Cheeger Gromoll type metrics on the tangent bundle

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1 Cheege Gomoll type metics on the tngent bundle Min Ion MUNTEANU Abstct In this ppe we study Riemnin metic on the tngent bundle T M) of Riemnnin mnifold M which genelizes the Cheege Gomoll metic nd comptible lmost complex stuctue which togethe with the metic confes to T M) stuctue of loclly confoml lmost Kählein mnifold. We found conditions unde which T M) is lmost Kählein, loclly confoml Kählein o Kählein o when T M) hs constnt sectionl cuvtue o constnt scl cuvtue MSC: 53B35, 53C07, 53C25, 53C55. Key wods: Riemnnin mnifold, Sski metic, Cheege Gomoll metic, tngent bundle, loclly confoml lmost) Kählein mnifold. 1 Peliminies Given Riemnnin mnifold M, g) one cn define sevel Riemnnin metics on the tngent bundle T M) of M. Mybe the best known exmple is the Sski metic g S intoduced in [20]. Although the Sski metic is ntully defined, it is vey igid. Fo exmple, the Sski metic is not, genelly, Einstein. O, the tngent bundle T M) with the Sski metic is neve loclly symmetic unless the metic g on the bse mnifold is flt see [12]). E.Musso & F.Ticei [15] hve poved tht the Sski metic hs constnt scl cuvtue if nd only if M, g) is loclly Euclidin. In the sme ppe, they hve given n explicit expession of positive definite Riemnnin metic intoduced by J.Cheege nd D.Gomoll in [9] nd clled this metic the Cheege-Gomoll metic. M.Sekizw see [21]), computed geometic objects elted to this metic. Lte, S.Gudmundson nd E.Kppos in [11], hve completed these esults nd hve shown tht the scl cuvtue of the Cheege Gomoll metic is neve constnt if the metic on the bse mnifold hs constnt sectionl cuvtue. Futhemoe, M.T.K.Abbssi & M.Sih hve poved tht T M) with the Cheege Gomoll metic is neve spce of constnt sectionl cuvtue cf. [2]). It is lso known tht the tngent bundle T M) of Riemnnin mnifold M, g) cn be ognized s n lmost Kählein mnifold see [10]) by using the decomposition of the tngent bundle to T M) into the veticl nd hoizontl distibutions, V T M nd HT M espectively the lst one being defined by the Levi Civit connection on M), the Sski metic nd n lmost complex stuctue defined by the bove splitting. A moe genel metic is given by M.Anstsiei in [6] which genelizes both of the Sski nd Cheege Gomoll metics: it peseves the othogonlity of the veticl 1

2 2 nd hoizontl distibutions, on the hoizontl distibution it is the sme s on the bse mnifold, nd finlly the Sski nd the Cheege Gomoll metic cn be obtined s pticul cses of this metic. A comptible lmost complex stuctue is lso intoduced nd hence T M) becomes loclly confoml lmost Kähein mnifold. On the othe hnd, V.Opoiu nd his collbotos constucted fmily of Riemnnin metics on the tngent bundles of Riemnnin mnifolds which possess inteesting geometic popeties cf. [16, 17, 18, 19]) fo exmple, the scl cuvtue of T M) cn be constnt lso fo non-flt bse mnifold with constnt sectionl cuvtue). Then M.T.K.Abbssi & M.Sih poved in [3] tht the consideed metics by Opoiu fom pticul subclss of the so-clled g-ntul metics on the tngent bundle see lso [1, 3, 4, 5, 13]). In this ppe we descibed fmily g of Riemnnin metics of Cheege Gomoll type, on the tngent bundle T M) of the Riemnnin mnifold M, g) nd comptible lmost complex stuctue J which bestow to T M) stuctue of loclly confoml lmost Kählein mnifold. We found n lmost Kählein stuctue on T M) nd we poved tht thee is no Cheege Gomoll type stuctue on T M) such tht the mnifold T M), g, J ) is Kählein. We studied the possibility fo the sectionl cuvtue on T M) to be constnt nd we found flt metic on T M) of couse of Cheege Gomoll type). Finlly, if M is el spce fom, we wee inteested to find when T M) endowed with the metic g hs constnt scl cuvtue. 2 On the Geomety of the Tngent Bundle T M) Let M, g) be Riemnnin mnifold nd let be its Levi Civit connection. Let τ : T M) M be the tngent bundle. If u T M) it is well known the following decomposition of the tngent spce T u T M) in u t T M)) T u T M) = V u T M) H u T M) whee V u T M) = ke τ,u is the veticl spce nd H u T M) is the hoizontl spce in u obtined by using. A cuve γ : I T M), t γt), V t)) is hoizontl if the vecto field V t) is pllel long γ = γ τ. A vecto on T M) is hoizontl if it is tngent to n hoizontl cuve nd veticl if it is tngent to fibe. Loclly, if U, x i ), i = 1,..., m, m = dim M, is locl cht in p M, conside τ 1 U), x i, y i ) locl cht on T M). If Γ k ij x) e the Chistoffel symbols, then δ i = x i Γ k ij x)yj in u, i = 1,..., m spn the spce H y k u T M), while y, i = 1,..., m spn i the veticl spce V u T M).) We hve obtined the hoizontl veticl) distibution HT M V T M) nd diect sum decomposition T T M = HT M V T M of the tngent bundle of T M). If X χm), denote by X H nd X V, espectively) the hoizontl lift nd the veticl lift, espectively) of X to T M). If u T M) then we conside the enegy density in u on T M), nmely t = 1 2 g τu)u, u).

3 3 2.1 The Cheege-Gomoll Stuctue The Cheege-Gomoll metic on T M) is given by g CGp,u) X H, Y H ) = g p X, Y ), g CGp,u) X H, Y V ) = 0 g CGp,u) X V, Y V ) = t g px, Y ) + g p X, u)g p Y, u)) 2.1) fo ny vectos X nd Y tngent to M. Moeove, n lmost complex stuctue J CG, comptible with the Chegee-Gomoll metic, cn be defined by the fomuls { JCG Xp,u) H = XV 1 1+ g px, u)u V J CG Xp,u) V = 1 XH 1 1+) g 2.2) px, u)u H whee = 1 + 2t nd X T p M). Remk tht J CG u H = u V nd J CG u V = u H. We get n lmost Hemitin mnifold T M), J CG, g CG ). If we denote by Ω CG the Kehle 2-fom nmely Ω CG U, V ) = g CG U, J CG V ), U, V χt M))) one cn pove the following Poposition 2.1 We hve whee ω Λ 1 T M)) is defined by ω p,u) X H ) = 0 nd ω p,u) X V ) = dω CG = ω Ω CG, 2.3) ) g p X, u), X T p M). 1 + Poof. A simple computtion gives the diffeentil of Ω CG : dω CG X H, Y H, Z H ) = dω CG X H, Y) H, Z V ) = dω CG X V, Y V, Z V ) = 0 dω CG X H, Y V, Z V ) = [gx, Y )gz, u) gx, Z)gY, u)] fo ny X, Y, Z χm). Hence the sttement. Remk 2.2 The lmost Hemitin mnifold T M), J CG, g CG ) is neve lmost Kehlein i.e. dω CG 0). Finlly, necessy condition fo the integbility of J CG is tht the bse mnifold M, g) is loclly Euclidin. 2.2 The Cheege Gomoll Type Stuctue A genel metic, let s cll it g, is in fct fmily of Riemnnin metics, depending on pmete, nd the Cheege-Gomoll metic is obtined by tking t) = 1 1+2t. It is defined by the following fomuls see lso [6]) g p,u) X H, Y H ) = g p X, Y ) g p,u) X H, Y V ) = 0 g p,u) X V, Y V ) = t) g p X, Y ) + g p X, u)g p Y, u) ), fo ll X, Y χm), whee : [0, + ) 0, + ). 2.4)

4 4 Poposition 2.3 see lso [14]) The metic defined bove cn be constuct by using the method descibed by Musso nd Ticei in [15]. We intend to find n lmost complex stuctue on T M), cll it J, comptible with the metic g. Inspied fom the pevious cses we look fo the lmost complex stuctue J in the following wy { J Xp,u) H = αxv + βg p X, u)u V J Xp,u) V = 2.5) γxh + ρg p X, u)u H whee X χm) nd α, β, γ nd ρ e smooth functions on T M) which will be detemined fom J 2 = I nd fom the comptibility conditions with the metic g. Following the computtions mde in [6] we get fist α = ± 1 nd γ =. Without lost of the genelity we cn tke α = 1 nd γ =. Then one ) obtins β = ) 2t + ɛ +2bt nd ρ = 1 2t + ɛ + 2bt whee ɛ = ±1. Remk 2.4 In this genel cse J is defined on T M) \ 0 the bundle of non zeo tngent vectos), but if we conside ɛ = 1 the pevious eltions define J on ll T M). Fom now on we will wok with ɛ = 1. We hve the lmost complex stuctue J J X H = 1 X V 1 J X V = ) 1+) gx, u)uv ) X H gx, u)uh. 2.6) One obtins n lmost Hemitin mnifold T M), g, J ). If we denote by Ω the Kähle 2-fom, Ω U, V ) = g U, J V ), U, V χt M)) one obtins Poposition 2.5 see lso [6]) The lmost Hemitin mnifold T M), g, J ) is loclly confoml lmost Kählein, tht is dω = ω Ω 2.7) whee ω is closed nd globlly defined 1 fom on T M) given by ωx H ) = 0 nd ωx V ) = 1 ) gx, u). 1 + As consequence one cn stte the following Theoem 2.6 The lmost Hemitin mnifold T M), g, J ) is lmost Kählein if nd only if t) = const e 1+2t t. 2.8) Poof. The esult is obtined by integting the eqution = We will tke ) = 2e 1 1+ if we sk 0) = 1.

5 5 2.3 The Integbility of J. In ode to hve n integble stuctue J on T M) we hve to compute the Nijenhuis tenso N J of J nd to sk tht it vnishes identiclly. Fo the integbility tenso N J we hve the following eltions N J X H, Y H ) ) = ) ) V gx, u)y gy, u)x + RXY u) V N J X V, Y V ) = R XY u 1+ gy, u)r Xuu + 1+ gx, u)r Y uu ) V gy, 2 1+) ) 1 V u)x gx, u)y. 2.9) The expession fo N J X H, Y V ) is vey complicted. Thus if J is integble then R XY u = ) ) gy, u)x gx, u)y ) fo evey X, Y χm) nd fo evey point u T M). It follows tht M is spce fom Mc) c is the constnt sectionl cuvtue of M). Consequently, ) = e ) ce 2 1) + k1 + )) with k positive el constnt nd c must be nonnegtive. Question: Cn T M), g, J ) be Kähle mnifold? If this hppens then we hve to find n ppopite constnt in 2.8) such tht the expession 2 1+) ) is lso constnt. Theoem 2.7 Thee is no Cheege Gomoll type stuctue on T M) such tht the mnifold T M), g, J ) is Kählein. Now we give Poposition 2.8 Let M, g) be Riemnnin mnifold nd let T M) be its tngent bundle equipped with the metic g. Then, the coesponding Levi Civit connection stisfies the following eltions: X H Y H = X Y ) H 1 2 R XY u) V X H Y V = X Y ) V + 2 R uy X) H X V Y H = 2 R uxy ) H 2.10) X Y V = L gx, u)y V + gy, u)x ) V + 1 L V 2 L gx, u)gy, u)u V, 2 gx, Y )u V whee L = t) 2t).

6 6 Poof. The sttement follows fom Koszul fomul mking usul computtions. Hving detemined Levi Civit connection, we cn compute now the Riemnnin cuvtue tenso R on T M). We give Poposition 2.9 The cuvtue tenso is given by R X H Y H Z H = R XY Z) H + 4 [R ur XZ uy R ury Z ux + 2R urxy uz] H [ ZR) XY u] V R X H Y Z V = [ R H XY Z + 4 R Y R uz Xu R XRuZ Y u) ] V + LgZ, u)rxy u) V L gr 2 XY u, Z)u V + 2 [ XR) uz Y Y R) uz X] H R X H Y Z H = V 2 [ XR) uy Z] H + [ RXZ Y 2 R XR uy Zu + LgY, u)r XZ u + 1 L gr 2 XZ u, Y )u ] V R X H Y Z V = V 2 R Y ZX) H 2 4 R uy R uz X) H + [ gz, u)ruy X) H gy, u)r uz X) H] + 4 R X V Y Z H = R V XY Z) H + 2 [gx, u)r uy Z gy, u)r ux Z] H [R uxr uy Z R uy R ux Z] H 2.11) R X V Y V Z V = F 1 t)gz, u) [ gx, u)y V gy, u)x V ] + +F 2 t) [ gx, Z)Y V gy, Z)X V ] + +F 3 t) [gx, Z)gY, u) gy, Z)gX, u)] u V, whee F 1 = L + L1 L), F 2 2 = L 2 1 L)2 nd F 2 3 = L L L 2. 4 In the following let Q U, V ) denote the sque of the e of the pllelogm with sides U nd V fo U, V χt M)), We hve Q U, V ) = g U, U)g V, V ) g U, V ) 2. Lemm 2.10 Let X, Y T p M be two othonoml vectos. Then { Q X H, Y H ) = 1, Q X H, Y V ) = t) 1 + gy, u) 2) Q X V, Y V ) = t) gx, u) 2 + gy, u) 2). 2.12) We compute now the sectionl cuvtue of the Riemnnin mnifold T M), g ), nmely K U, V ) = g R UV V,U) fo U, V χt M)). Q U,V ) Denote by T 0 M) = T M) \ 0 the tngent bundle of non-zeo vectos tngent to M. Fo given point p, u) T 0 M) conside n othonoml bsis {e i } i=1,m fo the tngent spce T p M) of M such tht e 1 = u u. Conside on T p,u)t M) the following vectos E i = e H i, i = 1, m, E m+1 = 1 ev 1, E m+k = 1 e V k, k = 2, m. 2.13)

7 7 It is esy to check tht {E 1,..., E 2m } is n othonoml bsis in T p,u) T M) with espect to the metic g ). We will wite the expessions of the sectionl cuvtue K in tems of this bsis. We hve K E i, E j ) = Ke i, e j ) 3t) 4 R eie j u 2, K E i, E m+1 ) = 0, K E i, E m+k ) = 1 4 R ue k e i 2, 2.14) K E m+1 E m+k ) = F2+2tF3 t), K E m+k E m+l ) = F 2 t), i, j = 1,..., m; k, l = 2,..., m. Hee denotes the nom of the vecto with espect to the metic g in point). Question: Cn we hve constnt sectionl cuvtue c on T M)? If this hppens, then it must be 0, so T M) is flt. One gets esily tht M is loclly Eucliden. Then, we should lso hve F 2 t) = 0. It follows, F 3 t) = 0 nd F 1 t) = 0. On the othe hnd n odiny diffeentil eqution occus: A simple computtion shows tht t) t) = t. e 2 1+2t t) = 0 1 +, 1 + 2t) 2 0 > ) Remk 2.11 The mnifold T M) equipped with the Cheege Gomoll hs non constnt sectionl cuvtue. Putting 0, such tht 0) = 1 we cn stte the following Theoem 2.12 Conside g 1 on T M) given by g 1 X H, Y H ) = gx, Y ), g 1 X H, Y V ) = 0 ) 2.16) g 1 X V, Y V ) = 4e2 1) 1+) gx, Y ) + gx, u)gy, u) 2 The mnifold T M), g 1 ) is flt. Let us now compe the scl cuvtues of M, g) nd T M), g ). Poposition 2.13 Let M, g) be Riemnnin mnifold nd endow the tngent bundle T M) with the metic g. Let scl nd scl be the scl cuvtues of g nd g espectively. The following eltion holds: scl = scl R ei e j u m mf 2 + 4tF 3 ), 2.17) whee {e i } i=1,...,m is locl othonoml fme on T M). i<j Poof. Using tht scl = i j Ke i, e j ) nd the fomul

8 8 we get the conclusion. m i,j=1 R ei ue j 2 = m i,j=1 R ei e j u 2 Conside M el spce fom with c the constnt sectionl cuvtue. Question: Could we find functions such tht T M) equipped with the metic g hs constnt scl cuvtue? Then T M), g ) hs constnt scl cuvtue if nd only if stisfies the following ODE: t) 2 t) 3 2 m m) t) t) 2 4 t c + 2 c t) 2 t) t c + 2 c t) 2 t) m) t t) t) t) m m) t) t) + 2 t t) t)) ) = const. which seems to be vey complicted to solve. Acknowledgements. This wok ws ptilly suppoted by Gnt CNCSIS 1463/ n.18/2005. Refeences [1] Abbssi M., Note on the Clssifiction Theoems of g-ntul Metics on the Tngent Bundle of Riemnnin Mnifold M, g), Commnet.Mth.Univ.Coline, 45 4)2004), [2] Abbssi M. nd Sih M., Killing Vecto Fields on Tngent Bundles with Cheege Gomoll Metic, Tsukub J. Mth., 272)2003) [3] Abbssi M. nd Sih M., On Some Heedity Popeties of Riemnnin g- Ntul Metics on Tngent Bundles of Riemnnin Mnifolds, Diffeentil Geomety nd Its Applictions ), [4] Abbssi M. nd Sih M., On Ntul Metics on Tngent Bundles of Riemnnin Mnifolds, Achivum Mthemticum Bno), ) 1, [5] Abbssi M. nd Sih M., On Riemnnin g-mtul Metics of the Fom.g s + b.g h + c.g v on the Tngent Bundle of Riemnnin Mnifold M, g), Medite. J. Mth., ), [6] Anstsiei M., Loclly Confoml Kehle Stuctues on Tngent Mnifold of Spce Fom, Libets Mth., ) [7] Bli D., Riemnnin Geomety of Contct nd Symplectic Mnifolds, Pogess in Mthemtics, Bikhäuse Boston, [8] Boeckx E. nd Vnhecke L., Hmonic nd Miniml Vecto Fields on Tngent nd Unit Tngent Bundles, Diffeentil Geomety nd Its Applictions, ),

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