Geometry of CR-manifolds of contact type

Geometry of CR-manifolds of contact type

Jong Taek Cho

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

Korea

e-mail : jtcho@chonnam.ac.kr

Abstract. We study on the geometry of CR-manifolds of contact type. This is a survey article whose contents are mainly based on the recent results in [14], [15] and [17].

1. Introduction

LetT M be the tangent bundle of a (2n+k)-manifold M. A CR-structure of type (n, k) onM is a complex ranknsubbundleH ⊂CT M=T M⊗C satisfying

(i)H ∩H¯={0},

(ii) [H,H]⊂ H(integrability).

We say (M,H) the CR manifold of type (n, k) and n the CR dimension and k the CR codimension of (M,H). In particular, a CR manifold of type (n,1) is contact type (or hypersurface type). Hereafter, we deal with only contact type CR manifold. Then there exists a unique subbundle D =Re{H ⊕H}, called the¯ Levi distribution(maximally holomorphic distribution) of (M,H), and a unique bundle map J such thatJ² =−I and H={X−iJX|X ∈D}. We call{D, J} the real representation of H. Let E ⊂ T^∗M be the conormal bundle of D. If M is an oriented CR-manifold thenEis a trivial bundle, henceT M admits globally defined a nowhere zero section, i.e, a real one-form onM such thatKer(η) =D. For{D, J}

we define the Levi form by

L:D×D→ F(M), L(X, Y) =−dη(X, JY)

whereF(M) denotes the algebra of differential functions onM. If the Levi form is hermitian, then the CR-structure is calledpseudo-hermitian, in addition, if the Levi form is non-degenerate (positive or negative definite, resp.), then the CR-structure is called a non-degenerate (strongly pseudo-convex, resp.) pseudo-hermitian CR structure.

Lee ([24]) introduced the notion ofpseudo-Einstein structure, namely, the pseu- dohermitian Ricci curvature tensor (of the Tanaka-Webster connection) is a scalar multiple of the Levi form, in a nondegenerate integrable CR manifold. This no- tion is a pseudo-homothetic (or D-homothetic) invariant. The pseudo-homothetic transformation is given by a special one of the gauge transformation (see section 2).

137

Also, in [14], [15] the present author defined another pseudo-homothetic invariant, i.e., the contact strongly pseudo-convex CR-space of constant pseudo-holomorphic sectional curvature(with respect to the Tanaka-Webster connection) (see section 3).

In this article, we arrange the recent results on the geometry of CR-manifolds of contact type.

2. Preliminaries

We start by collecting some fundamental materials about contact Riemannian geometry and contact strongly pseudo-convex CR-manifold. We refer to [4], [5] for further details. All manifolds in the present paper are assumed to be connected and of classC^∞.

A (2n+ 1)-dimensional manifold M²ⁿ⁺¹ is said to be a contact manifold if it admits a global one-formη such that η∧(dη)ⁿ 6= 0 everywhere. Given a contact form η, there exists a unique vector field ξ, called the characteristic vector field, satisfyingη(ξ) = 1 and dη(ξ, X) = 0 for any vector fieldX. It is well-known that there also exists a Riemannian metricgand a (1,1)-tensor fieldϕsuch that (1)

g(ϕX, ϕY) =g(X, Y)−η(X)η(Y), dη(X, Y) =g(X, ϕY), ϕ²X=−X+η(X)ξ, whereX andY are vector fields onM. From (1), it follows that

(2) ϕξ= 0, η◦ϕ= 0, η(X) =g(X, ξ).

A Riemannian manifoldM equipped with structure tensors (η, g) satisfying (1) is said to be a contact Riemannian manifold or contact metric manifold and it is denoted byM = (M;η, g). Given a contact Riemannian manifold M, we define a (1,1)-tensor field h by h = ¹₂Lξϕ, where L denotes Lie differentiation. Then we may observe thathis symmetric and satisfies

(3) hξ= 0, hϕ=−ϕh,

(4) ∇Xξ=−ϕX−ϕhX,

where ∇ is Levi-Civita connection. We denote by R the Riemannian curvature tensor. Along a trajectory of ξ, the Jacobi operator R_ξ =R(·, ξ)ξis a symmetric (1,1)-tensor field. We have

(5) ∇ξh=ϕ−ϕRξ−ϕh²,

(6) g(R(X, Y)ξ, Z) =g((∇Yϕ)X−(∇Xϕ)Y, Z) +g((∇Yϕh)X−(∇Xϕh)Y, Z) for all vector fieldsX, Y, ZonM. A contact Riemannian manifold for whichξ is a Killing vector field, is called aK-contact manifold. It is easy to see that a contact

Riemannian manifold is K-contact if and only ifh= 0. For a contact Riemannian manifoldM, one may define naturally an almost complex structureJ onM×Rby

J(X, fd

dt) = (ϕX−f ξ, η(X)d dt),

where X is a vector field tangent to M,t the coordinate ofRandf a function on M ×R. If the almost complex structure J is integrable,M is said to be normal or Sasakian. It is known thatM is normal if and only ifM satisfies

[ϕ, ϕ] + 2dη⊗ξ= 0,

where [ϕ, ϕ] is the Nijenhuis torsion ofϕ. A Sasakian manifold is also characterized by the condition

(7) (∇_Xϕ)Y =g(X, Y)ξ−η(Y)X

for all vector fieldsX andY on the manifold and this is equivalent to (8) R(X, Y)ξ=η(Y)X−η(X)Y

for all vector fieldsX andY.

For a contact Riemannian manifold M, the tangent spaceTpM of M at each point p∈ M is decomposed as TpM = Dp⊕ {ξ}p(direct sum), where we denote Dp={v∈TpM|η(v) = 0}. ThenD:p→Dpdefines a distribution orthogonal toξ.

The 2n-dimensional distribution (or subbundle)D is called thecontact distribution (or contact subbundle). For a given contact Riemannian manifoldM = (M;η, g) its associated almost CR-structure is given by the holomorphic subbundle

H={X−iJX:X∈D}

of the complexification T M^C of the tangent bundle T M, where J = ϕ|D, the restriction of ϕ to D. Then we see that each fiber Hx (x ∈ M) is of complex dimensionn andH ∩H¯ ={0}. Furthermore, we haveCD =H ⊕H. We say that¯ the associated CR-structure is integrable if [H,H]⊂ H. ForHwe define the Levi form by

L:D×D→ F(M), L(X, Y) =−dη(X, JY)

where F(M) denotes the algebra of differential functions on M. Then we see that the Levi form is Hermitian and positive definite, that is, the CR-structure isstrongly pseudo-convex, pseudo-Hermitian CR-structure. We call the pair (η, L) a strongly pseudo-convex, pseudo-hermitian structure on M. Since dη(ϕX, ϕY) =dη(X, Y), we see that [JX, JY]−[X, Y]∈Dand [JX, Y] + [X, JY]∈DforX, Y ∈D, further ifM satisfies the condition

[J, J](X, Y) = 0

for X, Y ∈ D, then the pair (η, L) is called a strongly pseudo-convex (integrable) CR-structure and (M;η, L) is called a strongly pseudo-convex CR-manifold. For

a given contact strongly pseudo-convex almost CR manifold M, the almost CR structure is integrable if and only ifM satisfies the integrability condition Ω = 0, where Ω is a (1,2)-tensor field onM defined by

(9) Ω(X, Y) = (∇Xϕ)Y −g(X+hX, Y)ξ−η(X)η(Y)ξ+η(Y)(X+hX) for all vector fields X, Y on M (see [30], Proposition 2.1]). Taking account of (7) and (9) we see that for a Sasakian manifold the associated CR structure is strongly pseudo-convex integrable (cf. [19]). It is well known that for 3-dimensional contact Riemannian manifolds their associated CR structures are always integrable (cf. [30]). One more class of contact Riemannian manifolds whose associated CR structures are integrable is so-called a (k, µ)-space, which is determined by the condition

R(X, Y)ξ= (kI+µh)(η(Y)X−η(X)Y)

for (k, µ)∈R². A (k, µ)-space remains invariant under a pseudo-homothetic trans- formation (see [6]). A pseudo-homothetic transformation (or D-homothetic trans- formation) is defined by a change of structure tensors of the form:

¯

η=aη, ξ¯= 1/aξ, ϕ¯=ϕ, g¯=ag+a(a−1)η⊗η, whereais a positive constant.

Now, we review the generalized Tanaka-Webster connection ([30]) on a con- tact strongly pseudo-convex almost CR-manifold M = (M;η, L). The generalized Tanaka-Webster connection ˆ∇ is defined by

∇ˆXY =∇XY +η(X)ϕY + (∇Xη)(Y)ξ−η(Y)∇Xξ for all vector fieldsX, Y onM. Together with (4), ˆ∇may be rewritten as

(10) ∇ˆXY =∇XY +A(X, Y), where we have put

(11) A(X, Y) =η(X)ϕY +η(Y)(ϕX+ϕhX)−g(ϕX+ϕhX, Y)ξ.

We see that the generalized Tanaka-Webster connection ˆ∇has the torsion Tˆ(X, Y) = 2g(X, ϕY)ξ+η(Y)ϕhX−η(X)ϕhY.

In particular, for aK-contact Riemannian manifold we get (12) A(X, Y) =η(X)ϕY +η(Y)ϕX−g(ϕX, Y)ξ.

Furthermore, it was proved that

Proposition 1. ([30])The generalized Tanaka-Webster connection∇ˆ on a contact Riemannian manifold M = (M;η, g) is the unique linear connection satisfying the following conditions:

(i) ˆ∇η= 0,∇ξˆ = 0;

(ii) ˆ∇g= 0;

(iii−1) ˆT(X, Y) = 2L(X, JY)ξ,X, Y ∈D;

(iii−2) ˆT(ξ, ϕY) =−ϕTˆ(ξ, Y),Y ∈D;

(iv) ( ˆ∇Xϕ)Y = Ω(X, Y),X, Y ∈T M.

The Tanaka-Webster connection ([29], [34]) on a nondegenerate integrable CR- manifold is defined as the unique linear connection satisfying (i), (ii), (iii-1), (iii-2) and Ω = 0. The metric affine connection ˆ∇ is a natural generalization of the Tanaka-Webster connection. In fact, in [5] the authors deal with the use of ˆ∇ in the non-integrable case. We refer to [5] for more details about pseudo-hermitian geometry in contact Riemannian manifolds.

Remark 1. The contact strongly pseudo-convex CR manifold is invariant under the pseudo-homothetic transformation. In fact, by direct computations we have

( ¯∇Xϕ)Y¯ = (∇Xϕ)Y + (a−1)η(Y)ϕ²X−(a−1)/ag(X, hY)ξ.

From this, we easily see that Ω = 0 implies ¯Ω = 0.

Now, we review the gauge transformation of contact Riemannian structure.

Given a contact formη, we consider a 1-form ¯η =ση for a positive smooth function σ. It is natural that one may consider the transformation (η, g) → (¯η,g). We¯ call this a gauge transformation of contact Riemannian structure. By assuming

¯

ϕ|D =ϕ|D the associated Riemannian structure ¯g of ¯η is determined in a natural way. Namely, we have

ξ¯= (1/σ)(ξ+ζ), ζ= (1/2σ)ϕgradσ, ϕ¯=ϕ+ (1/2σ)η⊗(gradσ−ξσ·ξ),

¯

g=σg−σ(η⊗ν+ν⊗η) +σ(σ−1 +kζk²)η⊗η,

where ν is dual to ζ with respect to g. We remark that particularly when σ is a constant then a gauge transformation reduces to a pseudo-homothetic transforma- tion.

Let ω be a nowhere vanishing (2n+ 1)-form on M and fix it. Let dM(g) = ((−1)ⁿ/2ⁿn!)η ∧(dη)ⁿ denote the volume element of (M, η, g). We define λ by dM(g) =±e^λω and θ ∈ Γ(D^∗) by θ(X) = Xλ for X ∈D. And [31] [32] defined

B∈Γ(D⊗D^∗3) by

(2n+ 4)g(B(X, Y)Z, W)

= (2n+ 4)g(R(X, Y)Z, W)−g(QY, Z)g(X, W) +g(QX, Z)g(Y, W)

−g(Y, Z)g(QX, W) +g(X, Z)g(QY, W) +g(QY, ϕZ)g(ϕX, W)

−g(QX, ϕZ)g(ϕY, W) +g((ϕQ+Qϕ)X, Y)g(ϕZ, W) +g(Y, ϕZ)g(ϕQX, W)−g(X, ϕZ)g(ϕQY, W)

−g(X, ϕY)g((ϕQ+Qϕ)Z, W)

+[ˆr/(2n+ 2)−4][g(Y, Z)g(X, W)−g(X, Z)g(Y, W)]

+[ˆr/(2n+ 2) + 2n][g(ϕY, Z)g(ϕX, W)−g(ϕX, Z)g(ϕY, W)

−2g(ϕX, Y)g(ϕZ, W)] + (2n−2)[g(ϕY, Z)g(ϕhX, W) (13)

−g(ϕX, Z)g(ϕhY, W) +g(ϕhY, Z)g(ϕX, W)−g(ϕhX, Z)g(ϕY, W)]

−6[g(hY, Z)g(X, W)−g(hX, Z)g(Y, W) +g(Y, Z)g(hX, W)

−g(X, Z)g(hY, W)] + (2n+ 4)[g(ϕhY, Z)g(ϕhX, W)

−g(ϕhX, Z)g(ϕhY, W)] +g(ϕ(∇ξh)Y, Z)g(X, W)

−g(ϕ(∇ξh)X, Z)g(Y, W) +g(Y, Z)g(ϕ(∇ξh)X, W)

−g(X, Z)g(ϕ(∇ξh)Y, W)−g(ϕY, Z)g((∇ξh)X, W) +g(ϕX, Z)g((∇ξh)Y, W)−g((∇ξh)Y, Z)g(ϕX, W)

+g((∇ξh)X, Z)g(ϕY, W)−(n+ 2)/(n+ 1)g(U(X, Y, Z;θ), W)),

where ˆr=r−g(Qξ, ξ) + 4nis the generalized Tanaka-Webster scalar curvature and U ∈Γ(D⊗D^∗3) defined by (7) in [32]. Moreover, the following was proved in [32].

Theorem 2. Let M = (M²ⁿ⁺¹, η, g) be a contact Riemannian manifold and letω be a fixed nowhere vanishing (2n+ 1)-form on M. Then B defined by (2.13) is a gauge invariant of type(1,3) of the contact Riemannian structure. Furthermore,

(i) IfB= 0, then the associated CR structure (η, J)is integrable.

(ii)IfΩ = 0, thenB reduces to the Chern-Morser-Tanaka invariant.

Remark 2. (1) If Ω = 0, then from the definition ofU we easily see that U = 0 (see (7) in [32]).

(2) Ifn= 1 (dimM=3), then we haveB= 0 (see Remark in [32]).

Moreover, we have

Proposition 3. ([14]) IfM is a normal contact Riemannian(orSasakian)man- ifold, then the the CR-invariant B coincides with C-Bochner curvature tensor.

Here, we recall the notion of a pseudo-homothetic transformation (or D- homothetic transformation) of a contact metric manifold. This transformation means a change of structure tensors of the form

(14) η¯=aη, ξ¯= 1/aξ, ϕ¯=ϕ, ¯g=ag+a(a−1)η⊗η,

where a is a positive constant. From (14), we have ¯h = (1/a)h. By using the well-known formula

2g(∇XY, Z) =Xg(Y, Z) +Y g(Z, X)−Zg(X, Y)−g(X,[Y, Z])

−g(Y,[X, Z]) +g(Z,[X, Y]) We have

(15) ∇¯XY =∇XY +C(X, Y), where Cis the (1,2)-type tensor defined by

C(X, Y) =−(a−1)[η(Y)ϕX+η(X)ϕY]−a−1

a g(ϕhX, Y)ξ.

Remark 3. (1) From (14) and the expression of the torsion tensor ˆT of the (gen- eralized) Tanaka-Webster connection, we easily see ˆT is pseudo-homothetically in- variant, namely, ¯ˆT = ˆT.

(2) The contact strongly pseudo-convex CR-manifold is invariant under the pseudo-homothetic transformation. In fact, by direct computations we have

( ¯∇_Xϕ)Y¯ = (∇_Xϕ)Y + (a−1)η(Y)ϕ²X−(a−1)/ag(X, hY)ξ.

From this, we easily see that Ω = 0 implies ¯Ω = 0.

3. Pseudo-Einstein contact strongly pseudo-convex CR-manifolds with B=0

We define the pseudo-hermitian curvature tensor of ˆR (with respect to the Tanaka-Webster connection) ˆ∇ by

R(X, Yˆ )Z= ˆ∇X( ˆ∇YZ)−∇ˆY( ˆ∇XZ)−∇ˆ_[X,Y]Z for all vector fieldsX, Y, Z inM. Then we have

Proposition 4. ([14], [15])

R(X, Yˆ )Z=−R(Y, X)Z,ˆ L( ˆR(X, Y)Z, W) =−L( ˆR(X, Y)W, Z).

The first identity follows trivially from the definition of ˆR. Since the connection is Levi-parallel, (i.e., ˆ∇L = 0), we have also the second one by a similar way as the case of Riemanian curvature tensor. We remark that the Tanaka-Webster connection is not torsion-free, the Jacobi- or Bianchi-type identies do not hold, in general. Before we study the curvature tensor ˆR, from (3), (10) and (11) we have

( ˆ∇Xh)Y =(∇Xh)Y +A(X, hY)−hA(X, Y)

=(∇Xh)Y + 2η(X)ϕhY +g((ϕh+ϕh²)X, Y)ξ (16)

+η(Y)(ϕhX+ϕh²X).

From the definition of ˆR, together with (10), taking account of ˆ∇η= 0, ˆ∇ξ= 0,

∇gˆ = 0, ˆ∇ϕ= 0 and (16), the straightforward computations yield R(X, Yˆ )Z=R(X, Y)Z+η(Z)³

ϕP(X, Y) +ϕ(A(X, hY)−A(Y, hX))

−ϕh(A(X, Y)−A(Y, X))´

−g

³

ϕP(X, Y) +ϕ(A(X, hY)−A(Y, hX))

−ϕh(A(X, Y)−A(Y, X)), Z

´ ξ

−2g(ϕX, Y)A(ξ, Z)−η(X)A(ϕhY, Z) +η(Y)A(ϕhX, Z)

−η(X)ϕA(Y, Z) +η(Y)ϕA(X, Z) +η(A(X, Z))(ϕY +ϕhY)

−η(A(Y, Z))(ϕX+ϕhX) +g(ϕX+ϕhX, A(Y, Z))ξ

−g(ϕY +ϕhY, A(X, Z))ξ.

By using (1), (2), (3) and (11) we have

(17) R(X, Yˆ )Z=R(X, Y)Z+B(X, Y)Z, where

B(X, Y)Z=η(Z)ϕP(X, Y)−g(ϕP(X, Y), Z)ξ−η(Z)[η(Y)(X+hX)

−η(X)(Y +hY)] +η(Y)g(X+hX, Z)ξ−η(X)g(Y +hY, Z)ξ +g(ϕY +ϕhY, Z)(ϕX+ϕhX)−g(ϕX+ϕhX, Z)(ϕY +ϕhY)

−2g(ϕX, Y)ϕZ

for all vector fieldsX, Y, Z in M. The pseudohermitian Ricci (curvature) tensor ˆρ of ˆ∇is

ˆ

ρ(X, Y) = trace of{V 7→R(V, X)Yˆ },

where X, Y are vector fields in M. From (17), by using (3) and (4), for an local orthonormal frame field{ei}, i= 1,2,· · ·,2n+ 1 we have

ˆ

ρ(X, Y) =ρ(X, Y) +

2n+1X

i=1

L(B(ei, X)Y, ei) (18)

=ρ(X, Y) + 2g(X, Y)−g(ϕ(∇ξh)X, Y)

where we have put ρ(X, Y) =g(QX, Y) and ˆρ(X, Y) =L( ˆQX, Y). Since∇_ξϕ= 0 (cf. [4] pp.67), from (3) we see that ˆQis the symmetric (1,1)-tensor field inM.

Now, we define the pseudo-Einstein structure in contact strongly pseudo-convex almost CR manifold (cf. [5], [14], [24]).

Definition 1. Let(M;η, L)be a contact strongly pseudo-convex almost CR mani- fold. Then the pseudohermitian structure (η, L)is said to be pseudo-Einstein if the pseudohermitian Ricci tensor is proportional to the Levi form, namely,

ˆ

ρ(X, Y) =λL(X, Y) or equivalently

ρ(X, Y) = (ˆr/2n−2)g(X, Y) +g(ϕ(∇ξh)X, Y) forX, Y ∈D, whereλ= ˆr/2n.

From the definition, in case thatM is K-contact, we know thath= 0, and ρ(X^T, Y^T) = (r−2n)/2ng(X^T, Y^T),

where X^T denotes the component ofX orthogonal toξ. SinceQξ = 2nξ, further we have

ρ(X, Y) = (r−2n

2n )g(X, Y) + (2n−r−2n

2n )η(X)η(Y) for any vector fields X, Y inM. Thus, we have

Proposition 5. In a K-contact manifold the pseudo-Einstein condition coincides with the η-Einstein condition.

We note that a K-contact manifold is calledη-Einstein if the Ricci tensor is of the form Q=aI+bη⊗ξ where a, b are constants (cf. pp. 285 in [36]). We also give

Definition 2. ([14], [15]) Let (M;η, L) be a contact strongly pseudo-convex CR manifold. ThenM is said to be of constant pseudo-holomorphic sectional curvature c with respect to the Tanaka-Webster connection or shortly, be of constant pseudo- holomorphic sectional curvaturec if M satisfies

L( ˆR(X, ϕX)ϕX, X) =c for any unit horizontal vector fieldX. If

L( ˆR(X, ϕX)ϕX, X) =f(p)

for any unit horizontal vector X(p) ∈ D(p), then M is said to have a pointwise constant pseudo-holomorphic sectional curvature.

From the definition of pseudo-homothetic transformation (or D-homothetic transformation) given in section 2, we have

(19) ∇¯XY =∇XY −(a−1)[η(Y)ϕX+η(X)ϕY]−(a−1)/ag(ϕhX, Y)ξ.

Furthermore, we have shown in [15] that

Proposition 6. The contact strongly pseudo-convex CR-space of (pointwise) con- stant pseudo-holomorphic sectional curvature is a pseudo-homothetic invariant.

Proof. From (19) and the definition of the curvature tensor, by long but tedious computations, we get

g( ¯R(X, ϕX)ϕX, X) =g(R(X, ϕX)ϕX, X)−(a−1)[3 +g(hX, X)]

−a−1

a [g(ϕhX, X)²+g(hX, X)(g(hX, X)−1)]

+(a−1)²

a g(hX, X)

for any unit horizontal vectorX∈D(p),p∈M. For any unit horizontal vectorX, from (17), we get

L( ¯ˆ¯ R(X,ϕX) ¯¯ ϕX, X) = 3 + ¯g( ¯R(X,ϕX¯ ) ¯ϕX, X)−g( ¯¯ ϕ¯hX, X)²−¯g(¯hX, X)². Hence, we have

L( ¯ˆ¯ R(X,ϕX¯ ) ¯ϕX, X) =aL( ˆR(X, ϕX)ϕX, X).

For a unit horizontal vector X, if we denote by ˆH(X) =L( ˆR(X, ϕX)ϕX, X) the pseudo-holomorphic sectional curvature, then this is rewritten by

H¯ˆ(X) =aH(Xˆ ).

Thus, we have proved.

We also have

Proposition 7. The generalized Tanaka-Webster connection is pseudo-homothetically invariant.

Proof. From (10), we have

∇¯ˆXY = ¯∇XY + ¯A(X, Y)

and

A(X, Y¯ ) =a(η(X)ϕY +η(Y)ϕX) +η(Y)ϕhX−g(ϕX, Y)ξ−1/ag(ϕhX, Y)ξ.

From this and (19), it follows that the generalized Tanaka-Webster connection is pseudo-homothetically invariant.

Corollary 8. The Tanaka-Webster curvature tensorRˆ and pseudohermitian Ricci tensor Sˆ are pseudo-homothetically invariant.

Remark 4. Together with the Definition 1 and the Corollary 8, we see that the pseudo-Einstein structure is a pseudo-homothetic invariant.

Since every 3-dimensional contact strongly pseudo-convex CR manifold has au- tomatically pseudo-Einstein structure (see, [24]), we assume that n≥2. Then we have ([14])

Theorem 9. A pseudo-Einstein contact strongly pseudo-convex CR manifold with B = 0has a pointwise constant pseudo-holomorphic sectional curvature

Hˆ = ˆr n(n+ 1) .

Proof. LetM be a pseudo-Einstein contact strongly pseudo-convex CR manifold.

Then from Definition 1, we see that Q= (ˆr/2n−2)I+ϕ(∇ξh) onD. From this, we also get

ϕQ+Qϕ= (ˆr/n−4)ϕ.

We suppose that B= 0. Then together with (14), for a unit vectorX, we have g(R(X, ϕX)ϕX, X) =¡ rˆ

n(n+ 1)−3¢

+g(hX, ϕX)²+g(hX, X)². But, for any unit horizontal vectorX, from (17), we obtain

g( ˆR(X, ϕX)ϕX, X) = 3 +g(R(X, ϕX)ϕX, X)−g(ϕhX, X)²−g(hX, X)². Hence, we have

g( ˆR(X, ϕX)ϕX, X) = ˆr n(n+ 1).

4. The Pseudo-Einstein unit tangent sphere bundles

The basic facts and fundamental formulae about tangent bundles are well-known (cf. [18], [23], [35]). We only briefly review of notations and their definitions. Let M = (M, G) be a n-dimensional Riemannian manifold T M denote its tangent bundle with the projectionπ:T M →M, π(x, u) =x. For a vectorX ∈TxM, we denote by X^H and X^V, the horizontal liftandthe vertical lift, respectively. Then we can define a Riemannian metric ˜g,Sasaki metriconT M in a natural way. That is,

˜

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 JX^H = X^V and JX^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]).

Now we consider the unit tangent sphere bundle (T1M, g⁰), which is isometrically embedded hypersurface in (T M,˜g) with unit normal vector field N = u^V. For X ∈TxM, we define thetangential lift ofX to (x, u)∈T1M by

X_(x,u)^T =X_(x,u)^V −G(X, u)N_(x,u).

Clearly, the tangent spaceT_(x,u)T1M spanned by vectors of the formX^H andX^T whereX ∈TxM. We put

ξ⁰=−JN, ϕ⁰=J−η⁰⊗N.

Then we find g⁰(X, ϕ⁰Y) = 2dη⁰(X, Y). By taking ξ = 2ξ⁰, η = ¹₂η⁰, ϕ=ϕ⁰, and g= ¹₄g⁰, we get the standard contact Riemannian structure (ϕ, ξ, η, g). Indeed, we easily check these tensors satisfy (1). The tensorsξandϕare explicitly given by

ξ= 2u^H, ϕX^T =−X^H+1

2G(X, u)ξ, (20)

ϕX^H=X^T

whereX andY are vector fields on M. From now we considerT1M = (T1M;η, g) with the standard contact Riemannian structure. We arrange fundamental formu- las, which is needed for the proof of our Theorem, without proofs (cf. [9], [10], [32], [33]). We denote by∇andR, the Levi-Civita connection and the Riemannian curvature tensor associated with g, respectively. We, also, denote by D and K, the Levi-Civita connection and the Riemannian curvature tensor associated with

G, respectively. Then we have

∇_XTY^T =−G(Y, u)X^T,

∇_XTY^H= 1

2(K(u, X)Y)^H, (21)

∇_XHY^T = (DXY)^T +1

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

∇_XHY^H= (DXY)^H−1

2(K(X, Y)u)^T, and

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)Zª_H +1

4

©[K(u, X), K(u, Y)]Zª_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)Xª_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)uª_T (22)

+1 2

©(DXK)(u, Y)ZªH

, R(X^H, Y^H)Z^T =©

K(X, Y)(Z−G(Z, u)u)ª_T +1

4

©K(Y, K(u, Z)X)u−K(X, K(u, Z)Y)uª_T +1

2

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

2

©K(u, K(X, Y)u)Zª_H

−1 4

©K(u, K(Y, Z)u)X−K(u, K(X, Z)u)Yª_H

+1 2

©(DZK)(X, Y)uª_T

for all vector fields X, Y and Z on M. Next, we calculate the Ricci tensorρ of (T1M, g) at the point (x, u)∈T1M. Let{E1,· · ·, En=u}be an orthonormal basis ofTxM. Then{2E₁^t,· · · ,2E_n−1^t ,2E₁^h,· · ·,2E_n−1^h ,2E_n^h=ξ}is an orthonormal basis forT(x,u)T1M andρis given by

ρ(A, B) =

n−1X

i=1

R(2E_i^t, A, B,2E_i^t) + Xn i=1

R(2E_i^h, A, B,2E_i^h).

From this, we obtain

ρ(X^t, Y^t) = (n−2)(G(X, Y)−G(X, u)G(Y, u)) +1

4 Xn i=1

G(K(u, X)Ei, K(u, Y)Ei), ρ(X^t, Y^h) =1

2

³

(∇uS)(X, Y)−(∇XS)(u, Y)

´ , (23)

ρ(X^h, Y^h) =ρ(X, Y)−1 2

Xn i=1

G(K(u, Ei)X, K(u, Ei)Y).

Contraction ofρwith respect to the metricg gives the scalar curvaturer:

r=

n−1X

i=1

ρ(2E_i^t,2E_i^t) + Xn i=1

ρ(2E_i^h,2E_i^h) (24)

=s+ (n−1)(n−2)−$(u, u)/4 where$(u, v) =P_n

i,j=1G(K(u, E_i)E_j, K(v, E_i)E_j).From the Definition 1 and (23) we have ([17])

Theorem 10. The standard contact strongly pseudo-convex CR-structure on the unit tangent sphere bundlesT1M of a space M with dimension≥3and of constant curvaturec admit the pseudo-Einstein structures if and only if c= 1.

In [2], the authors showed that the unit tangent sphere bundle of a space of constant curvature 1 has an pseudo-Einstein structure. The class of (k, µ)-spaces, mentioned in Preliminary, contains Sasakian spaces (κ = 1 and h = 0), further in [6] it was shown that the only unit tangent sphere bundles with this curvature property are precisely those of spaces of constant curvaturec(withk=c(2−c) and µ = −2c). Moreover, they ([6]) calculated the Ricci tensor for the non-Sasakian case, which gives

ρ(X, Y) = [2(n−1)−nµ]g(X, Y) + [2(n−1) +µ]g(hX, Y)

for any vector fieldsX, Y onD. We now suppose that a contact (κ, µ)-spaceM²ⁿ⁺¹ is a pseudo-Einstein space, then together with the 2nd equation of Definition 1, we have

(λ−2)I−µh= [2(n−1)−nµ]I+ [2(n−1) +µ]h

onD, where we have used∇ξh=µhϕ(see [6]). If we operateshagain, and taking the trace of this equation, then we get

(n−1)trace ofh²= 0,

which says that a non-Sasakian contact (κ, µ)-space is a pseudo-Einstein only in dimension three. Namely, we have

Theorem 11. ([17]) If contact(κ, µ)-spaceM²ⁿ⁺¹ (n≥2)is a pseudo-Einstein space, then M is a Sasakian space.

5. Pseudo-Einstein Hopf-hypersurfaces in a complex space form

LetM be an oriented real hypersurface of a K¨ahlerian manifold ˜M = ( ˜M;J,˜g) and N a global unit normal vector on M. By ˜∇, A we denote the Levi-Civita connection in ˜M and the shape operator with respect toN, respectively. Then the Gauss and Weingarten formulas are given respectively by

∇˜XY =∇XY +g(AX, Y)N, ∇˜XN =−AX

for any vector fieldsXandY tangent toM, wheregdenotes the Riemannian metric of M induced from ˜g. An eigenvector(resp. eigenvalue) of the shape operatorAis called a principal curvature vector(resp. principal curvature). We denote byVλ the eigenspace associated with an eigenvalue λ. For any vector fieldX tangent toM, we put

(25) JX=ϕX+η(X)N, JN =−ξ.

We easily see that the structure (η, ϕ, ξ, g) is an almost contact metric structure on M. Indeed, the structure tensors satisfy

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

From the condition ˜∇J = 0, the relations (25) and by making use of the Gauss and Weingarten formulas, we have

(∇Xϕ)Y =η(Y)AX−g(AX, Y)ξ, (26)

∇Xξ=ϕAX.

(27)

By using (26) and (27), we see that a real hypersurface in a K¨ahlerian man- ifold always has an integrable associated CR structure. Here, we note that the associated Levi form in a real hypersurface of a K¨ahlerian manifold is L(X, Y) = 1/2g(( ¯ϕA¯+ ¯Aϕ)X,¯ ϕY¯ ), where we denote by ¯A the restrictionAto D. Hence, we see that its associated CR-structure in general neither is pseudo-hermitian nor is non-degenerate. But, since a contact structure satisfies dη(X, Y) =g(X, ϕY) (the 2nd eq. in (1)), together with (27) we proved in [12], [16]

Proposition 12. Let M = (M;η, ϕ, g) be a real hypersurface of a K¨ahlerian manifold. The almost contact metric structure ofM is contact metric if and only if ϕA+Aϕ=±2ϕ, where±is determined by the orientation. Further, its associated CR-structure is pseudo-hermitian, strongly pseudo-convex.

Moreover, Remark 1 in [12] says that a geodesic hypersphere of radius ^π₄ inCPⁿ, a horosphere inCHⁿ, a unit sphereS²ⁿ⁻¹(1) or a generalized cylinderSⁿ⁻¹(1/2)× EⁿinCEⁿonly has a contact metric structure. Now we definethe pseudo-Einstein real hypersurface in a K¨ahlerian manifold by the same condition as in Definition 1. Let ˜M = ˜Mn(c) be a complex space form of constant holomorphic sectional curvature c and M a real hypersurface of ˜M. Then we have the following Gauss and Codazzi equations :

R(X, Y)Z= c

4{g(Y, Z)X−g(X, Z)Y

+g(ϕY, Z)ϕX−g(ϕX, Z)ϕY −2g(ϕX, Y)ϕZ}

(28)

+g(AY, Z)AX−g(AX, Z)AY,

(29) (∇XA)Y −(∇YA)X= c

4{η(X)ϕY −η(Y)ϕX−2g(ϕX, Y)ξ}

for any tangent vector fields X, Y, Z on M. From (28) we get for the Ricci tensorQof type (1,1):

(30) QX =c/4{(2n+ 1)X−3η(X)ξ}+hAX−A²X,

where h = trace of A denotes the mean curvature. From the definition of Lie derivative and using (26) and (27) we have

hX= 1/2(Lξϕ)X = 1/2(LξϕX−ϕLξX)

= 1/2([ξ, ϕX]−ϕ[ξ, X]) (31)

= 1/2

³

(∇ξϕ)X− ∇ϕXξ+ϕ∇Xξ

´

= 1/2(η(X)Aξ−ϕAϕX−AX).

By using (30) and (31), we have

Theorem 13. ([17]) LetM be a Hopf hypersurface of a complex space formM˜n(c) with constant holomorphic sectional curvaturecandn≥3. Suppose thatM admits a pseudo-Einstein structure. Then we have

(I) in case thatM˜n(c) =PnC,M is locally congruent to one of the following : (A1)a geodesic hypersphere of radius r, where0< r < ^π₂,

(A2)a tube of radius r over a totally geodesicPkC(1≤k≤n−2),0< r < ^π₂ andcot²r=q/p, wherea1= cotrof multiplicity2panda2=−tanrof multiplicity 2q,

(II) in case thatM˜n(c) =HnC,M is locally congruent to one of the following :

(A₀) a horosphere,

(A1) a geodesic hypersphere or a tube over a complex hyperbolic hyperplane Hn−1C,

(III) in case that M˜n(c) =EnC,M is locally congruent to one of the following : (A1) a sphereS²ⁿ⁻¹(r)of radius r∈R+,

(A2) a planeE²ⁿ⁻¹, (B)Eⁿ×Sⁿ⁻¹(n/2(n−1))

Remark 5. (1) In real hypersurfaces in a complex space form, Kon [22] defines he

“pseudo-Einstein space” by the same condition ofη-Eistein for (K-)contact geome- try. We call the space “η-Einstein space” to avoid the confusion. As compare with two notions, they are properly distinguished each other. Indeed, homogeneous tubes of type (B) with special radii inPnC areη-Einsteinean, not pseudo-Einteinean (cf.

[22]). We also find thatEⁿ×Sⁿ⁻¹(n/2(n−1)) inE_nC are a pseudo-Eintein space that is not anη-Einstein space.

(2) From the observations of the above, it leads naturally the following prob- lem:Is there a (non-Sasakian) contact pseudo-Einstein space of dimension≥5?

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