Symmetries on unit tangent bundles

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Symmetries on unit tangent bundles

Jong Taek Cho

Department of Mathematics, Chonnam National University, CNU The Institute of Basic Sciences, Kwangju 500-757, Korea

e-mail : [email protected] Sun Hyang Chun

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea e-mail : [email protected]

(2000 Mathematics Subject Classification : 53C15, 53C25, 53D10.)

Abstract. For a Riemannian manifoldM, its unit tangent bundleT1M admits a standard contact metric structure induced from an almost K¨ahler structure of its tangent bundle T M. In this article, we deal with several symmetries onT1M concerned with its contact metric structures.

1 Introduction

The local symmetry (∇R= 0) is one of the fundamental notions in Riemannian geometry. D. E. Blair ([4]) proved that the standard contact Riemannian structure (η, g, ϕ, ξ) of the unit tangent bundleT1M is locally symmetric if and only if the base manifold M is of constant curvature 0 or 2-dimensional and of constant curvature 1. Quite recently, for general contact Riemannian manifolds the first author and E. Boeckx [11] have proved that a locally symmetric contact metric space is either Sasakian and of constant curvature 1 or locally isometric to the unit tangent bundle of a Euclidean space (with its standard contact metric structure). (For Sasakian manifolds, the situation had been clear already for a long time by work of Okumura [23].) These results mean that local symmetry is too strong a condition to impose in contact geometry. In this context, T. Takahashi ([26]) introducedSasakian locally ϕ-symmetric spaces, which may be considered as the analogues of locally Hermi- tian symmetric spaces. He calls a Sasakian manifold locally ϕ-symmetric if the Riemannian curvature tensorRsatisfies (∗): g((∇_XR)(Y, Z)V, U) = 0 for all vector fields X, Y, Z, V and U orthogonal to ξ. He proves that this condition is equivalent to havingϕ-geodesic symmetries which are local automorphisms. Later, it was proved in [6] that the isometry property of theϕ-geodesic symmetry is already suf- Keywords: unit tangent bundles, the characteristic vector field, the flow-invariance, η-parallelity, Einstein-like structures.

153

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ficient. For the broader class of contact metric spaces, we have two generalizations for the notion of localϕ-symmetry. In [2] a contact Riemannian manifold is called locallyϕ-symmetric if it satisfies the condition (∗). In [7] the authors give a differ- ent definition for a locallyϕ-symmetric contact Riemannian manifold: they require the characteristic reflections (i.e., the reflections with respect to the integral curves of ξ) to be local isometries. This geometric definition leads to an infinite number of curvature conditions, including (∗). The symmetry (on a contact Riemannian manifold) defining in [2] is therefore calledlocalϕ-symmetry in the weak sense and the one in [7] is calledlocalϕ-symmetry in the strong sense. They ([7]) proved that T1M is of strong localϕ-symmetry if and only ifM is of constant curvature. This yields that a unit tangent bundleT1M of a space of constant curvature is weakly locallyϕ-symmetric.

Apart from the defining structure tensorsη,g,ϕandξ, two other operators play a fundamental role in contact Riemannian geometry, namely the structural operator h=¹₂L_ξϕand the characteristic Jacobi operator`=R(·, ξ)ξ, whereL_ξ denotes Lie differentiation in the characteristic directionξ. Special properties for the geometry of (M, G) will be reflected in special properties for the contact structure on T₁M and vice versa. In particular, the characteristic vector field ξ on T1M contains crucial information about M. In fact, all the geodesics in M are controlled by the geodesic flow onT1M which is precisely given byξ. An important topic in the study of the contact metric structure on unit tangent bundles has been to determine those Riemannian manifolds (M, G) for which the corresponding contact structure onT1M enjoys symmetry properties related to the geodesic flow.

A first symmetry type for the contact metric structure occurs when the geodesic flow, generated by ξ, leaves some structure tensors invariant. This is always the case for ξ and η since Lξξ = 0 and Lξη = 0. The metric g is left invariant by the flow ofξ (or equivalently, the flow consists of local isometries or ξ is a Killing vector field) if and only if the structural operator h vanishes. By definition, this corresponds precisely toLξϕ= 0, i.e., alsoϕis preserved under the geodesic flow.

Y. Tashiro proved in [27] that this happens for a unit tangent bundle (T1M;η, g, ϕ, ξ) if and only if (M, G) has constant curvature c = 1. In [12], we proved thatT1M satisfies the condition Lξh= 0 (Lξh⁰ = 0, respectively) if and only if (M, g) is of constant curvature c = 1 (c =−1,0,1, respectively, where h⁰ = ∇ξhdenotes the characteristic derivative) orT1M satisfies the conditionLξl= 0 if and only if (M, g) is of constant curvaturec= 0 or 1.

A second type of symmetry occurs when some structure tensors are covariantly parallel along the integral curves ofξ. On a contact metric space, it always holds

∇ξξ=∇ξη=∇ξg=∇ξϕ= 0, but the other structure tensors need not be parallel in theξ-direction. Recently, it was proved thatT1M satisfies the condition∇ξh= 0 or, equivalently, ∇ξ` = 0, if and only if (M, G) is of constant curvature c = 0 or c= 1 ([24], [25]). This result can easily be verified from the formulas in this paper.

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A final type of symmetry on contact Riemannian manifolds is the notion of η-parallelity. We call a (1,1)-tensorT η-parallel ifg((∇XT)Y, Z) = 0 for all vector fields X, Y, Z orthogonal toξ. A weak local ϕ-symmetry mentioned above means theη-paralleity of the curvature tensor R. In particular, the tensor ϕisη-parallel if and only if the contact structure is CR-integrable, (∇_Xϕ)Y =g(X+hX, Y)ξ− η(Y)(X+hX). On a unit tangent bundleT₁M, this occurs if and only if (M, G) is of constant curvature, as can easily be verified from the formulas further on. Contact metric spaces withη-parallel structural operatorhwere completely classified by the first author and E. Boeckx in [10]. As a corollary of that result, we obtain also that the standard contact Riemannian structure (η, g, ϕ, ξ) of a unit tangent bundle T1M hasη-parallel hif and only if (M, G) is of constant curvature. In the second part of Section , we determine the base manifold when the characteristic Jacobi operator `onT1M isη-parallel.

One may consider the Ricci-parallel condition (∇ρ= 0) which is deduced by the local symmetry. In fact, in [7] they determined the unit tangent bundle with the parallel Ricci tensor. There are two natural generalizations of a Ricci-parallel space, one is a space of Codazzi-type Ricci tensor ((∇Xρ)(Y, Z) = (∇Yρ)(X, Z) for all vector fieldsX, Y, Zon the manifold) the other is a space of cyclic parallel Ricci tensor ((∇_Xρ)(Y, Z) + (∇_Yρ)(Z, X) + (∇_Zρ)(X, Y) = 0 for all vector fieldsX, Y, Z on the manifold). In [8] they dealt with the two conditions for T₁M(c) of a space of constant curvaturec. In Section , we do the corresponding study forT₁M_n^C(4c) of ann-dimensional complex space formM_n^C(4c) of constant holomorphic sectional curvature 4c. Indeed, we prove that T1M_n^C(4c) (n ≥ 2) has Codazzi-type Ricci tensor if and only if c = 0, in this caseT1M_n^C(4c) is locally symmetric. Also, we prove thatT1M_n^C(4c) (n≥2) has a cyclic parallel Ricci tensor if and only ifc= 0 orc= 1. This yields thatT1CPn(4) of a complex projective spaceCPn(4) with the Fubini-Study metric is a proper example whose Ricci tensor is cyclic parallel and is not Ricci-parallel.

2 Preliminaries

All manifolds in the present paper are assumed to be connected and of class C^∞. We start by collecting some fundamental material about contact metric geometry. We refer to [3] for further details. A (2n+ 1)-dimensional manifoldM²ⁿ⁺¹ is said to be a contact manifold if it admits a global 1-formηsuch thatη∧(dη)ⁿ6= 0 everywhere. Given a contact formη, we have a unique vector fieldξ, thecharacter- istic vector field, satisfyingη(ξ) = 1 anddη(ξ, X) = 0 for any vector fieldX. It is well-known that there exists a Riemannian metricg and a (1,1)-tensor fieldϕsuch that

(1) η(X) =g(X, ξ), dη(X, Y) =g(X, ϕY), ϕ²X=−X+η(X)ξ

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whereX andY are vector fields onM. From (1) it follows that (2) ϕξ= 0, η◦ϕ= 0, g(ϕX, ϕY) =g(X, Y)−η(X)η(Y).

A Riemannian manifoldM equipped with structure tensors (η, g, ϕ, ξ) satisfying (1) is said to be a contact Riemannian manifold and is denoted byM = (M;η, g, ϕ, ξ).

Given a contact Riemannian manifold M, we define the structural operator h by h = ¹₂Lξϕ, where L denotes Lie differentiation. Then we may observe that h is symmetric and satisfies

hξ= 0 and hϕ=−ϕh, (3)

∇Xξ=−ϕX−ϕhX (4)

where∇is the Levi-Civita connection. From (3) and (4) we see that each trajectory ofξis a geodesic. We denote byRthe Riemannian curvature tensor defined by

R(X, Y)Z=∇X(∇YZ)− ∇Y(∇XZ)− ∇[X,Y]Z

for all vector fieldsX, Y, Z. Along a trajectory ofξ, the Jacobi operator`=R(·, ξ)ξ is a symmetric (1,1)-tensor field. We call it the characteristic Jacobi operator. We have

∇ξh=ϕ−ϕ`−ϕh², (5)

`=ϕ`ϕ−2(h²+ϕ²).

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A contact Riemannian manifold for whichξis Killing is called aK-contact manifold.

It is easy to see that a contact Riemannian manifold isK-contact if and only ifh= 0 or, equivalently,`=I−η⊗ξ.

3 The standard contact metric structure for the unit tangent bundles The basic facts and fundamental formulae about tangent bundles are well-known (cf. [18], [21], [29]). We only briefly review some notations and definitions. Let M = (M, G) be an n-dimensional Riemannian manifold and let T M denote its tangent bundle with the projection π : T M → M, π(x, u) = x. For a vector X ∈ TxM, we denote by X^H and X^V, the horizontal lift and the vertical lift, respectively. Then we can define a Riemannian metric ˜g, theSasaki metriconT M, in a natural way by

˜

g(X^H, Y^H) = ˜g(X^V, Y^V) =G(X, Y)◦π, ˜g(X^H, Y^V) = 0

for all vector fieldsX andY onM. Also, a natural almost complex structure tensor J of T M is defined by J X^H = X^V and J X^V = −X^H. Then we easily see that (T M; ˜g, J) is an almost Hermitian manifold. We note that J is integrable if and only if (M, G) is locally flat ([18]).

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Now we consider the unit tangent bundle (T1M,¯g), which is an isometrically embedded hypersurface in (T M,˜g) with unit normal vector field N = u^V. For X ∈T_xM, we define thetangential liftofX to (x, u)∈T₁M by

X_(x,u)^T =X_(x,u)^V −G(X, u)N(x,u).

Clearly, the tangent space T(x,u)T1M is spanned by vectors of the form X^H and X^T whereX ∈TxM. We put

ξ¯=−J N, ϕ¯=J−η¯⊗N.

Then we find ¯g(X,ϕY¯ ) = 2d¯η(X, Y). By taking ξ = 2 ¯ξ, η = ¹₂η,¯ ϕ = ¯ϕ, and g = ¹₄¯g, we get the standard contact Riemannian structure (ϕ, ξ, η, g). Indeed, we easily check that these tensors satisfy (1). Here we notice that ξ determines the geodesic flow. The tensorsξ andϕare explicitly given by

(7) ξ= 2u^H, ϕX^T =−X^H+1

2G(X, u)ξ, ϕX^H =X^T

where X and Y are vector fields on M. From now on, we consider T₁M = (T₁M;η, g) with the standard contact Riemannian structure. We list the fundamental formulae which we need for the proof of our theorems. They are derived in, e.g., [4], [7], [9], [24], [27]. The Levi-Civita connection ∇of (T₁M, g) is given by

∇_XTY^T =−G(Y, u)X^T,

∇_XTY^H= 1

2(K(u, X)Y)^H,

∇_XHY^T = (DXY)^T +1

2(K(u, Y)X)^H,

∇_XHY^H= (DXY)^H−1

2(K(X, Y)u)^T. (8)

For the Riemann curvature tensorR, we give only the two expressions we need for

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the characteristic Jacobi operator`:

R(X^T, Y^T)Z^T =−g(X^T, Z^T)Y^T+g(Z^T, Y^T)X^T, R(X^T, Y^T)Z^H =

K(X−G(X, u)u, Y −G(Y, u)u)Z ^H +1

4

[K(u, X), K(u, Y)]Z ^H R(X^H, Y^T)Z^T =−1

2

K(Y −G(Y, u)u, Z−G(Z, u)u)X}^H

−1

4{K(u, Y)K(u, Z)X ^H R(X^H, Y^T)Z^H =1

2

K(X, Z)(Y −G(Y, u)u)}^T −1 4

K(X, K(u, Y)Z)u ^T +1

2

(DXK)(u, Y)Z ^H, R(X^H, Y^H)Z^T =

K(X, Y)(Z−G(Z, u)u) ^T +1

4

K(Y, K(u, Z)X)u−K(X, K(u, Z)Y)u ^T +1

2

(DXK)(u, Z)Y −(DYK)(u, Z)X ^H, R(X^H, Y^H)Z^H = (K(X, Y)Z)^H+1

2

K(u, K(X, Y)u)Z ^H

−1 4

K(u, K(Y, Z)u)X−K(u, K(X, Z)u)Y ^H +1

2

(DZK)(X, Y)u ^T (9)

for all vector fieldsX,Y andZ onM.

In the above, we denote byDthe Levi-Civita connection and byKthe Riemann- ian curvature tensor associated withG. From (7) and (8), it follows

(10) ∇_XTξ=−2ϕX^T −(KuX)^H, ∇_XHξ=−(KuX)^T

whereKu=K(·, u)uis the Jacobi operator associated with the unit vectoru. From (4) and (10), it follows that

hX^T =X^T −(K_uX)^T, hX^H =−X^H+1

2G(X, u)ξ+ (K_uX)^H. (11)

Using the formulae (9), we get

`X^T = (K_u²X)^T + 2(K_u⁰X)^H,

`X^H = 4(K_uX)^H−3(K_u²X)^H+ 2(K_u⁰X)^T (12)

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whereK_u⁰ = (DuK)(·, u)uandK_u²=K(K(·, u)u, u)u. By using (5), (7) and (9) we obtain

h⁰X^T =−2(K_uX)^H+ 2(K_u²X)^H−2(K_u⁰X)^T, h⁰X^H=−2(KuX)^T + 2(K_u²X)^T + 2(K_u⁰X)^H (13)

where we puth⁰=∇ξh.

The above formulae (10)–(13) are also found in [9]. Finally, from (8) and (12) we compute

`⁰X^T = 4(K_u⁰KuX+KuK_u⁰X)^T + 4(K_u⁰⁰X+K_u²X−K_u³X)^H,

`⁰X^H= 8(K_u⁰X−K_u⁰K_uX−K_uK_u⁰X)^H+ 4(K_u⁰⁰X+K_u²X−K_u³X)^T (14)

where `⁰= (∇_ξR)(·, ξ)ξ.

4 Invariance under the geodesic flow and the η-parallelity

By using the fundamental formulae (10)–(14), we have the following theorems ([12]):

Theorem 1. Let T1M be the unit tangent bundle with the standard contact Riemannian structure(η, g, ϕ, ξ). ThenT1M satisfiesLξh= 0if and only if(M, G) is of constant curvature c= 1.

Theorem 2. Let T₁M be the unit tangent bundle with the standard contact Riemannian structure(η, g, ϕ, ξ). ThenT1M satisfiesLξ`= 0if and only if(M, G) is of constant curvature c= 0orc= 1.

Theorem 3. Let T₁M be the unit tangent bundle with the standard contact Riemannian structure(η, g, ϕ, ξ). ThenT₁M satisfiesL_ξh⁰= 0if and only if(M, G) is of constant curvature c=−1,c= 0 orc= 1.

We omit the detailed proofs of the above theorems. Instead of it, we note here that when (M, G) is of constant curvaturecthe following explicit expressions forh,

`,h⁰ and `⁰ from (11)–(14) are obtained:

hX^T = (1−c)X^T, hX^H = (c−1)(X^H−1

2G(X, u)ξ),

`X^T =c²X^T, `X^H= (4c−3c²)(X^H−1

2G(X, u)ξ), h⁰X^T = 2(c²−c)(X^H−1

2G(X, u)ξ), h⁰X^H= 2(c²−c)X^T,

`⁰X^T = 4(c²−c³)(X^H−1

2G(X, u)ξ), `⁰X^H= 4(c²−c³)X^T (15)

for vector fieldsX onM.

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If the characteristic Jacobi operator ` of a given contact Riemannian manifold satisfiesg((∇X¯`) ¯Y ,Z) = 0 for all vector fields ¯¯ X,Y¯ and ¯Z orthogonal toξ, then we say that`isη-parallel. Now, we determine the unit tangent bundles withη-parallel characteristic Jacobi operator by giving a short proof of:

Theorem 4. ([12]) Let T1M be the unit tangent bundle with the standard contact Riemannian structure(η, g, ϕ, ξ). Then the characteristic Jacobi operator`ofT1M isη-parallel if and only if(M, G) is of constant curvature.

Proof. From (8) and (12), we first compute the covariant derivatives of the characteristic Jacobi operator`:

(∇_XH`)Y^H=

4(D_XK)(Y, u)u−3(D_XK)(K_uY, u)u

−3Ku((DXK)(Y, u)u) +K(u, K_u⁰Y)X +K_u⁰(K(X, Y)u)^H

+

−2K(X, KuY)u+3

2K(X, K_u²Y)u + 2(D²_XuK)(Y, u)u+1

2K_u²(K(X, Y)u)T

, (16)

(∇_XH`)Y^T =1

2K(u, K_u²Y)X+ 2(D_Xu² K)(Y, u)u

−2Ku(K(u, Y)X) +3

2K_u²(K(u, Y)X)H

+

(D_XK)(K_uY, u)u+K_u((D_XK)(Y, u)u)

−K(X, K_u⁰Y)u−K_u⁰(K(u, Y)X)^T , (17)

(∇_XT`)Y^H =

4K(Y, X)u+ 4K(Y, u)X

+ 2K(u, X)KuY −3Ku(K(Y, X)u)

−3K_u(K(Y, u)X)−3K(K_uY, X)u

−3K(KuY, u)X−3

2K(u, X)K_u²Y

−2K_u(K(u, X)Y) +3

2K_u²(K(u, X)Y)^H +

2(DXK)(Y, u)u+ 2(DuK)(Y, X)u + 2(DuK)(Y, u)X−K_u⁰(K(u, X)Y)T

+G(X, u)

12(K_u²Y)^H−8(KuY)^H−6(K_u⁰Y)^T , (18)

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(∇_XT`)Y^T =

2(DXK)(Y, u)u+ 2(DuK)(Y, X)u + 2(D_uK)(Y, u)X+K(u, X)K_u⁰YH

+

K_u(K(Y, X)u) +K_u(K(Y, u)X) +K(K_uY, X)u+K(K_uY, u)X^T

−G(X, u)

4(K_u²Y)^T + 6(K_u⁰Y)^H +G(Y, u)

(K_u²X)^T + 2(K_u⁰X)^H . (19)

Now, we suppose that the Jacobi operator ` of T1M is η-parallel. Then from (16)–(19), we obtain several equations containing the curvature tensor K and it’s covariant derivatives. Then combining the equations obtained and using the 1st Bianchi identity, we can derive

0 = 2G((DXK)(Y, u)u, Z) + 2G((DuK)(Y, X)u, Z) + 4G((D_uK)(Y, u)X, Z)−G(K(K_u⁰Y, u)X, Z) + 2G((DYK)(X, u)u, Z) + 2G((DuK)(X, u)Y, Z).

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We suppose thatX =Y =Z are orthogonal tou. Then from (20), we find that (DXK)(·, X)X = 0 for all tangent vectorsX. From this, we conclude that the base manifold is locally symmetric. In the case when dim M = 2, we at once see that M is of constant curvature. Furthermore, using the fact that the base manifoldM is locally symmetric, we obtain

0 = 8G(K(Y, X)u, Z) + 8G(K(Y, u)X, Z)

−G(K_u²(K(Z, Y)u), X)−G(K(u, K_u²X)Y, Z).

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In (21), we putY =Z. ThenG(K(Y, X)Y, u) = 0 for any orthogonal tripleu, X, Y. By Cartan’s theorem ([14]), the base manifold (M, G) must have constant curvature if dimM ≥3. We conclude thatM is of constant curvature for all dimensions.

Conversely, we can use the expressions (15) to show that T1M has η-parallel characteristic Jacobi-operator`when the manifoldM is of constant curvaturec.

5 Einstein-like unit tangent bundles

By making use of the decomposition of the covariant derivative∇ρof the Ricci (0,2)-tensorρ, A. Gray([20]) introduced two interesting classesAandBof Riemann- ian manifolds which lie between the class of Ricci-parallel Riemannian manifolds and the one of Riemannian manifolds of constant scalar curvature, namely,

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1. the classAof Riemannian manifolds whose Ricci tensors are Codazzi tensors, i.e., (∇Xρ)(Y, Z) = (∇Yρ)(X, Z) for all vector fieldsX, Y, Zon the manifold.

2. the classBof Riemannian manifolds whose Ricci tensors are Cyclic parallel (or Killing tensors), i.e., (∇Xρ)(Y, Z) + (∇Yρ)(Z, X) + (∇Zρ)(X, Y) = 0 for all vector fieldsX, Y, Z on the manifold.

In ([7]) it was proved that the unit tangent bundlesT1M(c) of spaces of constant curvaturechas a Codazzi-type Ricci tensor if and only ifc= 0 or the base manifold is 2-dimensional and c= 1, i.e., if and only if T1M(c) is locally symmetric. Also, they proved that T1M(c) has a cyclic parallel Ricci tensor if and only ifM is 2- dimensional or c∈ {0,1}. In the former paper ([17]), we prove the corresponding results in the case that the base manifold is a complex space form. But, among the proofs of main Theorems A and B, we have some mistakes including the dimensional argument. Here, we state the corrected results:

Theorem 5. LetM_n^C(4c) be ann(≥2)-dimensional complex space form with constant holomorphic sectional curvature4c. ThenT1M_n^C(4c)is class ofAif and only if c= 0, in this caseM_n^C(4c)is locally symmetric.

Theorem 6. LetM_n^C(4c) be ann(≥2)-dimensional complex space form with constant holomorphic sectional curvature4c. ThenT1M_n^C(4c)is of classBif and only if c= 0 orc= 1.

In the above Theorem 6, T1CPn(4) of a complex projective spaceCPn(4) with the Fubini-Study metric is a proper example whose Ricci tensor is cyclic parallel and is not Ricci-parallel.

Remark 1. In the case of (real) 2-dimensional base manifoldM, we already know (cf. [15]) that T₁M ∈ Aif and only ifM is of constant curvature 0 or 1, or that T₁M ∈Bif and only ifM is of constant curvature.

From now, we right down the corrected version the proof. LetMnbe a complex n-dimensional K¨ahlerian manifold with almost complex structureJ and metricG.

For each 2-plane p in the tangent space Tx(M), the sectional curvature K(p) is defined by

K(p) =G(K(X, Y)Y, X),

where {X, Y} is an orthonormal basis for p. If p is invariant by J, then K(p) is called the holomorphic sectional curvature determined by p. The holomorphic sectional curvatureK(p) is given by

K(p) =G(K(X, J X)J X, X),

whereX is a unit vector inp. IfK(p) is a constant for all holomorphic planespin Tx(M) and for any pointx∈M, thenM is called aspace of constant holomorphic sectional curvatureor simply, acomplex space form. (Sometimes, a complex space form is defined as a simply connected and complete one.) Then it is well-known that

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the curvature tensor of a complex space form is expressed in a nice form. Namely, we have (cf. [30])

Proposition 7. A K¨ahlerian manifold M_n^C(4c) is of constant holomorphic sectional curvature4c (i.e.,M_n^C(4c)is a complex space form of constant holomorphic sectional curvature4c)if and only if

K(X, Y)Z=c

G(Y, Z)X−G(X, Z)Y

+G(J Y, Z)J X−G(J X, Z)J Y + 2G(X, J Y)J Z (22)

for any vector fields X, Y andZ on M.

Let (M_n^C(4c), G) be an-dimensional complex space form of constant holomorphic sectional curvature 4c. We consider the unit tangent bundle (T₁M_n^C(4c), g) of a complex space formM_n^C(4c). We compute the Levi Civita connection∇ and the Riemannian curvature tensor R of (T1M_n^C(4c), g). Namely, from (8), (9) and (22), we obtain

∇X^TY^T =−G(Y, u)X^T,

∇_XTY^H= c 2

n1

2G(X, Y)ξ−G(Y, u)X^H+G(J X, Y)(J u)^H +G(J Y, u)(J X)^H+ 2G(J X, u)(J Y)^Ho

,

∇_XHY^T = (D_XY)^T+ c 2

n1

2G(X, Y)ξ−G(X, u)Y^H+G(X, J Y)(J u)^H + 2G(J Y, u)(J X)^H+G(J X, u)(J Y)^Ho

,

∇_XHY^H= (DXY)^H−c 2

n

G(Y, u)X^T −G(X, u)Y^T + 2G(X, J Y)(J u)^T +G(J Y, u)(J X)^T−G(J X, u)(J Y)^To (23)

and further we have

R(X^T, Y^T)Z^T = (G(Y, Z)−G(Y, u)G(Z, u))X^T

−(G(X, Z)−G(X, u)G(Z, u))Y^T, (24)

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R(X^T, Y^T)Z^H = c−c²

4 n

(G(Y, Z)−G(Y, u)G(Z, u))

X^H−1

2G(X, u)ξ

−(G(X, Z)−G(X, u)G(Z, u)) Y^H−1

2G(Y, u)ξ

+ (G(J Y, Z) +G(Y, u)G(J Z, u))((J X)^H−G(X, u)(J u)^H)

−(G(J X, Z) +G(X, u)G(J Z, u))((J Y)^H−G(Y, u)(J u)^H)o + 2c

G(X, J Y)−G(X, u)G(J Y, u) +G(X, u)G(J X, u) (J Z)^H +c²

4 n1

2 G(J X, Z)G(J Y, u)−G(J Y, Z)G(J X, u)

−2G(J X, Y)G(J Z, u) ξ

−G(J Y, u)G(J Z, u)X^H+G(J X, u)G(J Z, u)Y^H + G(Y, Z)G(J X, u)−G(X, Z)G(J Y, u)

−2G(J X, Y)G(Z, u) (J u)^H

−G(J Y, u)G(Z, u)(J X)^H+G(J X, u)G(Z, u)(J Y)^Ho R(X^H, Y^T)Z^T =c²

4 −c 2

(G(X, Z)−G(X, u)G(Z, u)) Y^H−1

2G(Y, u)ξ +c

2(G(X, Y)−G(X, u)G(Y, u)) Z^H−1

2G(Z, u)ξ +c²

4 −c 2

(G(X, J Z) +G(J X, u)G(Z, u))((J Y)^H−G(Y, u)(J u)^H) +c

2(G(X, J Y) +G(J X, u)G(Y, u))((J Z)^H−G(Z, u)(J u)^H) +c²

8G(X, u)(G(Y, Z)−G(Y, u)G(Z, u))ξ

−c²

4G(J X, u)(G(Y, Z)−G(Y, u)G(Z, u))(J u)^H

−c(G(Y, J Z) +G(J Y, u)G(Z, u)−G(Y, u)G(J Z, u))(J X)^H

−c² 4

n1

2 −3G(X, J Z)G(J Y, u) +G(J X, u)G(Y, J Z) + 2G(J X, Y)G(J Z, u)

ξ

−4G(J Y, u)G(J Z, u)X^H+ 3G(J X, u)G(J Z, u)Y^H

+ 3G(X, Z)G(J Y, u)−G(J Y, Z)G(X, u) + 2G(X, Y)G(J Z, u) (J u)^H

−3G(X, u)G(J Z, u)(J Y)^H−2G(J X, u)G(J Y, u)Z^H

−2G(X, u)G(J Y, u)(J Z)^Ho

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R(X^H, Y^T)Z^H =c 2 −c²

4

(G(Y, Z)−G(Y, u)G(Z, u))X^T

−c²

4G(X, u)G(Z, u)Y^T

−c

2(G(X, Y)−G(X, u)G(Y, u))Z^T +c

2−c² 4

(G(Y, J Z)−G(Y, u)G(J Z, u))(J X)^T

−c²

4G(J X, u)G(Z, u)(J Y)^T

−c

2(G(J X, Y)−G(J X, u)G(Y, u))(J Z)^T

−3c²

4 G(J Y, u)G(J Z, u)X^T −c²

4G(J X, u)G(J Z, u)Y^T

−c²

2G(J X, u)G(J Y, u)Z^T +3c²

4 G(J Y, u)G(Z, u)(J X)^T +

c G(X, J Z) +c²

4G(X, u)G(J Z, u) (J Y)^T +c²

2G(X, u)G(J Y, u)(J Z)^T −c G(X, J Z)G(Y, u)(J u)^T +c²

4

n2G(X, Y)G(J Z, u) + 4G(X, Z)G(J Y, u) + 3G(Y, Z)G(J X, u) + 2G(X, J Y)G(Z, u) + 3G(J Y, Z)G(X, u)o

(J u)^T R(X^H, Y^H)Z^T =

c−c² 4

(G(Y, Z)−G(Y, u)G(Z, u))X^T

−(G(X, Z)−G(X, u)G(Z, u))Y^T +

c−c² 4

(G(J Y, Z)−G(J Y, u)G(Z, u))(J X)^T

−(G(J X, Z)−G(J X, u)G(Z, u))(J Y)^T + 2c G(X, J Y)(J Z)^T −2c G(X, J Y)G(Z, u)(J u)^T +c²

4 n

G(J X, u)G(J Z, u)Y^T −G(J Y, u)G(J Z, u)X^T

−G(X, u)G(J Z, u)(J Y)^T+G(Y, u)G(J Z, u)(J X)^T

−(G(X, Z)G(J Y, u)−G(Y, Z)G(J X, u) +G(X, J Z)G(Y, u)−G(Y, J Z)G(X, u))(J u)^T + 2(G(X, u)G(J Y, u)−G(Y, u)G(J X, u))(J Z)^To

(14)

R(X^H, Y^H)Z^H=

c G(Y, Z)−3c²

4 G(Y, u)G(Z, u) X^H

−

c G(X, Z)−3c²

4 G(X, u)G(Z, u) Y^H +3c²

8

G(X, Z)G(Y, u)−G(Y, Z)G(X, u) ξ +

c G(J Y, Z)−3c²

4 G(J Y, u)G(Z, u) (J X)^H

−

c G(J X, Z)−3c²

4 G(J X, u)G(Z, u) (J Y)^H +3c²

4

G(Y, Z)G(J X, u)−G(X, Z)G(J Y, u) (J u)^H

−3c²

4 G(J Y, u)G(J Z, u)X^H+3c²

4 G(J X, u)G(J Z, u)Y^H

−nc²

2G(J Y, u)G(Z, u)−5c²

4 G(Y, u)G(J Z, u)

−c²G(Y, J Z)o (J X)^H +nc²

2G(J X, u)G(Z, u)−5c²

4 G(X, u)G(J Z, u)

−c²G(X, J Z)o (J Y)^H

−n5c²

2 G(X, u)G(J Y, u)−5c²

2 G(J X, u)G(Y, u)

−(2c−2c²)G(X, J Y)o (J Z)^H

−n5c²

8 G(J Y, Z)G(J X, u)−5c²

8 G(J X, Z)G(J Y, u)

−5c²

4 G(J X, Y)G(J Z, u)o ξ

−n5c²

4 G(J Y, Z)G(X, u)−5c²

4 G(J X, Z)G(Y, u)

−5c²

2 G(J X, Y)G(Z, u)o (J u)^H

Next, we determine the Ricci tensorρof (T1M_n^C(4c), g) and its first covariant derivative. To calculate these tensors at the point (x, u)∈T1Mn(4c), let E1,· · ·, En = u, J E1,· · · , J En = J u be an orthonormal basis of TxM. Then 2E₁^T,· · ·,2E_n−1^T , 2(J E₁)^T,· · ·,2(J E_n)^T,2E₁^H,· · · ,2E_n^H=ξ,2(J E₁)^H,· · · ,2(J E_n)^His an orthonor-

(15)

mal basis forT(x,u)T1M. Thenρis given by ρ(X, Y) =

n−1

X

i=1

R(2E_i^T, X, Y,2E_i^T) +

n

X

i=1

R(2(J E_i)^T, X, Y,2(J E_i)^T)

+

n

X

i=1

R(2E_i^H, X, Y,2E_i^H) +

n

X

i=1

R(2(J E_i)^H, X, Y,2(J E_i)^H) (25)

Thus by using (24) we see that

ρ(X^T, Y^T) = (2n−2 +c²)(G(X, Y)−G(X, u)G(Y, u)) +c²(2n+ 5)G(J X, u)G(J Y, u),

ρ(X^H, Y^H) =c(2n+ 2−3c)G(X, Y)−c²(n+ 4)G(X, u)G(Y, u)

−c²(n+ 4)G(J X, u)G(J Y, u), ρ(X^T, Y^H) = 0.

(26)

Furthermore, by using (2) we obtain

(∇Z^Tρ)(X^T, Y^T) =c²(2n+ 5)

(G(J X, Z)−G(J X, u)G(Z, u) +G(X, u)G(J Z, u))G(J Y, u) + (G(J Y, Z)−G(J Y, u)G(Z, u) +G(Y, u)G(J Z, u))G(J X, u) , (∇Z^Tρ)(X^H, Y^H) =1

2c²(c−2)(n+ 4)

(G(X, Z)−G(X, u)G(Z, u))G(Y, u) + (G(Y, Z)−G(Y, u)G(Z, u))G(X, u) + (G(J X, Z)−G(J X, u)G(Z, u))G(J Y, u) + (G(J Y, Z)−G(J Y, u)G(Z, u))G(J X, u) , (∇_ZHρ)(X^T, Y^H) =c³

2(n+ 6)((G(X, Z)−G(X, u)G(Z, u))G(Y, u)

−c³(G(X, Y)−G(X, u)G(Y, u))G(Z, u) +c³

2(n+ 6)((G(X, J Z)−G(X, u)G(J Z, u))G(J Y, u)

−c³(G(X, J Y)−G(X, u)G(J Y, u))G(J Z, u) +c³

2(7n+ 22)

G(J X, u)G(Y, u)G(J Z, u)

−G(J X, u)G(J Y, u))G(Z, u) +c{c²−(n+ 1)c+ (n−1)}

G(X, Z)G(Y, u)−G(X, Y)G(Z, u)

−G(J X, Z)G(J Y, u) +G(J X, Y)G(J Z, u)

−c{(2n+ 9)c²−2(n+ 1)c+ 2(n−1)}G(J Y, Z)G(J X, u), (27)

(16)

(∇_ZTρ)(X^T, Y^H) = 0, (∇_ZHρ)(X^T, Y^T) = 0, (∇_ZHρ)(X^H, Y^H) = 0.

(28)

Then, taking account of (27), we have at once the following

Theorem 8. T₁M_n^C(4c) (n≥2) is Ricci-parallel(∇ρ= 0) if and only ifM is flat (c= 0).

Proof of Theorems 5 and 6

We first prove Theorem 5. Then, together with (27), we have

0 =(∇_ZTρ)(X^T, Y^T)−(∇_XTρ)(Z^T, Y^T)

=c²(2n+ 5)

2G(J X, Z)G(J Y, u) +G(J Y, Z)G(J X, u) +G(J X, Y)G(J Z, u) + 3 G(X, u)G(J Z, u)−G(J X, u)G(Z, u)

G(J Y, u) . (29)

Suppose thatT1M_n^C(4c) (n≥2) have Codazzi-type Ricci tensors. Then from (29) we havec²(2n+ 5) = 0,Hence, we see thatc= 0. Conversely, we easily can check thatT₁M_n(0) satisfies all possibilities:

(∇_ZTρ)(X^T, Y^T)−(∇_XTρ)(Z^T, Y^T) = 0, (∇Z^Tρ)(X^T, Y^H)−(∇X^Tρ)(Z^T, Y^H) = 0, (∇_ZTρ)(X^H, Y^H)−(∇_XHρ)(Z^T, Y^H) = 0, (∇_ZHρ)(X^H, Y^T)−(∇_XHρ)(Z^H, Y^T) = 0, (∇Z^Hρ)(X^T, Y^T)−(∇X^Tρ)(Z^H, Y^T) = 0, (∇_ZHρ)(X^H, Y^H)−(∇_XHρ)(Z^H, Y^H) = 0.

(30)

Before we prove Theorem 6, we remark in general that the polarization and the symmetry of ∇ρyields that the cyclic parallel Ricci tensor condition is equivalent to (∇Zρ)(Z, Z) = 0 for all vector fieldZ. Next, we prove Theorem 6. By using (27) and ¯X =X^T +Y^H, we obtain

(∇_XT+Y^Hρ)(X^T +Y^H, X^T +Y^H)

=(∇_XTρ)(X^T, X^T) + (∇_XTρ)(X^T, Y^H) + (∇_XTρ)(Y^H, X^T) + (∇_XTρ)(Y^H, Y^H) + (∇_YHρ)(X^T, X^T) + (∇_YHρ)(X^T, Y^H) + (∇_YHρ)(Y^H, X^T) + (∇_YHρ)(Y^H, Y^H)

=2c²(c−1)(n+ 4){(G(X, Y)−G(X, u)G(Y, u))G(Y, u) + (G(X, J Y)−G(X, u)G(J Y, u))G(J Y, u)}.

(31)

(17)

From (31), we see that it is necessary and sufficient condition forT1M_n^C(4c) (n≥2) to have cyclic parallel Ricci tensors that c = 0 orc = 1. This has completed the proof of Theorem 6.

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