Research Article | Open Access
On the M-Projective Curvature Tensor of -Contact Metric Manifolds
The object of the present paper is to study some curvature conditions on -contact metric manifolds.
The notion of the odd dimensional manifolds with contact and almost contact structures was initiated by Boothby and Wong in 1958 rather from topological point of view. Sasaki and Hatakeyama reinvestigated them using tensor calculus in 1961. Tanno  classified the connected almost contact metric manifolds whose automorphism groups possess the maximum dimension. For such a manifold, the sectional curvature of plain sections containing is a constant, say . He showed that they can be divided into three classes: (i) homogeneous normal contact Riemannian manifolds with , (ii) global Riemannian products of line or a circle with a Kähler manifold of constant holomorphic sectional curvature if , and (iii) a warped product space if . It is known that the manifolds of class (i) are characterized by admitting a Sasakian structure. Kenmotsu  characterized the differential geometric properties of the manifolds of class (iii); so the structure obtained is now known as Kenmotsu structure. In general, these structures are not Sasakian .
On the other hand in Pokhariyal and Mishra  defined a tensor field on a Riemannian manifold as where and . Such a tensor field is known as m-projective curvature tensor. Later, Ojha  defined and studied the properties of m-projective curvature tensor in Sasakian and Khler manifolds. He also showed that it bridges the gap between the conformal curvature tensor, conharmonic curvature tensor, and concircular curvature tensor on one side and H-projective curvature tensor on the other. Recently m-projective curvature tensor has been studied by Chaubey and Ojha , Singh et al. , Singh , and many others. Motivated by the above studies, in the present paper, we study flatness and symmetry property of -contact metric manifolds regarding m-projective curvature tensor. The present paper is organized as follows.
In this paper, we study the m-projective curvature tensor of -contact metric manifolds. In Section 2, some preliminary results are recalled. In Section 3, we study m-projectively semisymmetric -contact metric manifolds. Section 4 deals with m-projectively flat -contact metric manifolds. -m-projectively flat -contact metric manifolds are studied in Section 5 and obtained necessary and sufficient condition for an -contact metric manifold to be -m-projectively flat. In Section 6, m-projectively recurrent -contact metric manifolds are studied. Section 7 is devoted to the study of -contact metric manifolds satisfying . The last section deals with an -contact metric manifolds satisfying .
2. Contact Metric Manifolds
An odd dimensional differentiable manifold is said to admit an almost contact structure if there exist a tensor field of type (1, 1), a vector field , and a 1-form satisfying An almost contact metric structure is said to be normal if the induced almost complex structure on the product manifold defined by is integrable, where is tangent to , is coordinate of , and is smooth function on . Let be a compatible Riemannian metric with almost contact structure , that is, Then, becomes an almost contact metric manifold equipped with an almost contact metric structure . From (2) and (7), it can be easily seen that for all vector fields and . An almost contact metric structure becomes a contact metric structure if for all vector fields and . The 1-form is then a contact form, and is its characteristic vector field. We define a (1, 1)-tensor field by , where denotes the Lie-differentiation. Then, is symmetric and satisfies . We have and . Also, holds in a contact metric manifold.
A normal contact metric manifold is a Sasakian manifold. An almost contact metric manifold is Sasakian if and only if for all vector fields and , where is the Levi-Civita connection of the Riemannian metric . A contact metric manifold for which is a killing vector is said to be a K-contact manifold. A Sasakian manifold is K-contact, but the converse needs not be true. However, a 3-dimensional K-contact manifold is Sasakian . It is well known that the tangent sphere bundle of a flat Riemannian manifold admits a contact metric structure satisfying . On the other hand, on a Sasakian manifold the following holds: As a generalization of both and the Sasakian case; Blair et al.  considered the -nullity condition on a contact metric manifold and gave several reasons for studying it.
The -nullity distribution ([10, 11]) of contact metric manifold is defined by for all , where . A contact metric manifold with is called a -manifold. In particular, on a -manifold, we have On a -manifold, . If , the structure is Sasakian ( and is indeterminate), and if , the -nullity condition determines the curvature of completely . In fact, for a -manifold, the conditions of being a Sasakian manifold, a K-contact manifold, and are all equivalent.
In a -manifold, the following relations hold ([11, 12]): where is the Ricci tensor of type (0, 2), is the Ricci operator, that is, , and is the scalar curvature of the manifold. From (11), it follows that Also in a -manifold, holds.
The -nullity distribution of a Riemannian manifold  is defined by for all and being a constant. If the characteristic vector field , then we call a contact metric manifold an -contact metric manifold . If , then -contact metric manifold is Sasakian, and if , then -contact metric manifold is locally isometric to the product for and flat for . If , the scalar curvature is . If , then a -contact metric manifold reduces to a -contact metric manifolds. In , -contact metric manifold were studied in some detail.
For a -dimensional almost contact metric manifold, m-projective curvature tensor is given by  for arbitrary vector fields , , and , where is the Ricci tensor of type and is the Ricci operator, that is, .
The m-projective curvature tensor for an -contact metric manifold is given by
3. M-Projectively Semisymmetric -Contact Metric Manifolds
Definition 1. A -dimensional -contact metric manifold is said to be -projectively semisymmetric  if it satisfies , where is the Riemannian curvature tensor and is the m-projective curvature tensor of the manifold.
Theorem 2. An m-projectively semisymmetric -contact metric manifold is an Einstein manifold.
Proof. Suppose that an -contact metric manifold is -projectively semisymmetric. Then, we have The above equation can be written as follows: In view of (22), the above equation reduces to Now, taking the inner product of the above equation with and using (3) and (9), we get which on using (30), (32), (34), and (35), gives Putting in the above equation and taking summation over , , we get which shows that is an Einstein manifold. This completes the proof.
4. M-Projectively Flat -Contact Metric Manifolds
Theorem 3. An m-projectively flat -contact metric manifold is an Einstein manifold.
Proof. Let . Then, from (30), we have Let be an orthonormal basis of the tangent space at any point. Putting in the above equation and summing over , , we get which shows that is an Einstein manifold. This completes the proof.
5. -M-Projectively Flat -Contact Metric Manifolds
Definition 4. A -dimensional -contact metric manifold is said to be -m-projectively flat  if for all .
Theorem 5. A -dimensional -contact metric manifold is -m-projectively flat if and only if it is an -Einstein manifold.
Proof. Let . Then, in view if (30), we have By virtue of (9), (23), and (28), the above equation reduces to which by putting , gives Now, taking the inner product of above equation with , we get which shows that -contact metric manifold is an -Einstein manifold. Conversely, suppose that (47) is satisfied. Then, by virtue of (46) and (31), we have . This completes the proof.
6. M-Projectively Recurrent -Contact Metric Manifolds
Definition 6. A nonflat Riemannian manifold is said to be m-projectively recurrent if its m-projective curvature tensor satisfies the condition where is nonzero 1-form.
Theorem 7. If an -contact metric manifold is m-projectively recurrent, then it is an -Einstein manifold.
Proof. We define a function on , where the metric is extended to the inner product between the tensor fields. Then, we have
This can be written as
From the above equation, we have
Since the left-hand side of the above equation is identically zero and on , then
that is, 1-form is closed.
Now from we have In view of (52) and (54), we have Thus by virtue of Theorem 3, the above equation shows that is an -Einstein manifold. This completes the proof.
7. -Contact Metric Manifolds Satisfying
Theorem 8. If on an -contact metric manifold , then .
8. -Contact Metric Manifolds Satisfying
Theorem 9. On an -contact metric manifold, if , then .
Proof. Suppose that , then it can be written as which on using (33), takes the form Taking the inner product of above equation with , we get Now using (22), (28), and (29) in the above equation, we get Putting in the above equation and summing over , , we get This completes the proof.
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