Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 377–386

ON A SPECIAL CLASS OF HERMITIAN MANIFOLDS

J.C. MARRERO^{∗}  J. ROCHA^{∗}

Abstract

In this paper, we study a particular class of hermitian manifolds which we call special c′-hyperbolic hermitian manifolds with c′ ∈ IR, c′ ≠ 0. As main result, we prove that the universal covering space of a special c′-hyperbolic hermitian manifold is the product of a c-sasakian manifold with a hyperbolic space of odd dimension and of constant sectional curvature −(c′)^{2}. A.M.S. Classification: 53C15, 53C25, 53C55. Keywords: Hermitian manifolds, sasakian manifolds, kenmotsu manifolds, hyperbolic space.

I Introduction

An almost hermitian manifold V is called locally conformal Kähler if its metric is conformally related to a Kähler metric in some neighbourhood of every point of V (see [V1]). Examples of locally conformal Kähler manifolds are provided by the generalized Hopf manifolds which are locally conformal Kähler manifolds with parallel Lee form (see [V2]). The main non-Kähler examples of such manifolds are the Hopf manifolds (see [KN] and [V1]), which are defined as the quotients

H_{o}^{n} = \frac{(C^{n} − \{0\})}{Δ_{λ}},

being Δ_{λ} a cyclic group of transformations, and the nilmanifold N(r, 1) × S^{1}, where N(r, 1) = Γ(r, 1) \ H(r, 1) is a compact quotient of the generalized Heisenberg group H(r, 1) by a discrete subgroup Γ(r, 1) (see [CFL]). The product of a c-sasakian manifold N with the (2m+1)-dimensional hyperbolic space H_{c}^{2m+1} of constant sectional curvature −c^{2} also admits a locally conformal Kähler structure (J, g), where g is the product metric (see [MR]). In [MR] we have studied a particular class of locally conformal Kähler manifolds with similar properties to the locally conformal Kähler manifold N × H_{c}^{2m+1}.

On the other hand if c and c′ are arbitrary constants, c, c′ ≠ 0, then the almost hermitian structure (J, g) on the manifold N × H_{c′}^{2m+1} is hermitian but, in general, it is not locally conformal Kähler (see corollary III.1). In this paper we study a particular class of hermitian manifolds which we call special c′-hyperbolic hermitian manifolds, with c′ ∈ IR, c′ ≠ 0. These manifolds have similar properties to the product N × H_{c′}^{2m+1}. We generalize some facts obtained in [MR] and, as main result, we prove that the universal covering space of a special c′-hyperbolic hermitian manifold is almost complex isometric to the product of a c-sasakian manifold with a hyperbolic space of odd dimension and of constant sectional curvature −(c′)^{2} (see theorem VI.1).

∗Supported by the “Consejería de Educación del Gobierno de Canarias”.

This paper is in final form and no version of it will be submitted for publication elsewhere.

II Preliminaries

Let V be a C^{∞} almost hermitian manifold with metric g and almost complex structure J. Denote by 𝔛(V) the Lie algebra of C^{∞} vector fields on V. The almost hermitian manifold (V, J, g) is said to be: Hermitian if N_{J} = 0, where N_{J} denotes the Nijenhuis tensor of J; Locally conformal Kähler if it is hermitian, dω = 0 and dΩ = ω ∧ Ω, where ω and Ω are the Lee 1-form and the Kähler 2-form respectively of V (see [KN] and [V1]).

On the other hand if M is an almost contact metric manifold with metric g and almost contact structure (φ, ξ, η) then it is said to be c’-kenmotsu, where c′ ∈ IR, c′ ≠ 0, if N_{φ} + 2dη ⊗ ξ = 0, dϕ = −2c′η ∧ ϕ and dη = 0 being N_{φ} the Nijenhuis tensor of φ and ϕ the fundamental 2-form of M (see [JV]).

Let (H_{c′}^{2m+1}, (ds^{2})_{c′}) be the (2m + 1)-dimensional hyperbolic space, i.e.,

H_{c′}^{2m+1} = {(x_{1},…, x_{2m+1}) ∈ IR^{2m+1} / x_{2m+1} > 0}

and (ds^{2})_{c′} is the Riemannian metric on H_{c′}^{2m+1} given by

(ds^{2})_{c′} = \frac{1}{(c′x_{2m+1})^{2}} ∑_{i=1}^{2m+1}(dx_{i})^{2}, (c′ ≠ 0).

(H_{c′}^{2m+1}, (ds^{2})_{c′}) is a complete simply connected Riemannian manifold with constant sectional curvature −(c′)^{2}. Moreover, if (φ_{H_{c′}^{2m+1}}, ξ_{H_{c′}^{2m+1}}, η_{H_{c′}^{2m+1}}, (ds^{2})_{c′}) is the almost

contact metric structure on H_{c′}^{2m+1} defined by

\begin{matrix}φ_{H_{c′}^{2m+1}} = (c′x_{2m+1}) ∑_{i=1}^{m}(\frac{∂}{∂x_{i}} ⊗ α_{m+i} − \frac{∂}{∂x_{m+i}} ⊗ α_{i}), \\ ξ_{H_{c′}^{2m+1}} = (c′x_{2m+1})\frac{∂}{∂x_{2m+1}} , η_{H_{c′}^{2m+1}} = \frac{dx_{2m+1}}{(c′x_{2m+1})}, \end{matrix}  (II.1)

where α_{i}, i ∈ {1,…, 2m}, are the 1-forms on H_{c′}^{2m+1} given by

α_{i} = \frac{dx_{i}}{(c′x_{2m+1})},  (II.2)

then (φ_{H_{c′}^{2m+1}}, ξ_{H_{c′}^{2m+1}}, η_{H_{c′}^{2m+1}}, (ds^{2})_{c′}) is a c′-kenmotsu structure on H_{c′}^{2m+1} (see [ChG]).

An almost contact metric manifold (N, φ, ξ, η, g) is said to be c-sasakian (see [JV]), with c ∈ IR, c ≠ 0 if N_{φ} + 2dη ⊗ ξ = 0 and dη = cϕ. In [MR] we have proved that if (N, φ, ξ, η, g) and (N′, φ′, ξ′, η′, g′) are (2n − 1)-dimensional complete simply connected c-sasakian manifolds of constant φ-sectional curvature k then, N is almost contact isometric to N′. In fact, the unit sphere S^{2n−1} has a c-sasakian structure of constant φ-sectional curvature k, for all k > −3c^{2}; the (2n − 1)-dimensional real number space IR^{2n−1} is a c-sasakian manifold of constant φ-sectional curvature −3c^{2} and the product manifold IR × CD^{n−1}, where CD^{n−1} is a simply connected bounded complex domain in C^{n−1} with negative constant holomorphic sectional curvature, has a c-sasakian structure of constant φ-sectional curvature k, for all k < −3c^{2} (see [MR]). We shall denote by S^{2n−1}(c, k) (k > −3c^{2}), IR^{2n−1}(c) and (IR × CD^{n−1})(c, k) (k < −3c^{2}) respectively the c-sasakian manifolds with these structures.

III c’-Hyperbolic hermitian structures

In this section, we study a particular class of structures on an almost hermitian manifold which we call c′-hyperbolic hermitian structures.

Let (N, φ_{N}, ξ_{N}, η_{N}, g_{N}) be a (2n − 1)-dimensional c-sasakian manifold and (M, φ_{M}, ξ_{M}, η_{M}, g_{M}) a (2m + 1)-dimensional c′-kenmotsu manifold, with c, c′ ∈ IR, c ≠ 0, and c′ ≠ 0. Let us consider the product manifold V = N × M with the almost hermitian structure (J, g) defined by:

\begin{matrix}J(X, X′) = (φ_{N}X − η_{M}(X′) ξ_{N}, φ_{M}X′ + η_{N}(X) ξ_{M}) \\ g((X, X′), (Y, Y′)) = g_{N}(X, Y) + g_{M}(X′, Y′) \end{matrix} \}  (III.1)

where X, Y ∈ 𝔛(N) and X′ , Y ′ ∈ 𝔛(M).

Proposition III.1 The almost hermitian manifold (V, J, g) is a hermitian manifold with Lee 1-form

ω = −2 (c′ + \frac{n − 1}{n + m − 1}(c − c′)) π_{M}^{∗}η_{M}

where π_{M} : N × M ⟶ M is the canonical projection onto the second factor.

We remark that if c = c′ then (V, J, g) is a locally conformal Kähler manifold [MR]. Denote by l = −2(c′ + \frac{n−1}{n+m−1}(c − c′)). Suppose that l > 0. Thus we deduce that

l = ∥w∥. Moreover, using theorem 2.1 of [M] (see also [MR]) we have

Proposition III.2 Let (J, g) be the almost hermitian structure given by (III.1) on the product manifold N × M. Then, for every point (p, q) ∈ N × M there exists an open neighbourhood U of q in M and 2m independent 1-forms α_{1},…, α_{2m} on U, such that:

\begin{matrix}π_{U}^{∗}α_{j} ^{∘} J = π_{U}^{∗}α_{m+j}, & π_{U}^{∗}α_{m+j} ^{∘} J = −π_{U}^{∗}α_{j}, & j ∈ \{1,…, m\} \\ d(π_{U}^{∗}α_{i}) = \frac{c′}{l} π_{U}^{∗}α_{i} ∧ ω, & (π_{U}^{∗}α_{i})(B) = 0, & i ∈ \{1,…, 2m\} \end{matrix} \}  (III.2)

where π_{U} : N × U ⟶ U is the projection onto the second factor and ω, B are the Lee 1-form and the Lee vector field respectively of N × M.

The above result suggests us to consider the following particular class of hermitian structure:

Definition III.1 Let (V, J, g) be a (2n + 2m)-dimensional hermitian manifold with Lee 1-form ω ≠ 0 at every point and Lee vector field B and let α_{1}, .., α_{2m} be independent 1-forms on V, with m ≥ 0. We say that (J, g, α_{1},…, α_{2m}) is a c’-hyperbolic hermitian structure on V if

\begin{matrix}α_{j} ^{∘} J = α_{m+j}, & α_{m+j} ^{∘} J = −α_{j}, & j ∈ \{1,…, m\}, \\ dα_{i} = c′α_{i} ∧ u, & & i ∈ \{1, 2,…, 2m\} \\ α_{i}(B) = 0, & & i ∈ \{1, 2,…, 2m\} \end{matrix}  (III.3)

where c′ is a constant, c′ ≠ 0 and u is the unit Lee 1-form.

Remark If (N, φ_{N}, ξ_{N}, η_{N}, g_{N}) is a (2n − 1)-dimensional c-sasakian manifold and (M, φ_{M}, ξ_{M}, η_{M}, g_{M}) is a (2m + 1)-dimensional c′-kenmotsu manifold, with c, c′ ∈ IR, c ≠ 0, c′ ≠ 0 and c′m + c(n − 1) < 0, then, from proposition III.2, we deduce that for every point (p, q) ∈ N × M, there exists an open neighbourhood U of q in M and 2m 1-forms α_{1},…, α_{2m} on U, such that (J, g, π_{U}^{∗}α_{1},…, π_{U}^{∗}α_{2m}) is a c′-hyperbolic hermitian structure on N × U, where (J, g) is the almost hermitian structure given by (III.1) on the manifold N × M and π_{U} : N × U ⟶ U is the projection onto the second factor.

Now, let H_{c′}^{2m+1} be the (2m+1)-dimensional hyperbolic space. Denote by α_{1},…, α_{2m} the 1-forms on H_{c′}^{2m+1} given by (II.2) and by (φ_{H_{c′}^{2m+1}}, ξ_{H_{c′}^{2m+1}}, η_{H_{c′}^{2m+1}}, (ds^{2})_{c′}) the c′-

kenmotsu structure on H_{c′}^{2m+1} given by (II.1). Then, if N is a c-sasakian manifold, π_{H_{c′}^{2m+1}} : N × H_{c′}^{2m+1} ⟶ H_{c′}^{2m+1} is the projection onto the second factor, (J, g) is the

almost hermitian structure on N × H_{c′}^{2m+1} given by (III.1) and mc′ + (n − 1)c < 0, we obtain that

Corollary III.1 (J, g, π_{H_{c′}^{2m+1}}^{∗}α_{1},…, π_{H_{c′}^{2m+1}}^{∗}α_{2m}) is a c′-hyperbolic hermitian structure

on the hermitian manifold N × H_{c′}^{2m+1} and we have that

\begin{matrix}dΩ = −\frac{2c}{l}ω ∧ Ω − \frac{4(c′ − c)}{l} ∑_{j=1}^{m} ω ∧ π_{H_{c′}^{2m+1}}^{∗}α_{j} ∧ π_{H_{c′}^{2m+1}}^{∗}α_{m+j}, \\ ∇ω = −l c′ ∑_{j=1}^{2m}(π_{H_{c′}^{2m+1}}^{∗}α_{j}) ⊗ (π_{H_{c′}^{2m+1}}^{∗}α_{j}), \\ ∇π_{H_{c′}^{2m+1}}^{∗}α_{i} = \frac{c′}{l} (π_{H_{c′}^{2m+1}}^{∗}α_{i}) ⊗ ω, \end{matrix}  (III.4)

for i ∈ {1,…, 2m}, where ∇ is the Levi-Civita connection of the Riemannian metric g, Ω is the Kähler 2-form of N × H_{c′}^{2m+1} and ω is the Lee 1-form.

IV Special c’-hyperbolic hermitian manifolds

The results obtained in section III suggest us to introduce the following definition.

Definition IV.1 Let (J, g, α_{1},…, α_{2m}) be a c′-hyperbolic hermitian structure on a manifold V^{2n+2m} of dimension (2n + 2m), such that α_{1},…, α_{2m} are unit 1-forms. We say that V^{2n+2m} is a special c’-hyperbolic hermitian manifold if

\begin{matrix}dΩ = −\frac{2c}{l}ω ∧ Ω − \frac{4(c′ − c)}{l} ∑_{i=1}^{m} ω ∧ α_{i} ∧ α_{m+i} \\ ∇ω = −l c′ ∑_{j=1}^{2m} α_{j} ⊗ α_{j} \\ ∇α_{i} = \frac{c′}{l} α_{i} ⊗ ω \end{matrix} \}  (IV.1)

for i ∈ {1,…, 2m}, where ω is the Lee form of V^{2n+2m}, ∇ is the Levi-Civita connection of the metric g, l = ∥ω∥ ≠ 0 at every point, c′ is a constant, c′ ≠ 0 and c = c′ − \frac{1}{2}(\frac{n+m−1}{n−1})(l + 2c′).

Remark From (IV.1) we deduce that ω is a closed 1-form.

Let (V^{2n+2m}, J, g, α_{1}, .., α_{2m}) be a special c′-hyperbolic hermitian manifold and denote by A_{i}, with 1 ≤ i ≤ 2m, the vector fields on V^{2n+2m} given by

α_{i}(X) = g(X, A_{i})  (IV.2)

for all X ∈ 𝔛(V^{2n+2m}). From (III.3) and (IV.2), we obtain that

JA_{i} = −A_{m+i}, JA_{m+i} = A_{i}  (IV.3)

for i ∈ {1,…, m}. Moreover, using (IV.1) we have that the vector fields A_{i} and A_{j}, with i ≠ j, are orthogonal and that the Lee 1-form has constant norm.

We also remark that if c′ = −\frac{l}{2} then (V^{2n+2m}, J, g, α_{1},… α_{2m}) is an m-hyperbolic locally conformal Kähler manifold (see [MR]).

Let (V^{2n+2m}, J, g, α_{1},…, α_{2m}) be a special c′-hyperbolic hermitian manifold with Lee vector field B and Lee form ω. Then, in the rest of this paper, we shall use the following notation

l = ∥ω∥, u = \frac{ω}{l} , U = \frac{B}{l} , v = −u^{∘} J, V = JU.  (IV.4)

From (III.3), (IV.3) and (IV.4) we obtain that

\begin{matrix}u(V) = v(U) = u(A_{i}) = v(A_{i}) = 0 \\ α_{i}(U) = α_{i}(V) = 0 \end{matrix}  (IV.5)

for i ∈ {1,…, 2m}. Moreover, if Ω is the Kähler 2-form of V^{2n+2m} then, using that Ω is non-degenerate and (IV.5), we deduce that

Proposition IV.1 On a special c′-hyperbolic hermitian manifold V^{2n+2m}

Ω = ψ + 2(∑_{j=1}^{m}(α_{j} ∧ α_{m+j}) + v ∧ u),

where ψ is a 2-form of rank (2n − 2) such that:

\begin{matrix}ψ^{n−1} ∧ u ∧ v ∧ α_{1} ∧… ∧ α_{2m} ≠ 0, \\ ψ(X, A_{i}) = ψ(X, U) = ψ(X, V) = 0 \end{matrix}

for i ∈ {1,…, 2m}.

From (IV.1) and by a well-known result (see [KN], pg. 148) we have

Proposition IV.2 On a special c′-hyperbolic hermitian manifold V^{2n+2m}

(∇_{X}J)Y = −(\frac{c}{l}(ω(JY)X − ω(Y)JX − g(X, JY)B + g(X, Y)JB) +

+(\frac{c′ − c}{l}) ∑_{j=1}^{m}((ω(JY)α_{m+j}(X) + ω(Y)α_{j}(X))A_{m+j} +  (IV.6)

+(ω(JY)α_{j}(X) − ω(Y)α_{m+j}(X))A_{j} +

+(α_{j}(X)α_{j}(Y) + α_{m+j}(X)α_{m+j}(Y))JB − 2(α_{j} ∧ α_{m+j})(X, Y)B)).

From (IV.1) and (IV.6) we obtain some characterizations of special c′-hyperbolic hermitian manifolds.

Proposition IV.3 Let (J, g, α_{1}, .., α_{2m}) be a c′-hyperbolic hermitian structure on a (2n + 2m)-dimensional manifold V^{2n+2m} such that α_{1},…, α_{2m} are unit 1-forms. Then, (V^{2n+2m}, J, g, α_{1},…, α_{2m}) is a special c′-hyperbolic hermitian manifold if and only if l = ∥ω∥ is constant, dΩ is given as in (IV.1) and one of the four following relations holds

1. ∇u = −c′ ∑_{j=1}^{2m} α_{j} ⊗ α_{j},  ∇α_{i} = c′α_{i} ⊗ u,

2. ∇U = −c′ ∑_{j=1}^{2m} α_{j} ⊗ A_{j},  ∇A_{i} = c′α_{i} ⊗ U,

3. ∇V = c(J + v ⊗ U − u ⊗ V + ∑_{j=1}^{m}(α_{j} ⊗ A_{m+j} − α_{m+j} ⊗ A_{j})),
∇A_{i} = c′α_{i} ⊗ U,
4. ∇v = −cψ,  ∇α_{i} = c′α_{i} ⊗ u

for i ∈ {1,…, 2m}.

Next, using proposition IV.3, we deduce another results for a special c′-hyperbolic hermitian manifold V^{2n+2m}. Denote by L the Lie derivative on V^{2n+2m}.

Proposition IV.4 Let (V^{2n+2m}, J, g, α_{1},…, α_{2m}) be a special c′-hyperbolic hermitian manifold. Then, V is a Killing vector field for the metric g. Moreover, the following relations hold

[U, V] = 0, [V, A_{i}] = 0, [A_{i}, A_{j}] = 0, [U, A_{i}] = c′A_{i},  (IV.7)
L_{U}J = 0, L_{V} J = 0,  (IV.8)
L_{A_{k}}J = c′(v ⊗ A_{k} − u ⊗ A_{m+k}),  (IV.9)
L_{A_{m+k}}J = c′(v ⊗ A_{m+k} + u ⊗ A_{k}),  (IV.10)
L_{U}v = 0, L_{A_{i}}v = 0, dv = −c ψ  (IV.11)

for i, j ∈ {1,…, 2m} and k ∈ {1,…, m}.

If (V^{2n+2m}, J, g, α_{1} … α_{2m}) is a special c′-hyperbolic hermitian manifold with m ≥ 1 then, from (III.3) and (IV.11), we obtain that the volume element γ on V^{2n+2m} given by γ = α_{1} ∧…∧α_{2m} ∧u∧v ∧ψ^{n−1} is exact. In fact, γ = d((\frac{1}{−2c′m})α_{1} ∧…∧α_{2m} ∧v ∧ψ^{n−1}). Therefore,

Corollary IV.1 A compact manifold cannot admit a special c′-hyperbolic hermitian structure (J, g, α_{1}, .., α_{2m})with m ≥ 1.

V The curvature tensor on a special c’-hyperbolic hermitian manifold

In this section, we shall obtain some results about the Riemann curvature tensor and the sectional curvature of a special c′-hyperbolic hermitian manifold.

Let (V^{2n+2m}, J, g, α_{1},…, α_{2m}) be a (2n+2m)-dimensional special c′-hyperbolic hermitian manifold and let A_{i} be as in (IV.2) and l, u, U, v and V as in (IV.4). Then, using (III.3) and proposition IV.3, we have

Proposition V.1 On a special c′-hyperbolic hermitian manifold V^{2n+2m}

R(X, Y)U = −2(c′)^{2} ∑_{i=1}^{2m}(α_{i} ∧ u)(X, Y)A_{i},  (V.1)

R(X, Y)A_{i} = 2(c′)^{2}{(α_{i} ∧ u)(X, Y) U + ∑_{j=1}^{2m}(α_{i} ∧ α_{j})(X, Y) A_{j}},  (V.2)

where R is the Riemann curvature tensor of V^{2n+2m}, i ∈ {1,…, 2m}, X, Y ∈ 𝔛(V^{2n+2m}).

Let x be a point of V^{2n+2m}. Denote by K_{XY} the sectional curvature for the plane section in T_{x}M with orthonormal basis {X, Y }. Then, by using (V.1) and (V.2), we deduce

Corollary V.1 On a special c′-hyperbolic hermitian manifold V^{2n+2m}

K_{XU} = −(c′)^{2} ∑_{i=1}^{2m}(α_{i}(X))^{2},

K_{XA_{i}} = −(c′)^{2}{(u(X))^{2} + ∑_{j=1,j≠i}^{2m}(α_{j}(X))^{2}},

K_{UA_{i}} = K_{A_{i}A_{j}} = −(c′)^{2}

for i, j ∈ {1,…, 2m}.

VI The universal covering space of a special c’-hyper-bolic hermitian manifold

In this section we shall study the universal covering space of a special c′-hyperbolic hermitian manifold.

Let (V^{2n+2m}, J, g, α_{1},…, α_{2m}) be a special c′-hyperbolic hermitian manifold and let A_{i} be (1 ≤ i ≤ 2m) as in (IV.2) and l, u, U, v, V as in (IV.4). Denote by c = c′ − \frac{1}{2}(\frac{n+m−1}{n−1})(l + 2c′) and by 𝔉 the foliation given by u = 0, α_{i} = 0, 1 ≤ i ≤ 2m. 𝔉 defines on V^{2n+2m} a foliation of dimension (2n − 1), which we call the canonical foliation of V^{2n+2m}. Using (IV.7), proposition IV.3 and corollary V.1, we deduce

Proposition VI.1 The canonical foliation 𝔉 of a special c′-hyperbolic hermitian manifold is totally geodesic with integrable normal bundle. Moreover, if 𝔉^{⊥} is the foliation determined by the normal bundle of 𝔉, then 𝔉^{⊥} also is totally geodesic and its leaves are of constant sectional curvature −(c′)^{2}.

Let i : N ⟶ V^{2n+2m} be the immersion of a generic leaf N of the canonical foliation 𝔉. We define an almost contact metric structure (φ_{N}, ξ_{N}, η_{N}, g_{N}) on N by

φ_{N}X = JX + (i^{∗}v)(X)U |_{N}, ξ_{N} = −V |_{N}, η_{N} = −(i^{∗}v), g_{N} = i^{∗}g  (VI.1)

for all X ∈ 𝔛(N). Then, from (IV.8), (IV.11) and since N_{J} = 0, we have

Proposition VI.2 If c ≠ 0 the almost contact metric structure (φ_{N}, ξ_{N}, η_{N}, g_{N}) on N is c-sasakian.

Now, we consider the immersion j : M ⟶ V^{2n+2m} of a generic leaf M of the foliation 𝔉^{⊥} on V^{2n+2m}. We define an almost contact metric structure (φ_{M}, ξ_{M}, η_{M}, g_{M}) on M by

φ_{M}(Y) = JY + (j^{∗}u)(Y)V |_{M}, ξ_{M} = U |_{M}, η_{M} = (j^{∗}u), g_{M} = j^{∗}g,  (VI.2)

for all Y ∈ 𝔛(M). Then, using (IV.1), (IV.8) and since N_{J} = 0, we obtain

Proposition VI.3 The almost contact metric structure (φ_{M}, ξ_{M}, η_{M}, g_{M}) on M is c′- kenmotsu.

Next, we give the following definition

Definition VI.1 A special c′-hyperbolic hermitian manifold (V^{2n+2m}, J, g, α_{1},…, α_{2m}) of dimension (2n + 2m), with c = c′ − \frac{1}{2}(\frac{n+m−1}{n−1})(l + 2c′) ≠ 0 is called k-special c’- hyperbolic hermitian manifold (k ∈ IR) if every leaf N of the canonical foliation 𝔉 is of constant φ_{N}-sectional curvature k, where (φ_{N}, ξ_{N}, η_{N}, g_{N}) is the induced c-sasakian structure on N given by (VI.1).

If (V^{2n+2m}, J, g, α_{1},…, α_{2m}) is a k-special c′-hyperbolic hermitian manifold then V^{2n+2m} is said to have a k-special c’-hyperbolic hermitian structure (J, g, α_{1},…, α_{2m}).

Now, we introduce a definition which will be useful in the sequel. Let N, k be a (2n − 1)-dimensional manifold and a real number respectively and let (H_{c′}^{2m+1}, (ds^{2})_{c′}) be the (2m + 1)-dimensional hyperbolic space.

Definition VI.2 A distinguished special c’-hyperbolic hermitian (respectively distinguished k-special c’-hyperbolic hermitian) structure on V^{2n+2m} = N × H_{c′}^{2m+1} is a special c′-hyperbolic hermitian (respectively k-special c′-hyperbolic hermitian) structure (J, g, α_{1},…, α_{2m}) on V^{2n+2m} with c = c′ − \frac{1}{2}(\frac{n+m−1}{n−1})(l + 2c′) ≠ 0 and such that:

1. The metric g is of the form g = dσ^{2} + (ds^{2})_{c′} where dσ^{2} is a Riemann metric on N and

2. The unit Lee 1-form u and the 1-forms α_{i}, 1 ≤ i ≤ 2m, are given by

u = \frac{dx_{2m+1}}{c′x_{2m+1}} , α_{i} = \frac{dx_{i}}{c′x_{2m+1}}

where (x_{1},…, x_{2m+1}) are the usual coordinates on H_{c′}^{2m+1}.

From (IV.5) and propositions IV.4 and VI.2 we deduce

Proposition VI.4 If (J, g, α_{1},…, α_{2m}) is a distinguished special c′-hyperbolic hermitian structure on V^{2n+2m} = N × H_{c′}^{2m+1}, then the manifold N carries an induced csasakian structure (φ_{N}, ξ_{N}, η_{N}, g_{N}), where c = c′ − \frac{1}{2}(\frac{n+m−1}{n−1})(l + 2c′), and the almost hermitian structure (J, g) on V^{2n+2m} is given by (III.1). Moreover, if (J, g, α_{1},…, α_{2m}) is a distinguished k-special c′-hyperbolic hermitian structure on V^{2n+2m}, then N is of constant φ_{N}-sectional curvature k.

Next, we shall describe the universal covering space of a special c′-hyperbolic hermitian manifold.

Theorem VI.1 The universal covering space of a complete special c′-hyperbolic hermitian manifold (V^{2n+2m}, J, g, α_{1},…, α_{2m}) of dimension 2n + 2m with Lee form ω and such that c = c′ − \frac{1}{2}(\frac{n+m−1}{n−1})(l + 2c′) ≠ 0, is a product space \overline{V}^{2n+2m} = N × H_{c′}^{2m+1}, where N is the universal covering space of an arbitrary leaf of the canonical foliation of V^{2n+2m} and H_{c′}^{2m+1} is the (2m + 1)-dimensional hyperbolic space. The lift of the special c′-hyperbolic hermitian structure to \overline{V}^{2n+2m} gives a distinguished special c′-hyperbolic hermitian structure on \overline{V}^{2n+2m}. Moreover, if the structure of V^{2n+2m} is a k-special c′-hyperbolic hermitian structure we have:

1. If k > −3c^{2}, then \overline{V}^{2n+2m} is almost complex isometric to S^{2n−1}(c, k) × H_{c′}^{2m+1},

2. If k = −3c^{2}, then \overline{V}^{2n+2m} is almost complex isometric to IR^{2n−1}(c) × H_{c′}^{2m+1} and,

3. If k < −3c^{2}, then \overline{V}^{2n+2m} is almost complex isometric to (IR × CD^{n−1})(c, k) × H_{c′}^{2m+1}.

Proof This result follows using propositions VI.1 and VI.4, theorem A of [BH] and the following lemma VI.1.

Lemma VI.1 Let M be a (2m + 1)-dimensional complete, simply connected, Riemannian manifold of constant negative curvature −(c′)^{2} (c′ ≠ 0) and U, A_{i} vector fields on M such that {U, A_{1},…, A_{2m}} form an orthonormal basis for M and [U, A_{i}] = c′A_{i}, [A_{i}, A_{j}] = 0 for i, j ∈ {1,…, 2m}. Then, there is an isometry F of M to the (2m + 1)- dimensional hyperbolic space H_{c′}^{2m+1}, satisfying

F_{∗}U = (c′x_{2m+1})\frac{∂}{∂x_{2m+1}} , F_{∗}A_{i} = (c′x_{2m+1})\frac{∂}{∂x_{i}} ,

for i ∈ {1,…, 2m}, where (x_{1},…, x_{2m+1}) are the usual coordinates on H_{c′}^{2m+1}.

References

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[ChG] D. CHINEA, J.C. GONZALEZ, A classification of almost contact metric manifolds, Annali di Mat. Pura ed Appl. (IV) 156 (1990), 15–36.

[CFL] L.A. CORDERO, M. FERNANDEZ, M. DE LEON, Compact locally conformal Kähler nilmanifolds, Geometriae Dedicata 21 (1986), 187–192.

[JV] D. JANSSENS, L. VANHECKE, Almost contact structure and curvature tensors, Kodai Math. J. 4 (1981), 1–27.

[KN] S. KOBAYASHI, K.NOMIZU, Foundations of Differential Geometry, Vol. II, Intersc. Publ., New York (1969).

[M] J.C. MARRERO, The local structure of trans-sasakian manifolds, to appear in Ann. di Mat. Pura ed Appl. (IV), CLXI (1992).

[MR] J.C. MARRERO, J. ROCHA, Sasakian m-hyperbolic locally conformal Kähler manifolds, to appear in Rend. Mat. di Roma.

[V1] I. VAISMAN, On locally conformal almost Kähler manifolds, Israel J. Math. 24 (1976), 338–351.

[V2] I. VAISMAN, Generalized Hopf manifolds, Geometriae Dedicata 13 (1982), 231–255.

J.C. MARRERO and J. ROCHA

Dpto. MAT. FUNDAMENTAL, UNIV. de LA LAGUNA, TENERIFE, CANARY ISLAND, SPAIN E-MAIL: JSC@IAC.ES

Received 26 August 1992 Revised 3 January 1993