The Holomorphic Φ-Sectional Curvature of Tangent Sphere Bundles with Sasakian Structures

ANALELE S¸TIINT¸IFICE ALE UNIVERSITAT¸II˘ “AL.I. CUZA” DIN IAS¸I (S.N.) MATEMATICA,˘ Tomul LVII, 2011, Supliment DOI: 10.2478/v10157-011-0004-5

THE HOLOMORPHIC φ-SECTIONAL CURVATURE OF TANGENT SPHERE BUNDLES WITH SASAKIAN STRUCTURES

S. L. DRUT¸ A-ROMANIUC˘ and V. OPROIU

Abstract. We study the holomorphic φ-sectional curvature of the natural diagonal tangent sphere bundles with Sasakian structures, determined by Drut¸a-Romaniuc˘ and Oproiu. After finding the explicit expressions for the components of the curvature tensor field and of the curvature tensor field corresponding to the Sasakian space forms, we find that there are no tangent sphere bundles of natural diagonal lift type of constant holomorphic φ-sectional curvature. Mathematics Subject Classification 2000: 53C05, 53C15, 53C55. Key words: natural lift, tangent sphere bundle, Sasakian structure, holomorphic φ-sectional curvature.

1. Introduction The geometry of the tangent sphere bundles of constant radius r, endowed with the metrics induced by some Riemannian metrics from the ambi- ent tangent bundle, was studied by authors like Abbassi, Boeckx, Cal- varuso, Kowalski, Munteanu, Park, Sekigawa, Sekizawa (see [1], [2], [4]-[7], [9], [12]-[14], [16]). In the most part of the papers, such as [4], [5] and [16], the metric considered on the tangent bundle TM was the Sasaki metric, but Boeckx noticed that the unit tangent bundle equipped with the induced Cheeger-Gromoll metric is isometric to the tangent sphere bundle T √1 M with the induced Sasaki metric. Then, in 2000, Kowalski and 2 Sekizawa showed how the geometry of the tangent sphere bundles depends on the radius (see [10]). 76 S. L. DRUT¸ A-ROMANIUC˘ and V. OPROIU 2

The tangent sphere bundles TrM of constant radius r are hypersurfaces of the tangent bundles, consisting of the tangent vectors which have the norm equal to r only. Every almost Hermitian structure on the tangent bundle induces an almost contact structure on the tangent sphere bundle of constant radius r. Important results in this direction may be found in the recent surveys [1] and [9]. In the present paper we study the conditions under which the tangent sphere bundles of natural diagonal lift type with Sasakian structures, determined in [7], have constant holomorphic φ-sectional curvature, i.e. they are Sasakian space forms. We prove that this conditions are never satisfied, so there are no natural diagonal tangent sphere bundles of constant radius r, which are also Sasakian space forms. The manifolds, tensor fields and other geometric objects considered in this paper are assumed to be differentiable of class C∞ (i.e. smooth). The Einstein summation convention is used throughout this paper, the range of the indices h, i, j, k, l, m, r being always {1, . . . , n}.

2. Preliminary results Let (M, g) be a smooth n-dimensional Riemannian manifold and denote its tangent bundle by τ : TM → M. The total space TM has a structure of 2n- dimensional smooth manifold, induced from the smooth manifold structure of M. This structure is obtained by using local charts on TM induced from usual local charts on M. If (U, φ) = (U, x1, . . . , xn) is a local chart on M, then the corresponding induced local chart on TM is (τ −1(U), Φ) = (τ −1(U), x1, . . . , xn, y1, . . . , yn), where the local coordinates xi, yj, i, j = 1, . . . , n, are defined as follows. The first n local coordinates of a tangent vector y ∈ τ −1(U) are the local coordinates in the local chart (U, φ) of its base point, i.e. xi = xi ◦ τ, by an abuse of notation. The last n local coordinates yj, j = 1, . . . , n, of y ∈ τ −1(U) are the vector space coordinates of y with respect to the natural basis in Tτ(y)M defined by the local chart (U, φ). Due to this special structure of differentiable manifold for TM, it is possible to introduce the concept of M-tensor field on it (see [11]). Denote by ∇˙ the Levi Civita connection of the Riemannian metric g on M. Then we have the direct sum decomposition (2.1) TTM = VTM ⊕ HTM

of the tangent bundle to TM into the vertical distribution VTM = Ker τ∗ and the horizontal distribution HTM defined by ∇˙ (see [17]). The vertical 3 THE HOLOMORPHIC φ-SECTIONAL CURVATURE 77 and horizontal lifts of a vector field X on M will be denoted by XV and H { ∂ ∂ } −1 X respectively. The set of vector fields ∂y1 ,..., ∂yn on τ (U) defines a local frame field for VTM, and for HTM we have the local frame field { δ δ } δ ∂ − h ∂ h k h h δx1 ,..., δxn , where δxi = ∂xi Γ0i ∂yh , Γ0i = y Γki, and Γki(x) are the Christoffel symbols of g. { ∂ δ } { } The set ∂yi , δxj i,j=1,n, denoted also by ∂i, δj i,j=1,n, defines a local frame on TM, adapted to the direct sum decomposition (2.1). Extensive li- terature concerning canonical frame structures on TM defined by canonical N-connections can be found in [3]. Consider the energy density of the tangent vector y with respect to the Riemannian metric g 1 1 1 t = ∥y∥2 = g (y, y) = g (x)yiyk, y ∈ τ −1(U). 2 2 τ(y) 2 ik Obviously, we have t ∈ [0, ∞) for every y ∈ TM. The second author considered in [15] an (1,1)-tensor field J on the tangent bundle TM, obtained as natural 1-st order lift (see [8]) of the metric g from the base manifold to the tangent bundle TM:

H V V JXy = a1(t)Xy + b1(t)gτ(y)(X, y)yy , V − H − H JXy = a2(t)Xy b2(t)gτ(y)(X, y)yy , ∀ ∈ T 1 ∀ ∈ X 0 (M), y T M, a1, a2, b1, b2 being smooth functions of the energy density. The above (1,1)-tensor ﬁeld J deﬁnes an almost complex structure on the tangent bundle if and only if

1 b1 a2 = , b2 = − . a1 a1(a1 + 2tb1)

Then it was considered on TM a Riemannian metric Ge of natural diagonal lift type:

e H H G(Xy ,Yy ) = c1(t)gτ(y)(X,Y ) + d1(t)gτ(y)(X, y)gτ(y)(Y, y), e V V (2.2) G(Xy ,Yy ) = c2(t)gτ(y)(X,Y ) + d2(t)gτ(y)(X, y)gτ(y)(Y, y), e V H e H V G(Xy ,Yy ) = G(Xy ,Xy ) = 0, ∀ ∈ T 1 ∀ ∈ X,Y 0 (TM), y TM, where c1, c2, d1, d2 are smooth functions of the energy density on TM. The conditions for Ge to be a Riemannian 78 S. L. DRUT¸ A-ROMANIUC˘ and V. OPROIU 4

metric on TM (i.e. to be positive deﬁnite) are c1 > 0, c2 > 0, c1 + 2td1 > 0, c2 + 2td2 > 0 for every t ≥ 0. The Riemannian metric Ge is almost Hermitian with respect to the almost complex structure J if and only if c c c + 2td c + 2td 1 = 2 = λ, 1 1 = 2 2 = λ + 2tµ, a1 a2 a1 + 2tb1 a2 + 2tb2 where λ > 0, µ > 0 are functions of t. The symmetric matrix of type 2n × 2n ( ) ( ) e(1) G 0 c (t)g + d (t)g g 0 ij = 1 ij 1 0i 0j , e(2) 0 c (t)g + d (t)g g 0 Gij 2 ij 2 0i 0j e { } associated to the metric G in the adapted frame δj, ∂i i,j=1,n, has the inverse ( ) e ( ) Hkl 0 p (t)gkl + q (t)ykyl 0 (1) = 1 1 , e kl 0 p (t)gkl + q (t)ykyl 0 H(2) 2 2 kl where g are the entries of the inverse matrix of (gij)i,j=1,n, and p1, q1, p2, q2, are some real smooth functions of the energy density. More precisely, they may be expressed as rational functions of c1, d1, c2, d2 :

1 1 d1 d2 p1 = , p2 = , q1 = − , q2 = − . c1 c2 c1(c1 + 2td1) c2(c2 + 2td2) Let us recall some results proved in the papers [6] and [7]. Proposition 2.1 ([6]). The Levi-Civita connection ∇e associated to the Riemannian metric Ge from the tangent bundle TM has the form { ∇e V V V ∇e V ∇˙ V V H XV Y = Q(X ,Y ), XH Y = ( X Y ) + P (Y ,X ), ∇e H V H ∇e H ∇˙ H H H XV Y = P (X ,Y ), XH Y = ( X Y ) + S(X ,Y ), ∀ ∈ T 1 X,Y 0 (M), where the M-tensor ﬁelds Q, P, S, have the following components with { } respect to the adapted frame ∂i, δj i,j=1,n: 1 Qh = (∂ Ge(2) + ∂ Ge(2) − ∂ Ge(2))He kh, ij 2 i jk j ik k ij (2) 1 1 (2.3) P h = (∂ Ge(1) + Rl Ge(2))He kh,Sh = − (∂ Ge(2) + Rl Ge(2))He kh, ij 2 i jk 0jk li (1) ij 2 k ij 0ij lk (2) 5 THE HOLOMORPHIC φ-SECTIONAL CURVATURE 79

h Rkij being the components of the curvature tensor field of the Levi Civita ∇˙ h k h connection on the base manifold (M, g) and R0ij = y Rkij. The vector fields Q(XV ,Y V ) and S(XH ,Y H ) are vertically valued, while the vector field P (Y V ,XH ) is horizontally valued.

Using the relations (2.3), we may easily prove that the M-tensor ﬁelds Q, P, S, have invariant expressions of the forms

c′ c′ − 2d Q(XV ,Y V ) = 2 [g(y, X)Y V + g(y, Y )XV ] − 2 2 g(X,Y )yV 2c2 2(c2 + 2td2) c d′ − 2c′ d + 2 2 2 2 g(y, X)g(y, Y )yV , 2c2(c2 + 2td2) c′ d d P (XV ,Y H ) = 1 g(y, X)Y H + 1 g(y, Y )XH + 1 g(X,Y )yH 2c1 2c1 2(c1 + 2td1) c d′ − c′ d − d2 + 1 1 1 1 1 g(y, X)g(y, Y )yH 2c1(c1 + 2td1) c c d − 2 (R(X, y)Y )H − 2 1 g(X,R(Y, y)y)yH , 2c1 2c1(c1 + 2td1) d c′ S(XH ,Y H ) = − 1 [g(y, X) Y V + g(y, Y ) XV ] − 1 g(X,Y ) yV 2c2 2(c2 + 2td2) c d′ − 2d d 1 − 2 1 1 2 g(y, X)g(y, Y )yV − (R(X,Y )y)V , 2c2(c2 + 2td2) 2

∈ T 1 ∈ for every vector fields X,Y 0 (M) and every tangent vector y TM. Since from now on we shall work on the subset TrM of TM consisting of spheres of constant radius r, we shall consider only the tangent vectors r2 y for which the energy density t is equal to 2 , and the coefficients from the definition (2.2) of the metric Ge become constant. So we may consider them constant from the beginning. Then the M-tensor fields involved in the expression of the Levi-Civita connection become simpler:

V V d2 V Q(X ,Y ) = 2 g(X,Y )y , c2 + r d2 d P (XV ,Y H ) = 1 g(Y, y)XH 2c1 d1 − H + 2 [c1g(X,Y ) d1g(X, y)g(Y, y)]y 2c1(c1 + r d1) 80 S. L. DRUT¸ A-ROMANIUC˘ and V. OPROIU 6

− c2 H − c2d1 H (R(X, y)Y ) 2 g(X,R(Y, y)y)y , 2c1 2c1(c1 + r d1) d S(XH ,Y H ) = − 1 [g(X, y)Y V + g(Y, y)XV ] 2c2 d d 1 + 1 2 g(y, X)g(y, Y )yV − (R(X,Y )y)V . c2(c2 + 2td2) 2

3. Sasakian structures on tangent sphere bundles of constant radius r 2 Let TrM = {y ∈ TM : gτ(y)(y, y) = r }, with r ∈ (0, ∞), and the projection τ : TrM → M, τ = τ ◦ i, where i is the inclusion map of TrM into TM. Now we present the construction of a generator system for the tangent bundle to TrM. The horizontal lift of any vector field on M is tangent to TrM, but the vertical lift is not always tangent to TrM. The tangential lift of a vector X to (p, y) ∈ TrM is tangent to TrM and is defined by 1 XT = XV − g (X, y)yV . y y r2 τ(y) y ∈ T The tangential lift of the tangent vector y TrM, vanishes, i.e. yy = 0. T − 1 k The tangent bundle TTrM is spanned by δi and ∂j = ∂j r2 g0jy ∂k, { T } i, j, k = 1, n, although ∂j j=1,n are not independent. They fulfill the relation j T y ∂j = 0,

and in any point y ∈ TrM they span an (n − 1)-dimensional subspace of TyTrM. ′ e Denote by G the metric on TrM induced from the metric G defined on TM. If we consider only the tangent vectors y ∈ TrM, the energy density t r2 defined by them is equal to 2 , and the coefficients from the definition (2.2) e ′ of the metric G become constants. Thus, the metric G from TrM has the form   ′ H H G (Xy ,Yy ) = c1gτ(y)(X,Y ) + d1gτ(y)(X, y)gτ(y)(Y, y), (3.4) G′(XT ,Y T ) = c [g (X,Y ) − 1 g (X, y)g (Y, y)],  y y 2 τ(y) r2 τ(y) τ(y) ′ H T ′ T H G (Xy ,Yy ) = G (Yy ,Xy ) = 0,

for every vector ﬁelds X,Y on M and every tangent vector y, where c1, d1, c2 are constants. The conditions for G to be positive are c1 > 0, c2 > 2 0, c1 + r d1 > 0. 7 THE HOLOMORPHIC φ-SECTIONAL CURVATURE 81

Theorem 3.1 ([7]). The almost contact metric structure (φ, ξ, η, G) on TrM, with φ, ξ, η, and G given respectively by

a (3.5) φXH = a XT , φXT = −a XH + 2 g(X, y)yH , 1 2 r2 1 (3.6) ξ = yH , η(XT ) = 0, η(XH ) = 2αλg(X, y), 2λr2α (3.7) G = αG′

∈ T 1 for every tangent vector ﬁelds X,Y 0 (M), and every tangent vector 2 ∈ c1+r d1 ′ y TrM, where α = 4r2λ2 and the metric G is given by (3.4), is a contact metric structure, and it is Sasakian if and only if the base manifold 2 a1 has constant sectional curvature c = r2 .

Proposition 3.2 ([7]). The Levi-Civita connection ∇, associated to the Riemannian metric G on the tangent sphere bundle TrM of constant radius r has the expression { T h T T h T h ∇ T ∂ = A ∂ , ∇δ ∂ = Γ ∂ + B δh, ∂i j ij h i j ij h ji h h h T ∇ T δj = B δh, ∇δ δj = Γ δh + C ∂ , ∂i ij i ij ij h where the M-tensor ﬁelds involved as coeﬃcients have the expressions:

h − 1 h h − d1 h − h − 1 h Aij = 2 g0jδi ,Cij = (δj g0i δi g0j) R0ij, r 2c2 2 2 h h − 1 h d1 h d1 − 2c1 + r d1 h Bij = Pij 2 gi0P0j = δi g0j + 2 (gij 2 g0ig0j)y r 2c1 2(c1 + r d1) r c1 − c2 h k − c2d1 h k l Rjiky 2 Rikjly y y , 2c1 2c1(c1 + r d1)

i h i h where g0j = gj0 = y gij and P0j = y Pij.

We mention that A and C have these quite simple expressions, since they are the coeﬃcients of the tangential part of the Levi-Civita connection. All h T the terms containing y ∂h = 0 have been cancelled. 82 S. L. DRUT¸ A-ROMANIUC˘ and V. OPROIU 8

The invariant form of the Levi-Civita connection from TrM is   T 1 T ∇ T Y = − g(Y, y)X ,  X 2  r  H d1 H d1 H c2 H ∇ T Y = g(Y, y) X + g(X,Y )y − (R(X, y)Y )  X 2  2c1 2(c1 + d1r ) 2c1  d (2c + r2d )  − 1 1 1 g(X, y)g(Y, y)yH  2 2  2r c1(c1 + r d1)  c d  − 2 1 H 2 g(X,R(Y, y)y)y , 2c1(c1 + r d1)  T T d1 H d1 H ∇ H Y = (∇˙ Y ) + g(X, y)Y + g(X,Y )y  X X 2c 2(c + r2d )  1 1 1  c d (2c + r2d )  − 2 H − 1 1 1 H  (R(Y, y)X) 2 2 g(X, y)g(Y, y)y  2c1 2r c1(c1 + r d1)  c d  − 2 1 H  2 g(Y,R(X, y)y)y ,  2c1(c1 + r d1)  d 1 ∇ H ∇˙ H − 1 T − T − T XH Y = ( X Y ) [g(X, y) Y g(Y, y) X ] (R(X,Y )y) . 2c2 2

In the case where A, B, C are those given in Proposition 3.2, it can be shown (see e.g. [6]) that their covariant derivatives are:   1 ∇˙ Ah = 0, ∇˙ Ch = − ∇˙ Rh yl, k ij k ij 2 k lij c c d ∇˙ h − 2 ∇˙ h l − 2 1 ∇˙ l r h kBij = kRjily 2 kRiljry y y , 2c1 2c1(c1 + r d1)

∇˙ h ∇˙ where kRkij, kRhkij are the usual local coordinate expressions of the covariant derivatives of the curvature tensor field and the Riemann-Christoffel tensor field of the Levi Civita connection ∇˙ from the base manifold M. If ∇˙ h the base manifold (M, g) is locally symmetric then, obviously, kAij = ∇˙ h ∇˙ h 0, kBij = 0, kCij = 0. In the paper [6] we obtained by a standard straightforward computation the horizontal and tangential components of the curvature tensor field K, denoted by sequences of H and T , to indicate horizontal, or tangential argument on a certain position. For example, we may write

T h h T K(δi, δj)∂k = HHTHkijδh + HHTTkij∂h .

The expressions of the non-zero M-tensor ﬁelds which appear as coeﬃcients 9 THE HOLOMORPHIC φ-SECTIONAL CURVATURE 83 are

h h h l − h l h l HHHHkij = Rkij + BliCjk BljCik + BlkR0ij, 1 HHHT h = (∇˙ Rh yl − ∇˙ Rh yl), kij 2 j lik i ljk h c2 ∇˙ h − ∇˙ h l HHTHkij = ( jRikl iRjkl)y 2c1 c2d1 ∇˙ − ∇˙ l m h + 2 ( jRilkm iRjlkm)y y y 2c1(c1 + r d1) h T h h l − h l h l T HHTTkij∂h = (Rkij + CilBkj CjlBki + AlkR0ij)∂h , 1 TTHHh = ∂T Bh − ∂T Bh + BhBl − Bh Bl − (g Bh − g Bh ), kij i jk j ik il jk jl ik r2 i0 jk j0 ik 1 1 TTTT h = (g δh − g δh) + (δhg g − δhg g ), kij r2 jk i ik j r4 j 0i 0k i 0j 0k h c2 ∇˙ h l c2d1 ∇˙ l r h THHHkij = jRkily + 2 jRilkry y y , 2c1 2c1(c1 + r d1) h T T h h l − h l − ∇˙ h l T THHTkij∂h = (∂i Cjk + AilCjk CjlBik jRliky )∂h , h T h l h − l h h −∇˙ h THTHkij = ∂i Bkj + BkjBil AikBlj,THTTkij = jAik = 0.

h T h T We mention that from the ﬁnal values of HHTTkij∂h and THHTkij∂h , h h h obtained by replacing the expressions of Aij,Bij,Cij given in Proposition h T 3.2, we shall eliminate the terms containing y ∂h = 0.

4. The holomorphic φ-sectional curvature In this section we shall study the condition under which the Sasaki manifold (TrM, φ, ξ, η, G), with φ, ξ, η, and G given respectively by (3.5), (3.6), and (3.7), is a Sasaki space form, i.e. it has constant holomorphic φ-sectional curvature. The tensor ﬁeld corresponding to the curvature of the Sasakian space form (TrM, φ, ξ, η, G) is given by:

k + 3 K (X,Y )Z = [G(Y,Z)X − G(X,Z)Y ] 0 4 k − 1 + [η(X)η(Z)Y − η(Y )η(Z)X 4 + G(X,Z)η(Y )ξ − G(Y,Z)η(X)ξ + G(Z, φY )φX − G(Z, φX)φY + 2G(X, φY )φZ], 84 S. L. DRUT¸ A-ROMANIUC˘ and V. OPROIU 10

where k is a constant. { T } With respect to the generator system δi, ∂j i,j=1,...,n, the expressions are

T T T h T T T h K0(∂i , ∂j )∂k = TTTT0kij∂h ,K0(∂i , ∂j )δk = TTHH0kijδh, T T h T h T K0(∂i , δj)∂k = THTH0kijδh,K0(∂i , δj)δk = THHT0kij∂h , T h T h K0(δi, δj)∂k = HHTT0kij∂h ,K0(δi, δj)δk = HHHH0kijδh,

where the M-tensor ﬁelds involved as coeﬃcients have the expressions: k + 3 TTTT h = α(G′(2)δh − G′(2)δh), 0kij 4 kj i ki j − { [ ] h 2 k 1 ′(1) 1 h l − l h − h l − l h TTHH0kij = a2 αG g0i(δj y δjy ) g0j(δi y δiy ) 4 }kl r2 l h − l h + δjδi δiδj , − [ h −k + 3 ′(2) h k 1 1 ′(2) h THTH0kij = αG ki δj + α 2 G ki g0jy 4( ) 4 r( )] 1 1 + G′(2) g yh − δh + 2G′(2) g yh − δh , kj r2 0i i ij r2 0k k − [ h k + 3 ′(1) h k 1 − 2 h THHT0kij = αG kj δi + α 4λ αg0jg0kδi ( 4 ) 4 ] 1 − g yl − δl G′(1)δh + 2a2G′(2)δh , r2 0i i kl j 1 ij k k − 1 HHTT h = αa2(G′(2)δh − G′(2)δh), 0kij 4 1 kj i ki j k + 3 HHHH h = α(G′(1)δh − G′(1)δh) 0kij 4 kj i ki j k − 1 1 + α[4λ2αg (g δh − g δh) + (G′(1)g − G′(1)g )yh]. 4 0k 0i j 0j i r2 ki 0j kj 0i In the sequel we shall prove the main theorem of this paper.

Theorem 4.1. The Sasakian manifold (TrM, φ, ξ, η, G), with φ, ξ, η, and G given respectively by (3.5), (3.6), and (3.7), is never a Sasaki space form.

Proof. We have to study the vanishing conditions for all the components of the diﬀerence K − K0 with respect to the generator system { T } δi, ∂j i,j=1,...,n. 11 THE HOLOMORPHIC φ-SECTIONAL CURVATURE 85

2 h − h − c1(k+3)(c1+r d1) h The component THHTkij THHT0kij contains 16λ2r2 gjkδi 2 h − h c2(k+3)(c1+r d1) h and THTHkij THTH0kij contains the term 16λ2r2 (gik+g0ig0k)δj . Hence a necessary vanishing condition is k = −3. If (TrM, φ, ξ, η, G) is Sasakian, the base manifold has constant sectional curvature c. Taking this into account, and replacing the value −3 of k h − h into the expression of the component HHHHkij HHHH0kij, this will h − h contain the terms c(δi gjk δj gik), which vanish if and only if c = 0 (i.e. the base manifold is ﬂat). In this case, some components of the diﬀerence K −K0 vanish under certain very restrictive conditions, but the component h − h TTTTkij TTTT0kij is never vanishing, since it takes the form [ ] 1 1 1 δh(g − g g ) − δh(g − g g ) ≠ 0. r2 i jk r2 0j 0k j ik r2 0i 0k Thus the theorem is proved.

Acknowledgements. The ﬁrst author was supported by the program POSDRU/89/1.5/S/49944, ”AL.I. Cuza” University of Ia¸si,Romania.

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Received: 18.X.2010 Department of Sciences, Revised: 2.XII.2010 ”AL.I. Cuza” University of Ia¸si, 54 Lascar Catargi Street, Ia¸si, ROMANIA [email protected]

Faculty of Mathematics, ”AL.I. Cuza” University of Ia¸si, Bd. Carol I, No. 11, RO-700506 Ia¸si, ROMANIA [email protected]

and

Institute of Mathematics ”O. Mayer”, Romanian Academy, Ia¸siBranch, ROMANIA [email protected]