Table of Contents
ISRN Geometry
Volume 2013, Article ID 927159, 6 pages
http://dx.doi.org/10.1155/2013/927159
Research Article

On the Classification of Almost Kenmotsu Manifolds of Dimension 3

1Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, China
2School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China

Received 18 December 2012; Accepted 9 January 2013

Academic Editors: T. Friedrich, C. Qu, and E. H. Saidi

Copyright © 2013 Yaning Wang and Ximin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the classification of a 3-dimensional almost Kenmotsu manifold satisfying certain geometric conditions. Moreover, by applying our main classification theorem, we obtain some suffcient conditions for an almost Kenmotsu manifold of dimension 3 to be an Einstein-Weyl manifold.

1. Introduction

Contact metric manifolds known as a special class of almost contact metric manifolds are objects of increasing interest both from geometers and physicists [1] recently. We refer the reader to the recent monograph [2] for a wide and detailed overview of the results in this field. From (14) (see Section 3) we know that a normal almost contact metric manifold (which includes Sasakian and Kenmotsu manifolds as its special cases) of dimension 3 satisfies . But the above property need not be true in an almost contact metric manifold. Blair et al. [3] obtained a classification of 3-dimensional contact metric manifold with . However, in higher dimensions the classification of contact metric manifold with is still open. It is worthy to point out that Ghosh [4] recently proved that a contact metric manifold admitting the Einstein-Weyl structures (see Section 4) and is either a K-contact or an Einstein manifold.

On the other hand, in 1972, Kenmotsu [5] introduced a class of almost contact metric manifolds which are known as Kenmotsu manifolds nowadays. Recently, almost Kenmotsu manifolds satisfying -parallelism and locally symmetries are studied by Dileo and Pastore [6] and [7], respectively. We notice that Dileo and Pastore [8] complete the classification of 3-dimensional almost Kenmotsu manifold with the assumption that belongs to the -nullity distribution. However, to the best of our knowledge the study of 3-dimensional almost Kenmotsu manifolds is still lacking so far. The object of this paper is to classify the 3-dimensional almost Kenmotsu manifolds satisfying and other geometric conditions, providing some results which show the differences between almost Kenmotsu manifolds and the contact metric manifolds of dimension 3 [3, 9]. Moreover, by applying our main classification theorem, we obtain some suffcient conditions for an almost Kenmotsu manifold of dimension 3 to be an Einstein-Weyl manifold.

This paper is organized as the following way. In Section 2, we provide some basic formulas and properties of almost Kenmotsu manifolds. Section 3 is devoted to present our main theorems and their proofs. Finally, in Section 4, we prove that if an almost Kenmotsu manifold of dimension 3 is -Einstein with certain condition then it admits both Einstein-Weyl structures and .

2. Almost Kenmotsu Manifolds

First of all, we give some basic notions of almost Kenmotsu manifolds which follow from [5, 7]. An almost contact structure on a -dimensional smooth manifold is a triplet , where is a -tensor, a global vector field, and a 1-form, such that which implies that ,   and . It follows from [2, 10] that a Riemannian metric on is said to be compatible with the almost contact structure if An almost contact structure endowed with a compatible Riemannian metric is said to be an almost contact metric structure. The fundamental 2-form is defined by for any vector fields and on . An almost Kenmotsu manifold is defined as an almost contact metric manifold together with and . It is well known that the normality of almost contact structure is expressed by the vanishing of the tensor , where is the Nijenhuis tensor of . A normal almost Kenmotsu manifold is said to be a Kenmotsu manifold.

Now let be an almost Kenmotsu manifold. We denote by and on , where is the curvature tensor and is the Lie differentiation, respectively. Thus, the two -type tensors and are symmetric and satisfy We also have the following formulas following from [5, 7, 8]: for any , where ,  ,  , and denote the Ricci curvature tensor, Ricci operator, the Levi-Civita connection of , and the Lie algebra of vector fields in , respectively. From the above formulas we also have .

An almost contact manifold is said to be -Einstein if where and are both smooth functions on . It is easy to see that an -Einstein almost Kenmotsu manifold satisfies because of . We also recall that the -nullity distribution [11] is defined by where is a real number. When in (10) is a smooth functions, then the nullity distributions are called the generalized nullity distributions [12]. Also, the sectional curvature of a plane section spanned by and a vector orthogonal to is called a -sectional curvature and the sectional curvature of a plane section spanned by vectors and with orthogonal to is called a -sectional curvature [3].

A Riemanian manifold of dimension is conformally flat if and only if the Weyl tensor defined by vanishes for and for , where ,  , and denote the Ricci tensor, Ricci operator, and the scalar curvature, respectively.

3. A Classification Theorem

Let be a normal almost contact metric manifold of dimension 3, then we have . In fact, it follows from [13, 14] that satisfies where and are both constants and denotes the scalar curvature of . Then from the above equation it is easy to get Noticing that , then from (14) we know that . If (resp., ) then is just the 3-dimensional Sasakian (resp., Kenmotsu) manifold [13]. However, the condition need not be true in a contact metric manifold as well as an almost Kenmotsu manifold. In a contact metric manifold if the characteristic vector is a Killing vector field, then the manifold is said to be a K-contact manifold. A Sasakian manifold is always a K-contact one, but the converse need not hold only if is of dimension 3. We present the following result to characterize Kenmotsu manifold of dimension 3 which is analogous to the contact metric manifold of dimension 3.

Lemma 1. Let be an almost Kenmotsu manifold of dimension 3, then the following conditions are equivalent.(1) is a Kenmotsu manifold.(2).(3).

Proof. It follows from [7, 8] that an almost Kenmotsu is a Kenmotsu manifold if and only if Noticing that and (see [7]) for any , thus is a Kenmotsu manifold if and only if every nonvanishing vector field in contact distribution (defined by ) satisfies the following equations: On the other hand, on an almost Kenmotsu manifold by using (1)–(4) we obtain Comparing the above three equations with (16), we complete the proof of equivalence between (1) and (2). The equivalence between (2) and (3) follows from the fact that for any .

Theorem 2. Let be an almost Kenmotsu manifold of dimension 3 for which the characteristic vector field is an eigenvector field of the Ricci operator. If is conformally flat, then it satisfies for any , where denotes the scalar curvature of .

Proof. Let be an eigenvector field of the Ricci operator corresponding to the eigenvalue , that is, . Substituting the above equation into (6) implies that . Then we obtain Differentiating (21) along an arbitrary vector field and using (4) gives Since is conformally flat, then substituting into (12) and using gives for any . Using the symmetry , then from (23) we have for any . Substituting (22) into (24) gives for any . Replacing and by and , respectively, in (25) gives for any . Thus, it follows from (26) that and hence by substituting in (25) we obtain for any . Applying the exterior derivation on both sides of (27) and using the well-known Poincaré lemma , and then replacing , respectively, by in the resulting equation, we obtain Thus, substituting (28) into (27) gives (20).

Theorem 3. Let be a 3-dimensional almost Kenmotsu manifold satisfying . Then one of the following cases occurs.
Case  1. and hence is a Kenmotsu manifold.
Case  2. and hence the eigenvalues of are locally given by with coordinate on and a nonzero number.

Proof. For an almost Kenmotsu manifold the operator never vanishes. In fact, if , it follows from (6) that , there is a contradiction. Now we suppose that is an almost Kenmotsu manifold of dimension with . Noticing that then we have . Denoting by the projective component of on contact distribution , then from (6) we have . By using the hypothesis on the above equation we get , then we have It is well known that the curvature tensor of a 3-dimensional Riemannian manifold is given by for any , where denotes the scalar curvature of . Noticing (29) and replacing by in (30) yields Using and , it follows from (31) that for any , that is, On the other hand, substituting (32) into (5) gives that From (8) and (33) we obtain that
On the other hand, since is antisymmetric on then from (32) we have for any . Thus, noticing that , we obtain for any , that is, for any . Substituting the above equation into (31) gives Differentiating (35) along we obtain Also, it is well known that for any unit vector filed in contact distribution , where denotes the gradient of scalar curvature of . Now letting in (36) be unit vector fields in gives Similarly, we have Substituting (38) and (39) and into (37) implies By using (40) and taking an inner product with and , respectively, we obtain and for any , that is, is a constant on .
Case  1. If , then we find that is a constant on and hence from (6) we see that is also a constant on . Let be a unit eigenvector filed of with eigenvalue , that is, (and hence ), then is a constant since that is a constant. Using the first term of (34) gives ; thus, noting that is a constant then we obtain , that is, and hence from Lemma 1 we see that is a Kenmotsu manifold.
Case  2. If , it follows from Case  1 that , where denotes the eigenvalues of on . Locally, we can write and hence with coordinate on and a nonzero number following the fact that is a constant on contact distribution , which completes the proof.

Corollary 4. Let be an almost Kenmotsu manifold of dimension 3. Then the following assertions are equivalent:(a) is an -Einstein Kenmotsu manifold.(b).(c) belongs to the generalized -nullity distribution.Moreover, if one of the above conditions holds, then the -sectional curvature of is and the -sectional curvature is , where denotes the scalar curvature of .

Proof. Suppose that is an -Einstein almost Kenmotsu manifold of dimension 3; from (9) it is easy to see . If , then replacing by in (30) and using (35) we see that belongs to the generalized -nullity distribution, which means that . Now letting belongs to the generalized -nullity distribution, then by a straightforward calculation we know that is an eigenvector field of Ricci operator. Replacing by in (30) implies that .
Finally, if one of the above conditions holds, then from the above statements we have . We choose a unit nonvanishing vector field in contact distribution . Replacing in (30) gives an equation, taking an inner product with on the resulting equation and taking into account we obtain On the other hand, by the definition of Ricci curvature tensor we have Using for any (see Theorem 3), then it follows from (41) into (42) that . Then from (41) we know that the -sectional curvature of is , which completes the proof.

Kenmotsu [5] proved that if a Kenmotsu manifold is a space of constant -holomorphic sectional curvature then is a space of constant curvature and . Thus, the following result follows from Corollary 4 and Theorem 3.

Corollary 5. Let be an -Einstein almost Kenmotsu manifold of dimension 3 with . If the scalar curvature of is a constant, then is locally isometric to a hyperbolic space with constant scalar curvature .

4. Einstein-Weyl Structures

Recall that a Weyl structure [4, 15] on a Riemannian manifold of dimension is defined by the pair satisfying where is a unique torsion-free connection which preserves the conformal class on and is an 1-form on . It follows from (43) that where and denote the Levi-Civita connection of and the dual vector field of with respect to , respectively. The Weyl structure is said to be Einstein-Weyl if the symmetrized Ricci tensor associated with the Weyl connection is proportional to the Riemannian metric , that is, where denotes the Ricci tensor associated with and is a smooth function on . Notice that Narita [15] proved that an -Einstein almost contact metric manifold satisfying admits an Einstein-Weyl structure . However, (4) implies that an almost Kenmotsu manifold never satisfies Narita’s condition even if . Since then, we present the following sufficient conditions to characterize Einstein-Weyl structure on an almost Kenmotsu manifold of dimension 3.

Theorem 6. Let be an -Einstein almost Kenmotsu manifold of dimension 3 with , that is, . If is a constant , then admits an Einstein-Weyl structure .

Proof. We define -form by , where is a nonvanishing function on . Then the dual vector field of respective to is . We also define a connection on by then from (46) it is easy to verify that is torsion free and , that is, is a Weyl structure on . It follows from [15, 16] that for any . By using (4) then a simple computation gives that for any and where is a unit vector field on . Thus, substituting (48) and (49) into (47) yields that On the other hand, noticing Theorem 3 and Corollary 5, we know that and is a Kenmotsu manifold. We set or , then it follows from (50) that which completes the proof.

Remark 7. From Corollary 5 we know that an -Einstein almost Kenmotsu manifold of dimension 3 with is a Kenmotsu manifold and ; however, this property need not be true in higher dimensions more than 3. Thus, our result cannot be generalized to the case of higher dimensions.
A Weyl structure on a Riemannian manifold of dimension is defined by for a unique torsion-free connection and a 1-form . The Weyl structure is said to be an Einstein-Weyl structure if (45) holds for a smooth function on . For an Einstein-Weyl structure it follows from [4] that for any , where denotes the dual vector field of with respect to . Thus, a straightforward calculation which is similar to the proof of Theorem 6 gives the following.

Theorem 8. Let be an -Einstein almost Kenmotsu manifold of dimension 3 with , that is, . If is a constant , then admits both Einstein-Weyl structures .

Acknowledgments

The authors would like to thank the referee for his or her valuable suggestions and comments improving of this paper. This work is supported by the NSFC (no. 10931005) and the Natural Science Foundation of Guangdong Province of China (no. S2011010000471).

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