if and only if he Nijenhuis enso g ( φx, φy )= g ( X, Y ) η( X) η( Y) NXY, φφ, ( X,Y) dη( X, Y) ξ = [ ] vanishes, whee [, ] is a bace opeaion and d denoes he exeio deivaive. An almos conac meic sucue ( φξη,,,g) on M is said o be (a) Sasaian if Φ = dη and ( φξη,, ) is nomal, (b) cosymplecic if Φ and η ae closed and ( φξη,, ) is nomal. In [5], one of he pesen auho defined a new class of almos conac meic sucue on M which saisfies (*) d Φ = 0, ξ = λφx X and ( φξη,, ) is nomal fo a smooh funcion λ on M and denoes he Riemannian connecion fo g. Biefly, we denoe such a manifold by M *. I is easily seen ha M * is cosymplecic if λ = 0, and Sasaian if λ is a non-zeo consan. Theoem 1 [5]. On M *, we have (.) (.3) (.4) (.5) ( φ)( YZ, )= λ η( Y) g( XZ, ) η( g( XY, ), X { = { RX, ξ Y Xλ φy λ η YX g XY, ξ, ξλ = 0, = S ξ, X φx λ ( m 1) λ η( X), whee S is he Ricci cuvaue enso and R is he cuvaue enso defined by [ X Y] [ XY, ] RXYZ (, ) =, Z Z dimensional almos conac manifolds Le M * be a 3-dimensional manifold saisfying (*). I is well nown [] ha he confomal cuvaue enso of Weyl vanishes idenically fo 3-dimensional manifolds. Theefoe he cuvaue enso R of a 3- dimensional manifold M * is given by (3.1) RXYZ (, ) = SXZY (, ) SYZX (, ) g( XZQY, ) { g( YZQX, ) g( XZY, ) g ( YZX, ),

3 ON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS 31 whee is he scala cuvaue and Q is defined by g( QX, Y)= S( X, Y). Using (.3), (.5) and (3.1), we have (3.) = SXY, η X φyλ η Y φxλ g XY, X Y. λ λ 3 η η( ) If we subsiue (3.) ino (3.1), hen we ge (3.3) RXYZW (,,, )= g RXYZW (, ), = η( X) (( φ λ) g( Y, W) η( (( φx) λ) g ( Y, W) η( Y) (( φ λ) g( XW, ) η( (( φy) λ) g( XW, ) η( Y) (( φw) λ) g( X, η( W) (( φy) λ) g( X, η( X) (( φw) λ) g ( Y, η( W) (( φx) λ) g ( Y, λ g( XZ, ) g( YW, ) g( YZ, ) g( XW, ) 3λ η( X) ( Y W) η( Y) ( X W) η Z { g, g, η( Y) g( X, η( X) g( Y, η( W). { If we pu Y = ξ in (3.3), hen by (.3) we obain ha is (3.4) (3.5) o in local componens (3.6) whee λ = λ and he indices i, j,, un ove he ange {1,,...,m. Fom (3.5) o (3.6), we can calculae (3.7) whee λ = g i λ i ( Xλ) Φ( Z, W) λ { η( g( X, W) η( W) g( X, = λ { η( g( X, W) η( W) g( X, (( φw) λ) { η( X) η( g( X, (( φ λ) { η( X) η( W) g ( X, W), ( Xλ) Φ( Z, W)= (( φw) λ) { η( X) η( g( X,. Moeove we can easily see ha ( ){ φz λ η X η W g X, W, λ φ λ ( ηη ) φ λ, Φih= h i gi i ηη h gh ij Φij =( λφij) λ Φ i = φλ λ. = 4 λ φ λ, = 0 Lemma. In a 3-dimensional manifold M *, he funcion λ is consan if and only if φx λ fo all X. i

4 3 YOSHIO AGAOKA, BYUNG HAK KIM AND JIN HYUK CHOI If he Ricci cuvaue S on M is of he fom (3.8) SXY (, )= ag( XY, ) bη( X) η( Y), hen M is called an η-einsein space [1,6,7]. If M * is η-einsein, hen we have (3.9) 3a b = and (3.10) a b = 4λ by use of (.1), (3.) and (3.8). Hence we ge a = λ and b = 6λ. Theefoe he Ricci cuvaue S becomes (3.11) = ( ) SXY, λ g XY, 6λ η X η Y. If we pu Y = ξ in (3.11), hen we ge (3.1) = ( 6 ) φx λ λ η X fom (.5) and (3.11). If we se X = ξ in (3.1), hen i gives (3.13) =, 6λ ha is (3.14) ( φx) λ = 0 and ha (3.15) (, )= λ g( XY, ) SXY fom (3.11). We see ha λ is consan fom Lemma and (3.14). Since 3-dimensional Einsein space is a space of consan cuvaue, we obain he following heoem by using Lemma, (3.14) and (3.15). Theoem 3. Le M * be a 3-dimensional η-einsein manifold. Then M * is a space of consan cuvaue. Moeove M * is eihe Sasaian o cosymplecic manifold. In case λ = 0, since M * is a space of consan cuvaue, we have = 0 and hence RXYZ (, ) = 0, ha is M * is fla. On he ohe hand, E. M. Mosal obained he following esul (cf. [7]). Theoem 4. Le M be a complee and simply conneced Sasaian manifold. If M is Einsein and of posiive cuvaue, hen i is isomeic o he uni sphee. If λ is non-zeo consan, hen M * is Sasaian. Theefoe his fac and Theoems 3 and 4 educe

5 ON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS 33 Theoem 5. Le M * be a 3-dimensional η-einsein manifold. Then M * is eihe fla o isomeic o S 3 (1) if M * is complee and simply conneced. Acnowledgemen. The auhos would lie o expess hei hans o he efeee fo his caeful eading and helpful suggesions. Refeences 1. D. E. Blai, Riemannian geomey of conac and symplecic manifolds, PM03, Bihäuse, Belin (00).. B. Y. Chen, Geomey of submanifolds, Macel Dee, New Yo (1973). 3. F. Gouli-Andeou and P. J. Xenos, On a class of 3-dimensional conac meic manifolds, J. Geom., 63 (1998), J. B. Jun, I. B. Kim and U. K. Kim, On 3-dimensional almos conac meic manifolds, Kyungpoo Mah. J., 34 (1994), J. H. Kwon and B. H. Kim, A new class of almos conac Riemannian manifolds, Comm. Koean Mah. Soc., 8 (1993), S. Sasai, Almos conac manifolds, Lecue noes, Mahemaical Insiue, Tohou Univ. 1 (1965). 7. S. Tanno, Pomenades on sphees, Toyo Ins. Tech., Toyo (1996).