A dimensional restriction for a class of contact manifolds

In this work we consider a class of contact manifolds $(M,\eta)$ with an associated almost contact metric structure $(\phi, \xi, \eta,g)$. This class contains, for example, nearly cosymplectic manifolds and the manifolds in the class $C_9\oplus C_{10}$ defined by Chinea and Gonzalez. All manifolds in the class considered turn out to have dimension $4n+1$. Under the assumption that the sectional curvature of the horizontal $2$-planes is constant at one point, we obtain that these manifolds must have dimension $5$.


Introduction
A contact manifold is a C ∞ odd-dimensional manifold M 2n+1 together with a 1−form η, usually called a contact form on M , such that η ∧ (dη) n = 0 everywhere on M ; the contact distribution D is the vector subbundle of T M defined by D := ker η.
We shall denote by D p the fiber of D at a point p; moreover if X ∈ X(M ) is a vector field, we shall write X ∈ D to indicate that X is a section of D.
It is known that dη| Dp×Dp is non degenerate and In [3] Chern showed that the existence of a contact form η on a manifold M 2n+1 implies that the structural group of the tangent bundle T M can be reduced to the unitary group U (n) × 1. Such a reduction of the structural group of the tangent bundle of a manifold M 2n+1 is called an almost contact structure. In term of structure tensors we say that an almost contact structure on a manifold M 2n+1 is a triple (φ, ξ, η) consisting of a tensor field φ of type (1, 1), a vector field ξ and a 1−form η satisfying φ 2 = −I + η ⊗ ξ, η(ξ) = 1, see [2] p. 43. It then follows directly from the definition of almost contact structure that φξ = 0, η • φ = 0, and that the endomorphism φ has rank 2n. If, in addition, M is endowed with a Riemannian metric g such that then (φ, ξ, η, g) is said to be an almost contact metric structure on M. Thus, setting Y = ξ, we have immediately that η(X) = g(X, ξ).
In this case g is an associated metric and we speak of a contact metric structure; the vector field ξ is the Reeb vector field of M 2n+1 [2]. Of course, it is possible to have a contact manifold (M 2n+1 , η) with Reeb vector field ξ and an almost contact metric structure (φ, ξ, η, g) on M without dη(X, Y ) = g(X, φY ).
Here ∇ denotes the Levi-Civita connection of g and ∇ξ is the bundle endomorphism of T M defined by X → ∇ X ξ. A well-known example of this situation is given by the five-dimensional sphere S 5 . This is a consequence of the following theorem ( [2], Theorem 6.14) Theorem. Let i : M 2n+1 →M 2n+2 be a hypersurface of a nearly Kähler manifold (M 2n+2 , J,g). Then the induced almost contact structure (φ, ξ, η, g) satisfies (∇ X φ)X = 0 if and only if the second fundamental form σ is proportional to (η ⊗ η)Ji * ξ.
If we consider S 5 as a totally geodesic hypersurface of S 6 , we have that the nearly Kähler structure (J,g) on S 6 , defined as in Example 4.5.3 of [2], induces an almost contact metric structure (φ, ξ, η, g) on S 5 satisfying (∇ X φ)X = 0.
In the next section we will treat contact manifolds with an almost contact metric structure satisfying condition ( * ). Such manifolds will result of dimension 4n + 1, n 1. If we suppose that φ is η-parallel and the sectional curvature of the horizontal 2-planes is constant at one point, then we obtain that these manifolds have dimension 5 (Theorem 1).
It is well known that the contact condition imposes strong restrictions on the Riemannian curvature of an associated metric. For example Z. Olszak in [6] proves that if an associated metric has constant curvature, then c = 1 and g must be a Sasakian metric; earlier D.E. Blair in [1] showed that in dimension 5 there are no flat associated metrics. We obtain that this is sometimes true also in the case of non associated metrics; for example when g is the metric of a nearly cosymplectic structure, see Theorem 3 in Section 3.

A class of contact manifolds
Let (φ, ξ, η, g) be an almost contact metric structure on a contact manifold (M, η). We denote by A the vector bundle endomorphism ∇ξ : Even if η is a contact form, ξ in general is not the Reeb vector field of η.
Proposition 1. Let (φ, ξ, η, g) be an almost contact metric structure on a con- Then dim M = 4n + 1, n 1 and B : D → D is a bundle automorphism.
Proof. We know that if (M, η) is a contact manifold then dη| D×D is non degenerate. Thus equation (1) imply that B is an automorphism. The fact that dim M = 4n + 1 is an application of Lemma 1, point 2. 2. if A is non singular and skew-symmetric then dim D ≡ 0 (mod 4).
Proof. Let X 1 , .., X n ∈ D be vectors such that {X 1 , JX 1 , .., X n , JX n } is a basis of D. We begin by proving the existence of a vector Y ∈ D such that Y, JY, AY are linearly independent. If by contradiction AY ∈ span{Y, JY } for all Y ∈ D, and hence A is represented with respect to our basis by a block-diagonal matrix of the form it follows a 1 = b 1 = 0. This contradicts the hypothesis A = 0. Let Y ∈ D be such that Y, JY, AY are linearly independent. We can observe that JAY / ∈ span{Y, JY, AY }, so that JAY = W + Z, with W ∈ span{Y, JY, AY } and Z ∈ span{Y, JY, AY } ⊥ , Z = 0. Thus we found Z ∈ D orthogonal to Y, JY, AY such that < Z, JAY > = 0. Now we assume that A is non singular and skew-symmetric. Let X ∈ D be an eigenvector of the symmetric linear operator A 2 . Since A anti-commutes with J, we have that JX, AX, JAX are also eigenvectors of A 2 . Moreover the vectors X, JX, AX, JAX are pairwise orthogonal and hence dim D 4.
Assume dim D > 4. By the Spectral Theorem we can choose Y ∈ D eigenvector of A 2 orthogonal to X, JX, AX, JAX. We have that X, JX, AX, JAX, Y, JY, AY, JAY are eigenvectors of A 2 , pairwise orthogonal and hence dim D 8. Iterating this argument we obtain the assertion.
After these preliminaries we can state our main result that involve contact manifolds with an almost contact metric structure satisfying condition ( * ). Theorem 1. Let (φ, ξ, η, g) be an almost contact metric structure on a contact manifold (M 2n+1 , η) such that for each X, Y, Z ∈ D.
Suppose there exist p ∈ M and c ∈ R such that the sectional curvature K p (π) = c, for each 2−plane π of D p . Then dim M = 5. Moreover A p is an isomorphism if and only if c = 0.
Proof. For each vector field Z on M , we denote by Z H and Z V the components of Z in D and in its orthogonal complement D ⊥ respectively. We say that Z H is the horizontal part of Z and Z V the vertical part of Z. Let ∇ be the Levi-Civita connection of g. We define a new linear connectioñ Then for each X, Y ∈ D (∇ X φ)Y = 0, and hence for each X, Y, Z ∈ D we have that∇ X Y ∈ D and alsõ whereR is the curvature tensor of∇. On the other hand, for each X, Y, Z ∈ D we haveR Comparing this last equation with (4) we have If c = 0, i.e., all the sectional curvatures K p (π) with π ⊂ D p vanish, then for every X, Y, Z ∈ D p Consider Y ∈ D p such that AY = 0. Hence if we take Z = φAY we have for every X ∈ D p and thus A : D p → D p has rank 3. Then there exists X ∈ D p , X = 0 such that AX = 0. Then, by (6) and (1) we have that In conclusion, the equation (7) becomes g(AY, AY )AX = g(AX, AY )AY + g(AX, φAY )φAY, for every X ∈ D p , yielding rank(A) 2. Now the contact condition implies that dim(ker A) n. Thus 2n 2 + n, namely n 2 and hence dim M 5. On the other hand, observing that (2) also implies that B anti-commute with φ, by Proposition 1, we have that dim M 5. Now suppose c = 0. Then A : D p → D p is an isomorphism. Indeed, assume X ∈ D p such that AX = 0, and Y ∈ D p orthogonal to X, φX, BX (for example take Y = φBX). For X 1 , X 2 , X 3 ∈ D we set S(X 1 , X 2 , X 3 ) :=R(X 1 , X 2 )φX 3 − φ(R(X 1 , X 2 )X 3 ). = cg(X, X)φY, so that X = 0. Now, supposing that (2) holds, we apply Lemma 1; fix Y, Z ∈ D p such that Z ∈ span{Y, φY, AY } ⊥ and g(Z, φAY ) = 0, then the equation (5) becomes This implies that rank(A) 5, so that n 2. As before, we conclude that dim M = 5.
From the above proof, we see that in the case c = 0 one can obtain the assertion replacing the condition (2) with the weaker condition dη(φX, φY ) = −dη(X, Y ), i.e. we have the following Corollary 1. Let (φ, ξ, η, g) be an almost contact metric structure on a contact manifold (M 2n+1 , η) such that for each X, Y, Z ∈ D. We suppose there exists p ∈ M such that the sectional curvature K p (π) = 0, for each 2−plane π of D p . Then dim M = 5.
If there exists p ∈ M and c ∈ R such that the sectional curvature K p (π) = c, for each 2−plane π of D p , then dim M = 5.

Nearly cosymplectic case
In this section we will show that there does not exist a flat nearly cosymplectic manifold (M, φ, ξ, η, g) with η a contact form. If moreover η is a contact form, then (d) for all p ∈ M 2n+1 A p is an isomorphism that anti-commutes with φ, Proof. Let ∇ be the Levi-Civita connection of g. Since ξ is Killing, we have for all X, Y ∈ T M . By Lemma 3.1 of [5] we have that Then dη(φX, φY ) = g(AφX, φY ) = −g(AX, Y ) = −dη(X, Y ), from which it follows that dη(X, ξ) = −dη(φX, φξ) = 0.
As a consequence of (a) we have that A p is an isomorphism. Finally (e) follows from (d) and the equation due to H. Endo [5].
Hence, as a consequence of Theorem 1, we can state Theorem 3. Let (M 2n+1 , η) be a contact manifold endowed with a nearly cosymplectic structure (φ, ξ, η, g). Suppose there exist p ∈ M and c ∈ R such that for each 2−plane π of D p , K p (π) = c. Then c = 0 and dim M = 5. [5] determines the curvature tensor of a nearly cosymplectic manifold (M, φ, ξ, η, g) with pointwise constant φ-sectional curvature c