ON A CLASS OF CONTACT RIEMANNIAN MANIFOLDS

We determine a locally symmetric or a Ricci-parallel contact Riemannian manifold which satisfies a D-homothetically invariant condition.


Introduction.
In [8] Tanno proved that a locally symmetric K-contact Riemannian manifold is of constant curvature 1, which generalizes the corresponding result for a Sasakian manifold due to Okumura [6].For dimensions greater than or equal to 5 it was proved by Olszak [7] that there are no contact Riemannian structures of constant curvature unless the constant is 1 and in which case the structure is Sasakian.Further, Blair and Sharma [4] proved that a 3-dimensional locally symmetric contact Riemannian manifold is either flat or is Sasakian and of constant curvature 1.By the recent result [5] and private communication with Blair we know that the simply connected covering space of a complete 5-dimensional locally symmetric contact Riemannian manifold is either S 5 (1) or E 3 × S 2 (4).The question of the classification of locally symmetric contact Riemannian manifolds in higher dimensions is still open.
On the other hand, recently, Blair, Koufogiorgos and Papantoniou [3] introduced a class of contact Riemannian manifolds which is characterized by the equation where κ, µ are constant and 2h is the Lie derivative of φ in the direction ξ.It is remarkable that this class of spaces is invariant under D-homothetic deformations (see [3]).It was also proved in [3] that a Sasakian manifold, in particular, is determined by κ = 1 and further that this class contains the tangent sphere bundle of Riemannian manifolds of constant curvature.In this paper, we determine a locally symmetric or a Ricci-parallel contact Riemannian manifold which satisfies (1.1).More precisely, we prove the following two Theorems 1.1 and 1.2 in Sections 3 and 4.
Theorem 1.1.Let M be a contact Riemannian manifold satisfying (1.1).Suppose that M is locally symmetric.Then M is the product of flat (n+1)-dimensional manifold and an n-dimensional manifold of positive constant curvature equal to 4, or a space of constant curvature 1 and in which case the structure is Sasakian.Theorem 1.2.Let M be a contact Riemannian manifold satisfying (1.1).Suppose that M is Ricci-parallel.Then M is the product of flat (n+1)-dimensional manifold and an n-dimensional manifold of positive constant curvature equal to 4 or an Einstein-Sasakian manifold.

Preliminaries.
All manifolds in the present paper are assumed to be connected and of class C ∞ .A (2n + 1)-dimensional manifold M 2n+1 is said to be a contact manifold if it admits a global 1-form η such that η∧(dη) n ≠ 0 everywhere.Given a contact form η, we have a unique vector field ξ, which is called the characteristic vector field, satisfying η(ξ) = 1 and dη(ξ, X) = 0 for any vector field X.It is well known that there exists an associated Riemannian metric g and a (1, 1)-type tensor field φ such that where X and Y are vector fields on M. From (2.1) it follows that where ∇ is Levi-Civita connection.From (2.3) and (2.4), we see that each trajectory of ξ is a geodesic.A contact Riemannian manifold for which ξ is Killing is called a K-contact Riemannian manifold.It is easy to see that a contact Riemannian manifold is K-contact if and only if h = 0.For a contact Riemannian manifold M one may define naturally an almost complex structure J on M × R; where X is a vector field tangent to M, t the coordinate of R, and f a function on M ×R.
where [φ, φ] is the Nijenhuis torsion of φ.A Sasakian manifold is characterized by a condition for all vector fields X and Y on the manifold.We denote by R the Riemannian curvature tensor of M defined by for all vector fields X and Y .For a contact Riemannian manifold M, the tangent space where we denote A contact Riemannian manifold is said to be η-Einstein if where Q is the Ricci operator and a, b are smooth functions on M.
For more details about the fundamental properties on contact Riemannian manifolds we refer to [1,2].Blair [2] proved the following theorem.
Theorem 2.1.Let M = (M; η, g) be a contact Riemannian manifold and suppose that R(X, Y )ξ = 0 for all vector fields X, Y on M. Then M is locally the product of (n + 1)-dimensional flat manifold and an n-dimensional manifold of positive constant curvature 4.
Recently, Blair, Koufogiorgos, and Papantoniou [3] introduced a class of contact Riemannian manifolds which are characterized by equation (1.1).A D-homothetic deformation (cf.[9]) is defined by a change of structure tensors of the form where a is a positive constant.It was shown that [3] a contact Riemannian manifold M satisfying (1.1) is obtained by applying a D-homothetic deformation on a contact Riemannian manifold with R(X, Y )ξ = 0 and that the property (1.1) is invariant under the D-homothetic deformation.It is well known that the tangent sphere bundle of a flat Riemannian manifold admits a contact Riemannian structure satisfying R(X, Y )ξ = 0 [1, page 137].In [3] the authors classified the 3-dimensional case and showed that this class contains the tangent sphere bundles of Riemannian manifolds of constant sectional curvature.Furthermore in the same paper they showed that M satisfies for any vector fields X, Z on M. Here, we state some useful results in [3] to prove our Theorems 1.1 and 1.2.

Proof of Theorem 1.1.
Let M 2n+1 be a (2n+1)-dimensional contact Riemannian manifold which satisfies (1.1).Suppose that M is locally symmetric, that is, ∇R = 0.In view of the results of the Sasakian case [6] and the 3-dimensional contact Riemannian case [4], we now assume that n > 1 and M is non-Sasakian (κ ≠ 1).From hξ = 0, with (2.4) we have If we differentiate (1.1) covariantly, then using (2.4) we get for any vector fields X, Y on M. Putting Y = ξ, then with (2.2), (2.3), and (3.1) we have Together with (1.1) we have From (2.12) and (3.4) we have for any vector fields X, Z in M. If we put Z = ξ, then we have Since M is not Sasakian, we have µ = 0. Now, we consider the following equation in Theorem 2.3: where X λ ,Y λ ,Z λ ∈ D(λ).Differentiating (3.7) covariantly with respect to ( Together with Proposition 2.2 and using (3.7) again we get From (1.1), by using the property of the curvature tensor, we get By using (1.1), (2.1), and (3.10) we have and thus we have We may take an adapted orthonormal basis {ξ, e i ,φe i } such that hξ = 0, he i = λ i e i and hφe i = −λ i φe i , i = 1, 2,...,n at any point p ∈ M. Since g(φe i ,φV −λ ) = 0 and g(Y λ , ξ)g(ξ, φV −λ ) = 0, from (3.12) we have And hence, we obtain If we put φV −λ = Y λ in (3.14), then it follows that where X, Y are vector fields on M. Since n > 1 and κ = 1 − λ 2 , we conclude that κ = µ = 0, that is, M satisfies R(X, Y )ξ = 0 for any vector fields X, Y in M. Therefore by the results in [4,6] and Theorem 2.1 we have proved Theorem 1.1.

Proof of Theorem 1.2.
Let M be a contact Riemannian manifold which satisfies (1.1).Suppose that M is Ricci-parallel, that is, ∇Q = 0. From (1.1) and ( 2.3) we have From (2.4) and (4.1), we have Since M is Ricci-parallel, we have for any vector field Z on M. If we substitute Z with φZ, then by using (2.1) and (4.1), we obtain that If κ = 1 (h ≡ 0), then from (4.4) we see that M is Einstein-Sasakian and the scalar curvature τ = 2n(2n + 1).Now, we assume that κ ≠ 1, that is, M is non-Sasakian.Differentiating (2.14) covariantly, then it follows that and thus we get Together with (2.12) we have If we put Z = ξ in (4.7), then we have and hence we see that µ = 0 or 2(n − 1) + µ = 0. Now, we discuss our arguments divided into two cases: (i) µ = 0, (ii) 2(n − 1) + µ = 0.The case (i) µ = 0. Then (4.7) becomes Putting X = ξ, then by using (2.2) and (2.3) we get We apply φ and use (2.2), then we have which is impossible.Therefore, summing up all the arguments in this section we have Theorem 1.2.