On Almost Cosymplectic Manifolds with the Structure Vector Field ξ Belonging to the k-Nullity Distribution

Piotr Dacko

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract

Let M be an almost cosymplectic manifold whose structure vector field ξ belongs to the so-called k-nullity distribution. Then k ≤ 0 and the local structure of M is completely described. For a positive constant λ, it can be defined a Lie group G_{λ} endowed with a left invariant almost cosymplectic structure whose structure vector field ξ belongs to the (−λ^{2})-nullity distribution. It is proved that if k < 0, then M is locally isomorphic to G_{λ} with k = −λ^{2}; especially M is a locally homogeneous Riemannian manifold. We also show that in this case the manifold M is Ricci-pseudosymmetric. k = 0 if and only if M is locally a product of an open interval and an almost Kähler manifold.

Mathematics Subject Classification: 53C15, 53C25 Key words: almost cosymplectic manifold, k-nullity distribution, locally homogeneous Riemannian manifold, Ricci-pseudosymmetric manifold.

1 Preliminaries

An almost contact metric manifold M is by definition a (2n+1)-dimensional connected C^{∞}-manifold endowed with a quadruple (φ, ξ, η, g), where φ is a (1, 1)-tensor field, ξ a vector field, η a 1-form and g a Riemannian metric, such that (Blair [2])

φ^{2} = −I + η ⊗ ξ, η(ξ) = 1,
η(X) = g(X, ξ), g(φX, φY) = g(X, Y) − η(X)η(Y).

Here and in the sequel, X, Y, … denote arbitrary vector fields on M if it is not otherwise stated. Let Φ be the fundamental 2-form defined by Φ(X, Y) = g(φX, Y).

An almost contact metric manifold is said to be almost cosymplectic if both forms η and Φ are closed, i.e., dη = 0 and dΦ = 0 (Goldberg and Yano [9]). If an almost cosymplectic manifold is additionally normal, then it is called cosymplectic. An almost contact metric manifold is cosymplectic if and only if ∇φ = 0, ∇ being the operator of the covariant differentiation with respect to the Levi-Civita connection of g.

Balkan Journal of Geometry and Its Applications, Vol.5, No.2, 2000, pp. 47-60 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Let M be an almost cosymplectic manifold. By Frobenius’ Theorem, the 2ndimensional distribution D = Ker η is integrable. Let F be the foliation corresponding to D. The almost cosymplectic structure of M induces an almost Kähler structure (J̃, g̃) on any leaf M̃ of F. In fact, J̃X̃ = φX̃ and g̃(X̃, Ỹ) = g(X̃, Ỹ) for any vector fields X̃, Ỹ on M̃. If this structure is Kähler for any leaf M̃, then M is said to be almost cosymplectic with Kählerian leaves. An almost cosymplectic manifold has Kählerian leaves if and only if it fulfills the condition (Olszak [12])

(∇_{X}φ)Y = −g(φAX, Y)ξ + η(Y)φAX,

where A is a (1, 1)-tensor field defined by

A = −∇ξ. (1)

As it was already proved in [12],

g(AX, Y) = g(AY, X) (i.e., A is a symmetric linear operator),

Aφ + φA = 0, Aξ = 0, η ∘ A = 0. (2)

For any leaf M̃, the vector field ξ|_{M} is an orthonormal vector field of the leaf. Therefore, from (1) it follows that A|_{M} is the shape operator and H(X̃, Ỹ) = g̃(AX̃, Ỹ) is the second fundamental form of M̃.

Let k be a real constant. The k-nullity distribution N(k) on M is defined as the assignment M ∋ p → N_{p}(k), where N_{p}(k) is the k-nullity space at p defined as

N_{p}(k) = {z ∈ T_{p}M | R_{xy}z = k(g(y, z)x − g(x, z)y) for any x, y ∈ T_{p}M},

T_{p}M being the tangent space at p ∈ M [10]. Contact metric manifolds with the structure vector field ξ ∈ N(k) appeared and were investigated in the papers [15], [1], [13], [14], etc. H. Endo [6], [7], [8] studied almost cosymplectic manifolds with ξ ∈ N(k).

The purpose of this paper is to find the local structure of almost cosymplectic manifolds with the structure vector field ξ belonging to the k-nullity distribution and to prove that they are locally homogeneous. Moreover, we describe completely the curvature of this kind of manifolds and show that they are Ricci-pseudosymmetric.

2 Auxiliary results

Throughout this section, we assume that M is an almost cosymplectic manifold with the structure vector field ξ fulfilling the following relation

R_{XY} ξ = f(η(Y)X − η(X)Y) (3)

for any vector fields X, Y, where f is a certain function on M. In formula (3), f is not assumed to be a constant function. However, we shall show later that a function f realizing (3) must be constant.

We begin with the following lemmas.

Lemma 1 For the almost cosymplectic manifold, we have R_{ξZ}ξ = −(∇_{ξ}A)Z + A^{2}Z, (4) (∇_{X}A)Y − (∇_{Y} A)X = 0 if X, Y ⊥ ξ. (5) Proof. By direct calculations, using (1), we find

(6)  R_{XY} ξ = ∇_{X}∇_{Y} ξ − ∇_{Y} ∇_{X}ξ − ∇_{[X,Y]}ξ
= − ∇_{X}(AY) + ∇_{Y} (AX) + A[X, Y]
= − (∇_{X}A)Y + (∇_{Y} A)X.

¿From (6) with X = ξ, we have R_{ξY} ξ = −(∇_{ξ}A)Y + (∇_{Y} A)ξ. (7) Applying (1) and (2), we compute

(∇_{Y} A)ξ = ∇_{Y} Aξ − A∇_{Y} ξ = A^{2}Y,

which substituted into (7) gives (4). Finally, if X, Y ⊥ ξ, then R_{XY} ξ = 0 by (3). Therefore, formula (5) follows from (6). Lemma 2 For the almost cosymplectic manifold, tensor field A satisfies the following relations

(8)  A^{2}X = −fX + fη(X)ξ,
(9)  (∇_{ξ}A)X = 0,
(10)  (L_{ξ}A)X = 0,

where L_{ξ} indicates the Lie differentiation operator in the direction ξ. Moreover, function f is nonpositive. Proof. By (3), it follows that R_{ξφXφY ξ} + R_{ξXY ξ} = −2fη(X)η(Y) + 2fg(X, Y). (11) On the other hand, we have in general ([11], equ. (4.10))

R_{ξφXφY ξ} + R_{ξXY ξ} = −2g(∇_{X}ξ, ∇_{Y} ξ),

which in view of (1) and (2) turns into

R_{ξφXφY ξ} + R_{ξXY ξ} = −2g(A^{2}X, Y).

Comparing the last identity with (11), we obtain

g(A^{2}X, Y) = fη(X)η(Y) − fg(X, Y),

which implies (8). Note that (8) obviously implies f ≤ 0.

From (3) and (8), it follows that

R_{ξY} ξ = A^{2}Y,

which compared with (4) gives (9). Finally, making straightforward computations, we get

(L_{ξ}A)X = [ξ, AX] − A[ξ, X] = (∇_{ξ}A)X = 0,

where we have also used (1) and (9).

Lemma 3 For the almost cosymplectic manifold, we have

(12)  df(ξ) = 0,
(13)  g((∇_{X}A^{2})Y, Z) = −df(X)g(Y, Z),
(14) g((∇_{X}A)Z, AY) − g((∇_{Y} A)Z, AX) = − df(X)g(Y, Z) + df(Y)g(X, Z),

for any X, Y, Z orthogonal to ξ.

Proof. Differentiating covariantly (8) we obtain

(∇_{X}A^{2})Y = − df(X)Y + df(X)η(Y)ξ − fg(AX, Y)ξ − fη(Y)AX (15)

for any X, Y ∈ X (M). With X = ξ, this becomes, using also (9) and (2)

df(ξ)Y = df(ξ)η(Y)ξ,

for any Y ∈ X (M), which leads to (12).

From (15) it follows

g((∇_{X}A^{2})Y, Z) = −df(X)g(Y, Z)

for any X, Y, Z orthogonal to ξ, which is just (13).

Alternating (13) with respect to X, Y, we find

g((∇_{X}A^{2})Y, Z) − g((∇_{Y} A^{2})X, Z) = − df(X)g(Y, Z) + df(Y)g(X, Z) (16)

for any X, Y, Z ⊥ ξ. The left hand side of (16) equals

g((∇_{X}A)Z, AY) − g((∇_{Y} A)Z, AX) + g((∇_{X}A)Y − (∇_{Y} A)X, AZ),

which, by virtue of (5), reduces to

g((∇_{X}A)Z, AY) − g((∇_{Y} A)Z, AX).

Therefore, (16) implies (14).

Now we are going to prove that f is constant under the assumption (3). Since the proof in dimensions ≥ 5 differs from that in dimension 3, we present it in two parts.

Theorem 1 If an almost cosymplectic manifold satisfies the condition

R_{XY} ξ = f(η(Y)X − η(X)Y)

for any vector fields X, Y and a certain function f, then f must be constant.

Proof. The case dim M = 2n + 1 ≥ 5. By (12), it is suffcient to show that df(X) = 0 for any X ⊥ ξ. Restrict consideration to an arbitrary fixed point p ∈ M.

If f(p) = 0, then A^{2}(p) = 0, by (8). Since A(p) is symmetric, A(p) = 0 and consequently (∇A^{2})(p) = 0. Therefore, from (13) it follows that df(X)g(Y, Z) = 0 for any X, Y, Z ∈ T_{p}M, X, Y, Z ⊥ ξ. Hence df(X) = 0 for any X ∈ T_{p}M, X ⊥ ξ.

Now, let f(p) < 0 (note that always f(p) ≤ 0, see Proposition 2), and suppose λ = \sqrt{|f(p)|}. By (8), A^{2}X = λ^{2}X for any X ∈ T_{p}M, X ⊥ ξ. Since A anticommutes with φ (cf. (2)), both λ and −λ are eigenvalues of the same multiplicity equal to n (n ≥ 2 by the assumption). Let D_{−} and D_{+} denote eigenspaces corresponding to the eigenvalues −λ and λ, respectively.

Let X be an arbitrary vector from D_{−}. Take a unit vector Y ∈ D_{−} such that g(X, Y) = 0. By (14) with Z = Y, we obtain

−df(X) = −λ(g((∇_{X}A)Y, Y) − g((∇_{Y} A)Y, X))
= −λg((∇_{X}A)Y − (∇_{Y} A)X, Y) = 0,

where in the last equation we have used (5). In the same way we prove that df(X) = 0 if X ∈ D_{+}. Thus, the proof is complete in case of dim M ≥ 5.

The case dim M = 3. Define λ = \sqrt{|f|}. Let suppose that λ > 0 at some p ∈ M. Then, on a connected neighbourhood U of p, there exist an orthonormal frame (E_{0}, E_{1}, E_{2}), such that

E_{0} = ξ,  AE_{1} = −∇_{E_{1}}E_{0} = −λE_{1},  AE_{2} = −∇_{E_{2}}E_{0} = λE_{2}. (17)

In virtue of (17), we obtain

∇_{E_{0}}AE_{1} = −(E_{0}λ)E_{1} − λ∇_{E_{0}}E_{1}, (18)

By (9) , we have ∇_{E_{0}}AE_{1} = A∇_{E_{0}}E_{1}. (19)

It follows from (12) that E_{0}λ = 0. Therefore, (18), (19) lead to

A∇_{E_{0}}E_{1} = −λ∇_{E_{0}}E_{1}. (20)

In the same way, we get A∇_{E_{0}}E_{2} = λ∇_{E_{0}}E_{2}. (21)

The relation (20) (resp., (21)) shows that ∇_{E_{0}}E_{1} (resp., ∇_{E_{0}}E_{2}) is an eigenvector field of A corresponding to the eigenvalue function −λ (resp. λ). By (17), ∇_{E_{0}}E_{1} (resp. ∇_{E_{0}}E_{2}) is proportional to E_{1} (resp. E_{2}). Consequently, ∇_{E_{0}}E_{1} = ∇_{E_{0}}E_{2} = 0 on U. The last two formulas and (17) lead to

\begin{matrix}[E_{0}, E_{1}] = ∇_{E_{0}}E_{1} − ∇_{E_{1}}E_{0} = −λE_{1}, \\ [E_{0}, E_{2}] = ∇_{E_{0}}E_{2} − ∇_{E_{2}}E_{0} = λE_{2}, \end{matrix} (22)

where [·, ·] indicates the Poisson bracket of vector fields. The Jacobi’s identity and (22) give

(23)  [E_{0}, [E_{1}, E_{2}]] = [[E_{0}, E_{1}], E_{2}] + [E_{1}, [E_{0}, E_{2}]]
= dλ(E_{2})E_{1} + dλ(E_{1})E_{2}.

On the other hand, by (5)

(∇_{E_{1}}A)E_{2} − (∇_{E_{2}}A)E_{1} = 0,

or

∇_{E_{1}}AE_{2} − A∇_{E_{1}}E_{2} − ∇_{E_{2}}AE_{1} + A∇_{E_{2}}E_{1} = 0.

With the help of (17), we can rewrite the above formula in the following way

dλ(E_{1})E_{2} + dλ(E_{2})E_{1} + λ∇_{E_{1}}E_{2} + λ∇_{E_{2}}E_{1} = A[E_{1}, E_{2}].

Projecting the last equation onto E_{1} and taking into account (17) and the symmetry of A, we find

dλ(E_{2}) + λg(∇_{E_{1}}E_{2}, E_{1}) = −λg([E_{1}, E_{2}], E_{1}).

Because of

g(∇_{E_{1}}E_{2}, E_{1}) = g([E_{1}, E_{2}], E_{1}),

the previous formula reduces to

g([E_{1}, E_{2}], E_{1}) = −\frac{1}{2λ}dλ(E_{2}). (24)

Similarly, one obtains

g([E_{1}, E_{2}], E_{2}) = \frac{1}{2λ}dλ(E_{1}). (25)

We also claim the folowing

(26)  g([E_{1}, E_{2}], E_{0}) = g(∇_{E_{1}}E_{2} − ∇_{E_{2}}E_{1}, ξ)
= −g(∇_{E_{1}}ξ, E_{2}) + g(∇_{E_{2}}ξ, E_{1})
= −2dη(E_{1}, E_{2}) = 0.

From (24), (25), (26), we deduce that

[E_{1}, E_{2}] = −d(ln \sqrt{λ})(E_{2})E_{1} + d(ln \sqrt{λ})(E_{1})E_{2}.

Therefore,

(27)  [E_{0}, [E_{1}, E_{2}]] = [E_{0}, −d(ln \sqrt{λ})(E_{2})E_{1} + d(ln \sqrt{λ})(E_{1})E_{2}]
= − E_{0}(E_{2}(ln \sqrt{λ}))E_{1} + E_{0}(E_{1}(ln \sqrt{λ}))E_{2}
− E_{2}(ln \sqrt{λ})[E_{0}, E_{1}] + E_{1}(ln \sqrt{λ})[E_{0}, E_{2}].

By (22) and E_{0}(ln \sqrt{λ}) = 0, we have

E_{0}(E_{2}(ln \sqrt{λ})) = [E_{0}, E_{2}](ln \sqrt{λ}) = λE_{2}(ln \sqrt{λ}),
E_{0}(E_{1}(ln \sqrt{λ})) = [E_{0}, E_{1}](ln \sqrt{λ}) = −λE_{1}(ln \sqrt{λ}).

By virtue of these equalities and also (22), formula (27) reduces to [E_{0}, [E_{1}, E_{2}]] = 0. In this situation, (23) leads to E_{1}λ = 0, E_{2}λ = 0, which together with E_{0}λ = 0 yields λ = const ., and consequently f = const . on the neighbourhood U. Since M is connected, this property implies the constancy of f on M. This completes the proof in case of dim M = 3.

3 The local structure

In this section, we deal with almost cosymplectic manifolds M whose structure vector field ξ belongs to the k-nullity distribution, i.e., those satisfying the condition

R_{XY} ξ = k(η(Y)X − η(X)Y) (28)

for any vector fields X, Y, where k = const . ≤ 0.

At first, we consider the case k = 0, which is exceptional.

Theorem 2 For an almost cosymplectic manifold M, the structure vector field ξ belongs to the 0-nullity distribution if and only if M is locally the product of an open interval and an almost Kähler manifold.

Proof. Equality (8) shows that k = 0 if and only if A = 0. On the other hand, it is well-known that A = 0 (∇ξ = 0) if and only if M is locally the product of an open interval and an almost Kähler manifold.

In the rest of this section, we describe the local structure of almost cosymplectic manifolds satisfying (28) with k < 0.

Proposition 1 Let M be an almost cosymplectic manifold with the structure vector field ξ belonging to the k-nullity distribution with k < 0. Then M is almost cosymplectic with Kählerian leaves.

Proof. Using (28), we find that

R_{XY φZξ} − R_{φXφY φZξ} − R_{φXY Zξ} − R_{XφY Zξ}
= −2k(η(Y)g(X, φZ) − η(X)g(Y, φZ)).

The following curvature identity is already known for almost cosymplectic manifolds (see [11])

R_{XY φZξ} − R_{φXφY φZξ} − R_{φXY Zξ} − R_{XφY Zξ} = −2(∇_{AZ}Φ)(X, Y).

Comparing the last two relations, we obtain

k(η(Y)g(X, φZ) − η(X)g(Y, φZ)) = (∇_{AZ}Φ)(X, Y).

Putting AZ instead of Z in this equation and taking into account (8), we obtain

η(Y)g(X, φZ) − η(X)g(Y, φZ) = −(∇_{Z}Φ)(X, Y) + η(Z)(∇_{ξ}Φ)(X, Y).

Hence, since ∇_{ξ}Φ = 0 (see [11]), we have

(∇_{Z}φ)X = − g(φAZ, X)ξ + η(X)φAZ. (29)

Thus, by the result cited in the Introduction, M is almost cosymplectic with Kählerian leaves.

Formula (29) has already been obtained by Endo [6]. However, we included the proof for completness and since it differs in some details from that in [6]; for instance Endo does not use the notion of almost cosymplectic manifold with Kählerian leaves.

Lemma 4 For an almost cosymplectic manifold with the structure vector field ξ belonging to the k-nullity distribution with k < 0, we have

(30)  (L_{ξ}g)(X, Y) = −2g(AX, Y),
(31)  (L_{ξ}^{2}g)(X, Y) = − 4kg(X, Y) + 4kη(X)η(Y),

where L_{ξ}^{2} = L_{ξ} ∘ L_{ξ}.

Proof. Equation (30) holds realy for any almost cosymplectic manifold, as it is shown in [4]. Using (30), we find

(L_{ξ}^{2}g)(X, Y) = (L_{ξ}(L_{ξ}g))(X, Y) = 4g(A^{2}X, Y) − 2g((L_{ξ}A)X, Y).

This, in virtue of (8) and (10), turns into (31).

Theorem 3 Let M be an almost cosymplectic manifold and (φ, ξ, η, g) its almost cosymplectic structure. The structure vector field ξ of the manifold M belongs to the k-nullity distribution with k < 0 if and only if for any point p ∈ M there exists a coordinate neighbourhood (U, (t, x^{1} …, x^{2n})), p ∈ U, on which

(32)  ξ = \frac{∂}{∂t}, η = dt,
(33)  g = dt ⊗ dt + e^{2λt} ∑_{μ=1}^{n} dx^{μ} ⊗ dx^{μ} + e^{−2λt} ∑_{μ=1}^{n} dx^{n+μ} ⊗ dx^{n+μ},

(34)  φ = e^{2λt} ∑_{μ=1}^{n} dx^{μ} ⊗ \frac{∂}{∂x^{n+μ}} − e^{−2λt} ∑_{μ=1}^{n} dx^{n+μ} ⊗ \frac{∂}{∂x^{μ}} ,

where λ = \sqrt{|k|}.

Proof. Following the general procedure due to Olszak [12] and our Proposition 1, the local shape of the almost cosymplectic structure (φ, ξ, η, g) can be described in the following way. Let p ∈ M. There is a neighbourhood U of p such that U = (−a, a)×Ũ, where (−a, a) is an open interval with coordinate t and Ũ is a 2n-dimensional manifold, p = (0, p̃), p̃ ∈ Ũ. Moreover,on Ũ, there is a 1-parameter family of Kähler stuctures (J_{t}, G_{t}), −a < t < a, whose fundamental forms do not depend on the parameter t, and on U the almost cosymplectic structure is given by

ξ = \frac{∂}{∂t}, η = (dt, 0), φ = (0, J_{t}), g = dt ⊗ dt + G_{t}. (35)

Treating Ũ as a hypersurface given by t = 0, the Kähler structure (J, G) = (J_{0}, G_{0}) is just the induced Kähler structure on Ũ. Let Ã denote the operator A restricted to Ũ. Thus, Ã is the shape operator of Ũ.

Note that by (2) ÃJ + JÃ = 0. (36)

This and (8) imply that −λ, λ are constant eigenvalues of Ã both of the same multiplicity. Let D_{−}, D_{+} be the corresponding eigendistributions, then dim D_{−} = dim D_{+} = n.

Ã as the shape operator must satisfy Codazzi equation

R_{_{X}_{Y}}ξ = −( ˜∇_{X}Ã)Ỹ + ( ˜∇_{Y} Ã)X̃.

By applying (28), the last identity turns into

0 = −( ˜∇_{X}Ã)Ỹ + ( ˜∇_{Y} Ã)X̃.

Being a Codazzi tensor field and having two different constant eigenvalues, the tensor field Ã must be parallel. Equivalently, the distributions D_{−}, D_{+} are parallel and the Riemann metric G is locally productable. We restrict (Ũ, G) to the Riemann product neighbourhood of p̃, Ũ = Ũ_{1} × Ũ_{2}, G = G_{1} × G_{2}, p̃ = (p̃_{1}, p̃_{2}), p̃_{1} ∈ Ũ_{1}, p̃_{2} ∈ Ũ_{2}, Ũ_{1}, Ũ_{2} being the integral submanifolds of D_{−}, D_{+}, respectively, and G_{1}, G_{2} are the Riemann metrics induced on Ũ_{1}, Ũ_{2}.

The second fundamental form H has also the product shape, precisely

H(X̃_{1}, Ỹ_{1}) = −λG_{1}(X̃_{1}, Ỹ_{1}), H(X̃_{2}, Ỹ_{2}) = λG_{2}(X̃_{2}, Ỹ_{2}), H(X̃_{1}, Ỹ_{2}) = 0. (37)

In the above and in what follows, we denote by X̃_{1}, Ỹ_{1}, Z̃_{1} … and X̃_{2}, Ỹ_{2}, Z̃_{2} … vector fields tangent to Ũ_{1} and Ũ_{2}, respectively.

Denote by R̃^{1} the curvature tensor of G_{1}. As it is well-known, for the product Riemann metric, we have

R̃_{_{X_{1}}_{Y_{1}}}^{1} Z̃_{1} = R̃_{_{X_{1}}_{Y_{1}}} Z̃_{1}, R̃_{_{X_{1}}_{Y_{1}}}^{1} Z̃_{2} = 0.

By (36), JZ̃_{1} is tangent to Ũ_{2} and consequently

R̃_{_{X_{1}}_{Y_{1}}}^{1} Z̃_{1} = R̃_{_{X_{1}}_{Y_{1}}} Z̃_{1} = −JR̃_{_{X_{1}}_{Y_{1}}}JZ̃_{1} = 0.

This means that G_{1} is locally flat. Analogously, we prove that G_{2} is locally flat, too.

Restrict Ũ_{1} to a neighbourhood of p̃_{1} with coordinates (x^{1}, …, x^{n}) , and Ũ_{2} to a neighbourhood of p̃_{2} with coordinates (x^{n+1}, …, x^{2n}) such that

G_{1} = ∑_{μ=1}^{n} dx^{μ} ⊗ dx^{μ}, G_{2} = ∑_{μ=1}^{n} dx^{n+μ} ⊗ dx^{n+μ}. (38)

Note that the coefficients of J with respect to the coordinates (x^{1}, …, x^{2n}) on Ũ are constant. This provides the orthonormal frame

(\frac{∂}{∂x^{1}} , …, \frac{∂}{∂x^{n}} , J \frac{∂}{∂x^{1}} , …, J \frac{∂}{∂x^{n}} )

to be holonomic. We change linearly coordinates (x^{n+1}, …, x^{2n}) to the new ones, which are denoted by the same letters, in the way that

J \frac{∂}{∂x^{μ}} = \frac{∂}{∂x^{n+μ}} , (39)

and the local expression of G_{2} given in (38) is still fulfilled. For the fundamental form Ω (Ω(X̃, Ỹ) = G(JX̃, Ỹ)) of (J, G), by (39), we have

Ω = 2 ∑_{μ=1}^{n} dx^{μ} ∧ dx^{n+μ}. (40)

Now, we back to the metric g on U = (−a, a)×Ũ_{1} ×Ũ_{2}. Let g_{ij} be the components of g with respect to the coordinates (x^{0} = t, x^{1}, …, x^{2n}). By (35), we have g_{00} = 1, g_{i0} = 0, 1 ≤ i ≤ 2n. As concerns the functions g_{ij}, 1 ≤ i, j ≤ 2n, by (31), they must satisfy the following 2-nd order differential equation

\frac{∂^{2}g_{ij}}{∂t^{2}} = 4λ^{2}g_{ij}, (41)

which has the only solution of the form

(42) g_{ij} = α_{ij} cosh(2λt) − \frac{1}{λ}β_{ij} sinh(2λt)

where

(43)  α_{ij}(x^{1}, …, x^{2n}) = g_{ij}(0, x^{1}, …, x^{2n}),
(44)  β_{ij}(x^{1}, …, x^{2n}) = −\frac{1}{2} \frac{∂g_{ij}}{∂t} (0, x^{1}, …, x^{2n}).

In view of (43), α_{ij} are just the components of the induced metric G = G_{0} on Ũ. In virtue of (30), the second fundamental form H is given by

H(X̃, Ỹ) = g(ÃX̃, Ỹ) = g(AX̃, Ỹ) = −\frac{1}{2}(L_{ξ}g)(X̃, Ỹ).

By (44), β_{ij} are the components of H on Ũ.

On virtue of formulas (38), (37), we have

α_{ij} = δ_{ij}, β_{μν} = −λδ_{μν}, β_{(n+μ)(n+ν)} = λδ_{μν},

otherwise β_{ij} = 0. Consequently, with the help of (42), we obtain

g_{μν} = e^{2λt}δ_{μν}, g_{(n+μ)(n+ν)} = e^{−2λt}δ_{μν}, (45)

otherwise g_{ij} = 0. This proves formula (33).

Formulas (35), (40) lead to

Φ_{μ(n+ν)} = −Φ_{(n+ν)μ} = δ_{μν} (46)

and Φ_{ij} = 0 in other cases. With the help of (45), (46) and φ_{i}^{j} = ∑_{s} Φ_{is}g^{sj}, we compute the components of the operator φ

φ_{ν}^{n+μ} = e^{2λt}δ_{ν}^{μ}, φ_{n+ν}^{μ} = −e^{−2λt}δ_{ν}^{μ}, (47)

otherwise φ_{i}^{j} = 0. This gives (34).

Conversely, let us assume that an almost cosymplectic structure is given locally by (32)-(34). Define a local orthonormal frame (E_{0}, E_{1}, …, E_{2n}) by putting

E_{0} = ξ = \frac{∂}{∂t}, E_{μ} = e^{−λt} \frac{∂}{∂x^{μ}} , E_{n+μ} = e^{λt} \frac{∂}{∂x^{n+μ}}

for μ = 1, …, n. The non-zero Lie brackets [E_{i}, E_{j}] are

[E_{0}, E_{μ}] = −λE_{μ}, [E_{0}, E_{n+μ}] = λE_{n+μ}. (48)

Moreover, the non-zero covariant derivatives are

\begin{matrix}∇_{E_{μ}}E_{0} = λE_{μ}, & ∇_{E_{μ}}E_{μ} = −λE_{0}, \\ ∇_{E_{n+μ}}E_{0} = −λE_{n+μ}, & ∇_{E_{n+μ}}E_{n+μ} = λE_{0}. \end{matrix} (49)

With help of (48) and (49), we find

R_{E_{i}E_{j}}E_{0} = 0, R_{E_{0}E_{j}}E_{0} = λ^{2}E_{j}

for any i, j = 1, …, 2n, i < j. Consequently, we have (28) with k = −λ^{2}.

4 Homogeneity

Let λ be a real positive number and G_{λ} be the solvable non-nilpotent Lie group whose underlying manifold is the Cartesian space R^{2n+1} and the multiplication is defined by the formula

(t, x^{1}, …, x^{n}, x^{n+1}, …, x^{2n}) ∗ (s, y^{1}, …, y^{n}, y^{n+1}, …, y^{2n}) =

(t + s, e^{−λt}y^{1} + x^{1}, …, e^{−λt}y^{n} + x^{n}, e^{λt}y^{n+1} + x^{n+1}, …, e^{λt}y^{2n} + x^{2n})

for any (t, x^{1}, …, x^{2n}), (s, y^{1}, …, y^{2n}) ∈ R^{2n+1}. As a basis of the Lie algebra of G_{λ}, we take the following left invariant vector fields

E_{0} = \frac{∂}{∂t}, E_{μ} = e^{−λt} \frac{∂}{∂x^{μ}} , E_{n+μ} = e^{λt} \frac{∂}{∂x^{n+μ}} . (50)

With respect to this base, we define a left invariant almost contact metric structure (φ, ξ, η, g) on G_{λ} as follows

\begin{matrix}ξ = E_{0},  g(E_{i}, E_{j}) = δ_{ij},  η(·) = g(ξ, ·), \\ φE_{0} = 0,  φE_{μ} = E_{n+μ},  φE_{n+μ} = −E_{μ}. \end{matrix} (51)

In terms of the global coordinates (t, x^{1}, …, x^{2n}) on G_{λ}, the structure (φ, ξ, η, g) is given just by the formulas (32)–(34). By Theorem 3, the quadruple (φ, ξ, η, g) becomes an almost cosymplectic structure on G_{λ} whose structure vector field ξ belongs to the k-nullity distribution with k = −λ^{2}.

To formulate our next theorem, we need the following notion. Let M, M′ be almost cosymplectic manifolds and (φ, ξ, η, g), (φ′, ξ′, η′, g′) their almost cosymplectic structures, respectively. We say that M, M′ are locally isomorphic if for any points p ∈ M, p′ ∈ M′ there exist neighborhhoods U, U′ of p, p′, respectively, and a diffeomorphism ψ : U → U′, ψ(p) = p′, such that

g = ψ^{∗}g′, ψ_{∗} ∘ φ = φ′ ∘ ψ_{∗}, ψ_{∗}ξ = ξ′, η = ψ^{∗}η′.

By virtue of Theorem 3 and the above construction, the following theorem is true:

Theorem 4 Let M be an almost cosymplectic manifold whose structure vector field ξ belongs to the k-nullity distribution with k < 0. Then M is locally isomorphic with G_{λ} endowed with the structure (φ, ξ, η, g) as in (51) and k = −λ^{2}.

As a consequence of the last theorem, we have following:

Theorem 5 An almost cosymplectic manifold M whose structure vector field ξ belongs to the k-nullity distribution with k < 0 is a locally homogeneous Riemaniann manifold. If M is additionally simply connected and complete, then it is Riemannian homogeneous.

Remark. In the paper [3], there was defined a wider class of solvable non-nilpotent Lie groups G(k_{1}, …, k_{n}) admitting left invariant non-cosymplectic almost cosymplectic structures. The importance of this class stem from the fact that these Lie groups admit discretes subgroups for which coset spaces are compact and inherit almost cosymplectic structures. It is worthwhile to notice that our Lie group G_{λ} endowed with the almost cosymplectic structure (51) belong to this class; indeed, the group G_{λ} is in fact the Lie group G(k_{1}, …, k_{n}) from [3] with k_{1} = … = k_{n} = −λ.

5 Curvature properties

In this section, we give a complete description of the curvature R of an almost cosymplectic manifold M satisfying (28) with k < 0, and next we prove that it is Riccipseudosymmetric.

First, let R_{XY} and X ∧ Y be the endomorphisms acting on vector fields as follows

R_{XY} Z = [∇_{X}, ∇_{Y}]Z − ∇_{[X,Y]}Z,
(X ∧ Y)Z = g(Y, Z)X − g(X, Z)Y.

We extend R_{XY} and X ∧ Y to derivations of the tensor algebra on M, so that for the Ricci curvature tensor S, we have

(R_{XY} S)(Z, W) = − S(R_{XY} Z, W) − S(Z, R_{XY} W),
((X ∧ Y)S)(Z, W) = − S((X ∧ Y)Z, W) − S(Z, (X ∧ Y)W),

respectively.

Following R. Deszcz [5], M is said Ricci-pseudosymmetric if its Ricci tensor S satisfies the condition R_{XY} S = L (X ∧ Y)S (52)

for a certain function L and any vector fields X, Y.

Lemma 5 For an almost cosymplectic manifold satisfying (28) with k = −λ^{2} < 0, the curvature tensor R and the Ricci tensor S are given by

(53) R_{XY} = − (AX) ∧ (AY) − λ^{2}η ⊗ ((X ∧ Y)ξ) − λ^{2}(η ∘ (X ∧ Y)) ⊗ ξ,
(54)  S = −2nλ^{2}η ⊗ η.

Proof. In virtue of the results of the previous section, it is sufficient to prove the assertion only in case of M = G_{λ} and the almost cosymplectic structure is defined as in (51). We denote the Lie algebra of G_{λ} by g_{λ}. Note that (48)-(49) enable us to write the following

\begin{matrix}∇_{X}Y = − η(Y)AX + g(AX, Y)ξ, \\ [X, Y] = − η(Y)AX + η(X)AY, \end{matrix} (55)

for X, Y ∈ g_{λ}. As a consequence of the above, we also have

\begin{matrix}∇_{X}AY = ∇_{Y} AX = λ^{2}(g(X, Y) − η(X)η(Y))ξ, \\ ∇_{AX}Y = − λ^{2}η(Y)X + λ^{2}g(X, Y)ξ. \end{matrix} (56)

For the curvature, using (55) and (56), we find

R_{XY} Z = −g(AY, Z)AX + g(AX, Z)AY − λ^{2}η(Y)η(Z)X
+ λ^{2}η(X)η(Z)Y + λ^{2}η(Y)g(Z, X)ξ − λ^{2}η(X)g(Z, Y)ξ,

which is just equivalent to (53). Hence, we find (54).

Theorem 6 Let M be an almost cosymplectic manifold whose structure vector field ξ belongs to the k-nullity distribution with k = −λ^{2} < 0. Then M is Ricci-pseudosym-metric, precisely, it realizes (52) with L = −λ^{2}.

Proof. Using (54), we compute R_{XY} S. Namely, we have

(R_{XY} S)(Z, W) = −2nλ^{2}(R_{XY} η)(Z)η(W) − 2nλ^{2}η(Z)(R_{XY} η)(W). (57)

Note that

(R_{XY} η)(Z) = − λ^{2}g(Z, X)η(Y) + λ^{2}g(Z, Y)η(X),

which (57) turns into

(R_{XY} S)(Z, W) = 2nλ^{4}g(Z, X)η(Y)η(W) − 2nλ^{4}g(Z, Y)η(X)η(W)
+2nλ^{4}g(W, X)η(Y)η(Z) − 2nλ^{4}g(W, Y)η(X)η(Z)
= −λ^{2}((X ∧ Y)S)(Z, W).

This completes the proof.

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Institute of Mathematics, Wrocław University of Technology Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland