A classification of certain
almost Kenmotsu manifolds
Abstract. We study homothetic deformations of almost Kenmotsu structures. We characterize almost contact metric manifolds which are integrable almost Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under homothetic deformations.
If the canonical connection associated to the structure has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator defined by .
In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost Kenmotsu structure. In the case of almost Kenmotsu spaces, this classification gives rise to a scalar invariant depending on the real numbers and .
2010 Mathematics Subject Classification: 53C15; 53C25.
Key words and phrases: almost Kenmotsu manifolds, homothetic deformations, manifolds, parallel structures, nullity distributions.
Introduction
Almost Kenmotsu manifolds are a special class of almost contact metric manifolds, recently investigated in [16, 14, 7, 8, 9]. An almost contact metric manifold is said to be an almost Kenmotsu manifold if and , where is the fundamental form associated to the structure. Normal almost Kenmotsu manifolds are known as Kenmotsu manifolds [13]: they set up one of the three classes of almost contact metric manifolds whose automorphism group attains the maximum dimension [19].
The class of almost Kenmotsu manifolds is not invariant with respect to homothetic deformations, that is changes of the structure tensors of the form
(1) 
where is a positive constant. These deformations were introduced by Tanno in [18] and largely studied for the class of contact metric manifolds. Indeed, for an almost contact metric structure, such a change preserves the property of being contact metric, Kcontact, Sasakian or strongly pseudoconvex , and the property for the characteristic vector field of a contact metric structure to belong to the nullity distribution. In [4] E. Boeckx provides a full classification of nonSasakian contact metric spaces up to homothetic deformations. He associates to each nonSasakian space an invariant depending on the real numbers , and provides an explicit example of such a space for every dimension and for every value of the invariant.
In this paper we consider the class of almost Kenmotsu manifolds [16, 14, 12]. They are almost contact metric manifolds with structure such that and , being a nonzero real constant. Applying deformation (1), one obtains an almost Kenmotsu structure.
After some preliminaries on general properties of almost Kenmotsu manifolds, dealing with the LeviCivita connection and the Riemannian curvature, also under the hypothesis of local symmetry, we shall focus on some properties which are invariant under homothetic deformations. The first one is the parallelism of the operator , where denotes the Lie derivative. The vanishing of the covariant derivative is also an invariant property. If both these conditions are satisfied and , then the spectrum of is of type , each being a positive constant. Denoting by the distribution of the eigenvectors of with eigenvalue and orthogonal to , and by and the eigendistributions with eigenvalues and respectively, the manifold is locally the warped product
where is an open interval, , and are integral submanifolds of the distributions , and . The warping functions are , and , with , and positive constants. Moreover, is an almost Kähler manifold and the structure is integrable if and only if is a simple eigenvalue or is a Kähler manifold (Theorem 4).
As a special case, we shall consider almost Kenmotsu manifolds whose characteristic vector field belongs to the nullity distribution, that is, for some real numbers , , the Riemannian curvature satisfies
(2) 
for all vector fields and . Applying a homothetic deformation, condition (2) is preserved up to a change of the real numbers . We shall see that, for an almost Kenmotsu space, the operator is parallel and . We also prove that . If , then . If then , the structure is integrable and the Riemannian curvature is completely determined (Theorem 5).
In order to obtain a local classification of the above manifolds, up to homothetic deformations, we consider in section 3 an invariant linear connection, called the canonical connection, which was introduced in [8] for almost Kenmotsu manifolds. The existence of this connection characterizes almost contact metric manifolds which are integrable almost Kenmotsu manifolds; it can be viewed as the analogue of the TanakaWebster connection in contact geometry. In [5] E. Boeckx and J. T. Cho study TanakaWebster parallel spaces, i.e. integrable contact metric manifolds for which the TanakaWebster connection has parallel torsion and curvature tensors; they prove that these spaces are Sasakian locally symmetric spaces or nonSasakian contact metric manifolds such that the characteristic vector field belongs to the nullity distribution.
Considering the canonical connection of a integrable almost Kenmotsu manifold, we prove that the torsion is parallel with respect to if and only if the tensor field is parallel and satisfies . If, furthermore, the curvature tensor satisfies , then vanishes and this occurs if and only if is a simple eigenvalue of or the integral submanifolds of the distribution have constant Riemannian curvature (Theorem 7). For a fixed dimension of the manifold, supposing and , we prove that the local geometry is completely determined, up to homothetic deformations, by the spectrum of the operator (Theorem 9). In particular, the manifold is locally equivalent to a solvable nonnilpotent Lie group, which is a subgroup of the affine group , endowed with a left invariant almost Kenmotsu structure, whose canonical connection coincides with the left invariant linear connection.
Applying the above classification to almost Kenmotsu spaces, with nonvanishing , we obtain a scalar invariant , depending on the real numbers and . Together with the dimension of the manifold, determines the local structure up to homothetic deformations. We also show that such a manifold is locally conformal to an almost cosymplectic manifold whose characteristic vector field belongs to the nullity distribution, with .
1 Preliminaries
An almost contact metric manifold is a differentiable manifold endowed with a structure , given by a tensor field of type , a vector field , a form and a Riemannian metric satisfying
Such a structure is said to be integrable if the associated almost structure is integrable, where is the dimensional distribution orthogonal to and is the restriction of to . The structure is normal if the tensor field identically vanishes, where is the Nijenhuis torsion of . It is well known that normal almost contact metric manifolds are manifolds [11]. We refer to [2, 3] for more details.
An almost Kenmotsu manifold is an almost contact metric manifold with structure such that
(3) 
where is a nonzero real constant and is the fundamental form defined by for any vector fields and . Normal almost Kenmotsu manifolds are known as Kenmotsu manifolds. Let us consider the tensor field
This operator satisfies , it is symmetric and anticommutes with . If is an eigenvector of with eigenvalue , then is an eigenvector with eigenvalue , and thus and have the same multiplicity. If , we denote by the distribution of the eigenvectors of with eigenvalue ; if , we denote by the distribution of the eigenvectors of with eigenvalue and orthogonal to , which has even rank.
The LeviCivita connection of satisfies , which implies that and for any . Moreover, for any vector field , or equivalently,
(4) 
for all vector fields . From (3) it follows that the distribution is integrable with almost Kähler leaves. The mean curvature vector field of the integral manifolds of is and these manifolds are totally umbilical if and only if [14].
An almost Kenmotsu structure is integrable if and only if the tensor vanishes on , or equivalently, the integral manifolds of are Kähler manifolds. In terms of the LeviCivita connection, the integrability of the structure can be characterized by the condition
(5) 
for all vector fields , which is equivalent to the parallelism of the tensor field , that is for any vector fields orthogonal to .
Analogously, the operator is said to be parallel if for every vector fields orthogonal to , and this is equivalent to requiring that
(6) 
for every vector fields .
Most of the results proved in [7] for the class of almost Kenmotsu manifolds can be generalized to the class of almost Kenmotsu manifolds. We omit the proofs since they are similar.
Theorem 1
Let be an almost Kenmotsu manifold such that . Then is locally a warped product , where is an almost Kähler manifold, is an open interval with coordinate , , for some positive constant .
Proposition 1
Let be an almost Kenmotsu manifold such that the integral manifolds of are Kähler. Then, is an Kenmotsu manifold if and only if , or equivalently, . Therefore, a dimensional almost Kenmotsu manifold such that is an Kenmotsu manifold.
Consequently, an Kenmotsu manifold is a warped product of type , where is an open interval, is a Kähler manifold and , for some positive constant .
As regards the Riemannian curvature of an almost Kenmotsu manifold, an easy computation shows that
(7) 
for every vector fields , which implies that
If the almost Kenmotsu manifold is locally symmetric, then the operator satisfies , and for any unit eigenvector of with eigenvalue , the sectional curvature is given by
which implies that . The geometry of a locally symmetric almost Kenmotsu manifold is quite different in the two cases with vanishing or nonvanishing . Indeed, we have the following results.
Theorem 2
Let be a locally symmetric almost Kenmotsu manifold. Then, is an Kenmotsu manifold if and only if ; in this case the manifold has constant sectional curvature .
Given an almost Kenmotsu manifold of constant curvature , it can be proved that , and the above Theorem implies that the structure is normal and . In the case of nonvanishing we have
Theorem 3
Let be a locally symmetric almost Kenmotsu manifold with . Then the operator admits the eigenvalues and . If, moreover, the Riemannian curvature satisfies for any , then the spectrum of is , with as simple eigenvalue. The distributions and are integrable with totally geodesic leaves and is locally isometric to the Riemannian product of an dimensional manifold of constant curvature and a flat dimensional manifold.
In the following we consider almost Kenmotsu manifolds with . Notice that if is an almost Kenmotsu structure with , then is an almost Kenmotsu structure with .
2 homothetic deformations
Let be an almost Kenmotsu manifold and the almost Kenmotsu structure obtained by the homothetic deformation (1). Notice that the operators and associated to these structures coincide. Let and be the LeviCivita connections of and respectively. We prove that for all vector fields ,
(8) 
Indeed, applying the Koszul formula and , we have
and using (4), we obtain (8). The covariant derivatives of and satisfy
for all vector fields and , so that the property for the tensor fields and to be parallel and the vanishing of the covariant derivative are invariant under homothetic deformations.
An easy computation shows that the Riemannian curvature tensors and of and are related by the following formula:
(9)  
for every vector fields . It follows that for every vector fields . If belongs to the nullity distribution, i.e. the Riemannian curvature tensor satisfies (2), then belongs to the nullity distribution, with
Let us analyze now the geometry of almost Kenmotsu manifolds such that is parallel and satisfies .
Theorem 4
Let be an almost Kenmotsu manifold such that is parallel and . Then the eigenvalues of the operator are constant. Let be the spectrum of , with . Then is locally the warped product
(10) 
where is an open interval, , and are integral submanifolds of the distributions , and respectively. The warping functions are , and , with , and positive constants. Finally, is an almost Kähler manifold and the structure is integrable if and only if is a simple eigenvalue or is a Kähler manifold.
Proof. The result is proved in [9] for almost Kenmotsu manifolds, corresponding to the case . Let us consider an almost Kenmotsu structure , with , such that is parallel and . Applying the homothetic deformation (1) with , we obtain an almost Kenmotsu structure such that is parallel and , and the result applies to this structure. In particular, the distributions , and are integrable and for any distinct eigenvalues of , the distribution is integrable with totally geodesic leaves with respect to ; (8) implies that such leaves are totally geodesic also with respect to .
Let us consider an eigenvalue of . We prove that the leaves of the distribution are totally umbilical. Indeed, since is totally geodesic, choosing a local orthonormal frame of , the second fundamental form satisfies ; the mean curvature vector field is and, for any , we have , so that the leaves of are totally umbilical. Since the orthogonal distribution is integrable with totally geodesic leaves, then is locally a warped product such that and (see [10]). We denote by and the Riemannian metrics on and respectively, such that the warped metric is given by . The projection is a Riemannian submersion with horizontal distribution and vertical distribution . The mean curvature vector field of the immersed submanifold is related to ([1], 9.104) and thus, . If is the multiplicity of , we choose local coordinates on such that and for any . Hence, we get , .
Now, let us consider . The distribution is integrable with totally geodesic leaves in and is integrable with totally umbilical leaves in . Since is a totally geodesic submanifold of , these distributions are respectively totally geodesic and totally umbilical in and, arguing as above, is locally a warped product. This argument can be applied to each distribution and , , obtaining that is locally the warped product
where and , with and positive constants. The manifold is a totally geodesic submanifold of and it is an integral submanifold of the distribution . By Theorem 1, is locally a warped product of an open interval and an almost Kähler manifold , with , .
Under the hypotheses of the above Theorem, applying (6), we have
(11) 
for any . Now, if we suppose that , with simple eigenvalue, then and thus, from (11) and (7) it follows that
Hence, we have
Proposition 2
Let be an almost Kenmotsu manifold such that is parallel and . If , with simple eigenvalue, then belongs to the nullity distribution, with and .
As regards almost Kenmotsu spaces we have the following result.
Theorem 5
Let be an almost Kenmotsu manifold such that belongs to the nullity distribution. Then .
If , then and is locally a warped product , where is an almost Kähler manifold, is an open interval with coordinate , , for some positive constant .
If , then , and , with as simple eigenvalue and . The operator is parallel and satisfies . The integral manifolds of are Kähler manifolds. The distributions and are integrable with totally umbilical leaves; the distributions and are integrable with totally geodesic leaves. Finally, is locally isometric to the warped products
where is a space of constant curvature , tangent to the distribution , is the hyperbolic space of constant curvature , tangent to the distribution , and , with positive constants.
Proof. The result is proved in [8] for almost Kenmotsu manifolds. Let us consider an almost Kenmotsu structure , with , such that belongs to the nullity distribution. Applying the homothetic deformation (1) with , we obtain an almost Kenmotsu structure such that belongs to the nullity distribution, with , and . Then . If , or equivalently , then and we apply Theorem 1.
If then , and , with as simple eigenvalue and . The tensor fields and are parallel and , since these properties are invariant under homothetic deformations; in particular, the integral manifolds of are Kähler manifolds. From Theorem 4 it follows that is locally the warped product
where is an open interval, and are integral submanifolds of the distributions and respectively, and , with positive constants.
We compute now the Riemannian curvature of . Recall that the integral submanifolds of the distribution have constant Riemannian curvature with respect to the deformed Riemannian metric . Let us compute the relation between the curvature tensors and of and respectively. Combining (7) with the nullity condition, , we get
and thus, applying (9), we obtain
for any . On the distribution we have and applying the above formula, for any , we get
Therefore, the leaves of the distribution have constant Riemannian curvature with respect to and analogously, the leaves of the distribution have constant Riemannian curvature . Then, is locally isometric to the warped products
We prove that the fibers of the two warped products are flat Riemannian spaces. Denote by and the Riemannian metrics on and respectively, such that the first warped metric is given by . Applying Proposition 7.42 in [17], for any , we have
On the other hand, and . Then, . Analogously, the fibers of the second warped product are flat Riemannian spaces.
Under the hypotheses of the above Theorem, if then both the distributions and are integrable with totally geodesic leaves and the manifold turns out to be locally isometric to the Riemannian product , which is locally symmetric. Conversely, supposing that is locally symmetric, then, by Theorem 3, and is locally isometric to . Hence, we have
Corollary 1
Let be an almost Kenmotsu manifold such that and belongs to the nullity distribution, . Then is locally symmetric if and only if , or equivalently , in which case the manifold is locally isometric to .
As another consequence of Theorem 5, we can obtain more information on the Riemannian curvature of an almost Kenmotsu manifold such that is parallel and , as in the hypotheses of Theorem 4. Indeed, for any eigenvalue of the operator , the distribution is integrable with totally geodesic leaves which inherit an almost Kenmotsu structure from . If , then the distribution reduces to and the leaves are local warped products , where is a Kähler manifold in hypothesis of integrability. If then, by Proposition 2, the leaves of are almost Kenmotsu manifolds with characteristic vector field belonging to the nullity distribution, with and . By Theorem 5, the leaves of have constant Riemannian curvature and the leaves of have constant Riemannian curvature .
3 The canonical connection
Theorem 6
Let be an almost contact metric manifold. Then is a integrable almost Kenmotsu manifold if and only if there exists a linear connection such that the tensor fields , , are parallel with respect to and the torsion satisfies:

, for any ,

, for any ,

is selfadjoint.
The connection is invariant under homothetic deformations and it is uniquely determined by
(12) 
where is the LeviCivita connection. The connection will be called the canonical connection associated to the structure .
Proof. The result of existence and uniqueness of the connection is proved in [8] for almost Kenmotsu manifolds. Let be the almost contact metric structure obtained from through deformation (1) with . Then is a integrable almost Kenmotsu structure if and only if is a integrable almost Kenmotsu structure, and this is equivalent to the existence of a unique linear connection such that the tensor fields , and are parallel with respect to , and the torsion vanishes on and satisfies

, for any ,

is selfadjoint with respect to .
The parallelism of , , is equivalent to the parallelism of , , and ) is obviously equivalent to b). Moreover, for any vector fields , we have
If is selfadjoint with respect to , then is selfadjoint with respect to since . Hence, ) implies c). Analogously, one verifies that c) implies ).
Denoting by the LeviCivita connection of , for any vector fields and , we have
and applying (8) with , we get (12). Finally, the connection is invariant under homothetic deformations. Indeed, if is the canonical connection associated to the almost Kenmotsu structure , it can be easily verified that