Certain Results On $N(\Kappa)$-contact Metric Manifolds

TL;DR

This paper classifies certain $N(ppa)$-contact metric manifolds satisfying specific curvature conditions, revealing their geometric structure and providing an example.

Contribution

It provides a classification of $N(ppa)$-contact metric manifolds under various curvature constraints, including Weyl-pseudosymmetry.

Findings

Weyl-pseudosymmetric $N(ppa)$-contact metric manifolds are either locally isometric to a product space or $ta$-Einstein.

The paper offers explicit classifications based on curvature conditions.

An explicit example of such a manifold is constructed.

Abstract

In this paper, $N(\kappa)$-contact metric manifolds satisfying the conditions $\widetilde{C}(\xi,X)\cdot\widetilde{C}=0$, $\widetilde{C}(\xi,X)\cdot R=0$, $\widetilde{C}(\xi,X)\cdot S=0$, $\widetilde{C}(\xi,X)\cdot C=0$, $C\cdot S=0$ and $R\cdot C=f_{C}Q(g,C)$ have been investigated and obtained their classification. Among others it is shown that a Weyl-pseudosymmetric $N(\kappa)$-contact metric manifold is either locally isometric to the Riemannian product $E^{n+1}(0)\times S^{n}(4)$ or an $\eta$-Einstein manifold. Finally, an example is given.

Equations

$$R(X,Y)=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]},$$

$$(X{\wedge }_{g}Y)Z=g(Y,Z)X-g(X,Z)Y,$$

$$g(QX,Y)=S(X,Y),$$

$$S^2(X,Y)=S(QX,Y).$$

$$\tilde{C}(X,Y)=\left(R-\frac{r}{m(m-1)}{\wedge }_g\right)\left(X,Y\right),$$

$$C(X,Y)=R(X,Y)-\frac{1}{m-2}\left\{(X\wedge QY)+(QX\wedge Y)-\frac{r}{m-1}(X\wedge Y)\right\}.$$

$$(R(X,Y)\cdot T)(X_1,\cdots ,X_l)=$$

$$\cdots -T(X_1,\cdots ,X_{l-1},R(X,Y)X_l),$$

$$Q(g,T)(X_1,\cdots ,X_l;X,Y)=$$

$$\cdots -T(X_1,\cdots ,X_{l-1},(X\wedge Y)X_l),$$

$$R\cdot C=f_{C}Q(g,C),$$

$$(C(X,Y)\cdot S)(Z,W)=-S(C(X,Y)Z,W)-S(Z,C(X,Y)W).$$

$$\eta (X)=g(X,\xi ),d\eta (X,Y)=g(X,\varphi Y),{\varphi }^2(X)=-X+\eta (X)\xi ,$$

$$\varphi \xi =0,\eta \circ \varphi =0,g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y),$$

$$h\xi =0,h\varphi =-\varphi h,Tr.h=Tr.\varphi h=0.$$

$${\nabla }_X\xi =-\varphi X-\varphi hX.$$

$$R(X,Y)\xi =\eta (Y)X-\eta (X)Y.$$

$$N(\kappa ):p\to N_p(\kappa )=\left\{Z\in T_{p}M:R(X,Y)Z=\kappa [g(Y,Z)X-g(X,Z)Y]\right\},$$

$$R(X,Y)\xi =\kappa [\eta (Y)X-\eta (X)Y]$$

$$N(\kappa ,\mu ):p\to N_p(\kappa ,\mu )=\{Z\in T_{p}M:R(X,Y)Z=\kappa [g(Y,Z)X-g(X,Z)Y]$$

$$+\mu [g(Y,Z)hX-g(X,Z)hY],$$

$$R(X,Y)\xi =k[\eta (Y)X-\eta (X)Y]+\mu [\eta (Y)hX-\eta (X)hY].$$

$$Q\varphi -\varphi Q=4(n-1)h\varphi ,$$

$$h^2=(\kappa -1){\varphi }^2,\kappa \le 1,$$

$$Q\xi =2n\kappa \xi ,$$

$$R(\xi ,X,Y)=\kappa [g(X,Y)\xi -\eta (Y)X].$$

$$Tr.h^2=2n(1-\kappa ),$$

$$S(X,\varphi Y)+S(\varphi X,Y)=2(2n-2)g(\varphi X,hY),$$

$$S(\varphi X,\varphi Y)=S(X,Y)-2nk\eta (X)\eta (Y)-2(2n-2)g(hX,Y),$$

$$S(X,\xi )=2n\kappa \eta (X),$$

$$Q\varphi +\varphi Q=2\varphi Q+2(2n-2)h\varphi ,$$

Full text

1. Introduction

Let $M$ be an $m(\geq 3)$-dimensional connected smooth Riemannian manifold endowed with the Riemannian metric $g$, Levi-Civita connection $\nabla$. Let $R$, $S$, $r$ be respectively the Riemannian curvature tensor, Ricci tensor and scalar curvature of $M$ such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for any vector field $X$, $Y$, $Z$ $\in\chi(M)$, $\chi(M)$ being the Lie algebra of all smooth vector fields on $M$. Then the concircular curvature $\widetilde{C}$ and Weyl-conformal curvature tensor $C$ on $M$ are given by [15]

[TABLE]

[TABLE]

For a $(0,l)$-tensor field $T$, $l\geq 1$, on $(M,g)$, the tensors $R\cdot T$ and $Q(g,T)$ are defined as

[TABLE]

[TABLE]

respectively [10].

If the tensors $R\cdot C$ and $Q(g,C)$ are linearly dependent then $M$ is called Weyl-pseudosymmetric, that is,

[TABLE]

holding on the set $U_{C}=\left\{x\in M\left|C\right.\neq 0\,\,at\,\,x\right\}$, where $f_{C}$ is some function on $U_{C}$. If $R\cdot C=0$ then $M$ is called Weyl-semisymmetric ([9], [10], [11]). If $\nabla C=0$ then $M$ is called conformally symmetric [8]. It is well-known that a conformally symmetric manifold is Weyl-semisymmetric. Furthermore, we define the tensor $C\cdot S$ on $M$

[TABLE]

A contact metric manifold $M$ of dimension $m(=2n+1),(n>1)$ is a quadruple $(\phi,\xi,\eta,g)$ such that

[TABLE]

[TABLE]

for all $X,Y\in\chi(M)$.

On a contact metric manifold $M$, a tensor field $h$ is defined by $h=\frac{1}{2}\ell_{\xi}\phi$, where $\ell$ denotes the operator of Lie differentiation. Then $h$ is symmetric and satisfies ([1])

[TABLE]

[TABLE]

If $\xi$ is a Killing vector field then $M$ is said to be a $K$-contact metric manifold, and $M$ is Sasakian if and only if

[TABLE]

The notion of $\kappa$-nullity distribution was introduced by Tanno [14] for a real number $\kappa$ as a distribution

[TABLE]

for any $X,Y\in T_{p}M$. Hence if $\xi\in N(\kappa)$ then

[TABLE]

holds. Thus a contact metric manifold $M$ for which $\xi\in N(\kappa)$ is called a $N(\kappa)$-contact metric manifold. From (1.15) and (1.17) it follows that a $N(\kappa)$-contact metric manifold is a Sasakian if and only if $\kappa=1$. On the other-hand if $\kappa=0$, then the manifold is locally isometric to the product $E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat for $n=1$ [2]. Also in a $N(\kappa)$-contact metric manifold, $\kappa$ is always a constant such that $\kappa\leq 1$ [14]. Throughout the paper by $M$ we mean a $(2n+1),(n>1)$-dimensional $N(\kappa)$-contact metric manifold unless otherwise stated.

The paper is structured as follows: Section 2 deals with some requisitory curvature properties of $N(\kappa)$-contact metric manifold. Section 3 is concerned with main results and it is shown that a $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$ satisfies $\widetilde{C}(\xi,X)\cdot\widetilde{C}=0$ (resp., $\widetilde{C}(\xi,X)\cdot C=0$, $\widetilde{C}(\xi,X)\cdot S=0$) if and only if the manifold is either $N(1-\frac{1}{n})$-contact metric manifold or locally isometric to the hyperbolic space $H^{2n+1}(-\kappa)$ (resp., Einstein manifold, $\eta$-Einstein manifold). Also it is shown that if $M$ is a Weyl-pseudosymmetric then it is either locally isometric to the Riemannian product $E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat for $n=1$ or $\eta$-Einstein manifold.

2. $N(k)$-Contact Metric Manifolds

Generalizing the notion of $N(\kappa)$-contact metric manifold in 1995, Blair, Koufogiorgos and Papantoniou [3] introduced the notion of $N(\kappa,\mu)$-contact metric manifold, for real numbers $\kappa$ and $\mu$ as a distribution :

[TABLE]

[TABLE]

for any $X,Y\in T_{p}M$. Hence if the characteristic vector field $\xi$ belongs to the $(\kappa,\mu)$-nullity distribution, then

[TABLE]

A contact metric manifold $M$ satisfying the relation (2.2) is called a $N(\kappa,\mu)$-contact metric manifold or simply a $(\kappa,\mu)$-contact metric manifold. In particular, if $\mu=0$, then the relation (2.2) reduces to (1.17) and hence a $N(\kappa)$-contact metric manifold is a $N(\kappa,0)$-contact metric manifold.

Let $M$ be a $N(\kappa)$-contact metric manifold. Then the following relations hold ([13], [14], [19], [20], [21]):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In view of (1.1) and (1.2), it follows from (2.1), (2.2), (2.3) and (2.4) that in a $N(\kappa)$-contact metric manifold, the following relations hold:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for any vector field $X,Y\in\chi(M)$. Also in a $N(\kappa)$-contact metric manifold the scalar curvature $r$ is given by ([3], [13])

[TABLE]

In [7] Boeckx introduced an invariant on a non-Sasakian contact metric manifold as

[TABLE]

and showed that for two non-Sasakian $(\kappa,\mu)$-manifolds $M_{1}$ and $M_{2}$, we have $I_{M_{1}}=I_{M_{2}}$ if and only if up to a $D$-homothetic deformation, the two manifolds are locally isometric as contact metric manifolds. Thus, we see that from all non-Sasakian $(\kappa,\mu)$-manifolds of dimension $(2n+1)$ and for every possible value of the invariant $I_{M}$, one $(\kappa,\mu)$-manifold $M$ can be obtained. For $I_{M}>-1$ such examples may be found from the standard contact metric structure on the tangent sphere bundle of a manifold of constant curvature $c$ where we have $I_{M}=\frac{1+c}{1-c}$. Boeckx also gives a Lie algebraic construction for any odd dimension and value of $I_{M}<-1$.

Using this invariant, Blair, Kim and Tripathi [4] constructed an example of a $(2n+1)$-dimensional $N(1-\frac{1}{n})$-contact metric manifold $n>1$. Since the Boeckx invariant for a $(1-\frac{1}{n},0)$-manifold is $\sqrt{n}>-1$, we consider the tangent sphere bundle of an $(n+1)$-dimensional manifold of constant curvature $c$ so chosen that the resulting $D$-homothetic deformation will be a $(1-\frac{1}{n},0)$-manifold. That is, for $k=c(2-c)$ and $\mu=-2c$, we solve

[TABLE]

for $a$ and $c$. We have

[TABLE]

and taking $c$ and $a$ to be these values we obtain $N(1-\frac{1}{n})$-contact metric manifold.

Definition 2.1**.**

A $N(\kappa)$-contact metric manifold $M$ is said to be $\eta$-Einstein if its Ricci tensor $S$ of type $(0,2)$ is of the form

[TABLE]

where $c_{1},c_{2}$ are smooth functions on $M$. Contracting (2.15), we have

[TABLE]

Beside this, taking $X=Y=\xi$ in (2.15) and using (2.10) we also have

[TABLE]

Hence in view of (2.15),(2.16) and (2.17), we have the result.

Proposition 2.1**.**

In an $\eta$-Einstein $N(\kappa)$-contact metric manifold $M$, the Ricci tensor $S$ is of the form

[TABLE]

Proposition 2.2**.**

[2]** A contact metric manifold $M$ satisfying the condition $R(X,Y)\xi=0$ for all $X,Y$ is locally isometric to the Riemannian product of a fat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of positive curvature $4$, i.e., $E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat for $n=1$.

Proposition 2.3**.**

[6]** Let $M$ be an $\eta$-Einstein manifold of dimension $(2n+1)$, $(n\geq 1)$, if $\xi$ belongs to the $\kappa$-nullity distribution, then $\kappa=1$ and the structure is Sasakian.

Proposition 2.4**.**

[14]** Let $M$ be an Einstein manifold of dimension $(2n+1)$, $(n\geq 2)$, if $\xi$ belongs to the $\kappa$-nullity distribution, then $\kappa=1$ and the structure is Sasakian.

3. Main results

In this section, we focus on the characterization of $N(\kappa)$-contact metric manifolds satisfying the condition $\widetilde{C}(\xi,X)\cdot\widetilde{C}=0$, $\widetilde{C}(\xi,X)\cdot R=0$, $\widetilde{C}(\xi,X)\cdot S=0$, $\widetilde{C}(\xi,X)\cdot C=0$, $C\cdot S=0$ and $R\cdot C=f_{C}Q(g,C)$ and deduce some results. For equivalency of several curvature restrictions on a semi-Riemannian manifold, we refer to the reader to see [16], [17].

Theorem 3.1**.**

A $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$, satisfies

[TABLE]

if and only if either the manifold is $N(1-\frac{1}{n})$-contact metric manifold or it is locally isometric to the hyperbolic space $H^{2n+1}(-\kappa)$.

Proof. In view of (1.17) and (2.6), equation (1.5) reduces

[TABLE]

[TABLE]

The condition $\widetilde{C}(\xi,X)\cdot\widetilde{C}=0$ implies that

[TABLE]

Using (3.1) and (3.2) in (3.3), we have

[TABLE]

So either $r=2n\kappa(2n+1)$, or

[TABLE]

If $r=2n\kappa(2n+1)$, comparing the value of $r=2n(2n-2+\kappa)$, we get $\kappa=(1-\frac{1}{n})$. Hence, the manifold is $N(1-\frac{1}{n})$-contact metric manifold. That is, it is locally isometric to Example 2.1.

Also from (1.5) and (3.5), we obtain

[TABLE]

This implies that $M$ is of constant curvature $(-\kappa)$. Consequently it is locally isometric to the hyperbolic space $H^{2n+1}(-\kappa)$. Conversely, if the manifold is $H^{2n+1}(-\kappa)$, then (3.6) holds, which yields $r=2n\kappa(2n+1)$. Then from (3.1) it follows that $\widetilde{C}(\xi,X)=0$. Again if $k=(1-\frac{1}{n})$, then in view of (2.14) we have $r=2(n-1)(2n+1)$ and hence it follows from (3.1) that $\widetilde{C}(\xi,X)=0$. Consequently, $\widetilde{C}(\xi,X)\cdot R=0$ since $\widetilde{C}(\xi,X)$ acts as a derivation. Hence $\widetilde{C}(\xi,X)\cdot\widetilde{C}=0$. This proves the theorem.

Corollary 3.1**.**

A $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$ $(n>1)$, satisfies

[TABLE]

if and only if either the manifold is $N(1-\frac{1}{n})$-contact metric manifold or it is locally isometric to the hyperbolic space $H^{2n+1}(-\kappa)$.

Corollary 3.2**.**

In a $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$ $(n>1)$, we have $R\cdot\widetilde{C}=R\cdot R$.

In particular, if we consider a $3$-dimensional $N(\kappa)$-contact metric manifold, then $n=1$ and in that case, we have $\kappa=0$. Hence, in view of Proposition 2.2, we have the following result.

Corollary 3.3**.**

A $3$-dimensional $N(\kappa)$-contact metric manifolds satisfies $\widetilde{C}(\xi,X)\cdot\widetilde{C}=0$ if and only if the manifold is flat.

In[5] it is proved that a $3$-dimensional $N(\kappa)$-contact metric manifolds is either Sasakian, flat or locally isometric to a left invariant metric on the Lie group $SU(2)$ or $SL(2,\Re)$. Hence we have the following result.

Corollary 3.4**.**

If a $3$-dimensional $N(\kappa)$-contact metric manifold satisfies the condition $\widetilde{C}(\xi,X)\cdot\widetilde{C}=0$ then the manifold is either Sasakian, flat or locally isometric to a left invariant metric on the Lie group $SU(2)$ or $SL(2,\Re)$.

Corollary 3.5**.**

If a $3$-dimensional $N(\kappa)$-contact metric manifold satisfies the condition $\widetilde{C}(\xi,X)\cdot R=0$ then the manifold is either Sasakian, flat or locally isometric to a left invariant metric on the Lie group $SU(2)$ or $SL(2,\Re)$.

Theorem 3.2**.**

A $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$ $(n>1)$, satisfies

[TABLE]

if and only if either the manifold is $N(1-\frac{1}{n})$-contact metric manifold or it is an Einstein manifold.

Proof. The condition $\widetilde{C}(\xi,X)\cdot S=0$ implies that

[TABLE]

With the help of (3.1), we get from (3.7) that

[TABLE]

Making use of (2.10), we have

[TABLE]

From (3.9), we obtain that either $r=2n\kappa(2n+1)$ or, $S(X,Y)=2n\kappa g(X,Y)$. If $r=2n\kappa(2n+1)$, comparing the value of $r=2n(2n-2+\kappa)$ we get $\kappa=(1-\frac{1}{n})$. Hence, the manifold is $N(1-\frac{1}{n})$-contact metric manifold. This completes the proof.

In particular, if we consider a $3$-dimensional $N(\kappa)$-contact metric manifold, then $n=1$ and in that case, we have $\kappa=0$. Hence, in view of Proposition 2.2, we have the following result.

Corollary 3.6**.**

A $3$-dimensional $N(\kappa)$-contact metric manifolds satisfies $\widetilde{C}(\xi,X)\cdot S=0$ if and only if the manifold is flat.

Therefore, a $N(\kappa)$-contact metric manifold satisfying $\widetilde{C}(\xi,X)\cdot S=0$ is an Einstein manifold. Therefore, in view of Proposition 2.4, the manifold is a Sasakian manifold. In view of the above discussions, we can state the following:

Theorem 3.3**.**

A $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$ $(n>1)$, satisfies

[TABLE]

if and only if the manifold is an Einstein-Sasakian manifold.

Next, we have following theorem

Theorem 3.4**.**

A $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$ $(n>1)$, satisfies

[TABLE]

if and only if either the manifold is $N(1-\frac{1}{n})$-contact metric manifold or it is an $\eta$-Einstein manifold.

Proof. The condition $\widetilde{C}(\xi,X)\cdot C=0$ implies that

[TABLE]

By virtue of (3.1), we get from (3.10) that

[TABLE]

[TABLE]

[TABLE]

Thus either $r=2n\kappa(2n+1)$ or,

[TABLE]

[TABLE]

[TABLE]

If $r=2n\kappa(2n+1)$, comparing the value of $r=2n(2n-2+\kappa)$, we get $\kappa=(1-\frac{1}{n})$. Hence, the manifold is $N(1-\frac{1}{n})$-contact metric manifold. Again, taking the inner product of (3.12) with $\xi$, we obtain

[TABLE]

[TABLE]

[TABLE]

Using (1.2), (1.6), (2.6) and (2.10) in (3.13), we have

[TABLE]

where $\alpha=(\kappa-\frac{r}{2n(2n+1)})$ and $\beta=(\kappa+\frac{r}{2n(2n+1)})$. This completes the proof.

In particular, if we consider $3$-dimensional $N(\kappa)$-contact metric manifold, then $n=1$ and in that case, we have $\kappa=0$. Hence, in view of Proposition 2.2, we have the following result.

Corollary 3.7**.**

A $3$-dimensional $N(\kappa)$-contact metric manifold is $\widetilde{C}(\xi,X)\cdot C=0$ if and only if the manifold is flat.

Using the Proposition 2.3, we have the following:

Theorem 3.5**.**

A $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold satisfies the condition $\widetilde{C}(\xi,X)\cdot C=0$, $(n>1)$, then the manifold is a Sasakian manifold.

Theorem 3.6**.**

If a $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$ $(n>1)$, satisfies the condition $C\cdot S=0$, then

[TABLE]

Proof. In view of (1.10) the condition $C\cdot S=0$ implies that

[TABLE]

where $X,Y,Z,W\in\chi(M)$. Taking $X=\xi$ in (3.15), we get

[TABLE]

By virtue of (1.2),(1.6),(2.6) and (2.10), equation (3.16) reduces

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So, replacing $W$ with $\xi$ in (3.17) and using (2.10), we have

[TABLE]

[TABLE]

This completes the proof.

Lemma 3.1**.**

[12]** Let $A$ be a symmetric $(0,2)$-tensor at point $x$ of a semi-Riemannian manifold $(M^{n},g)$, $(n>1)$, and let $T=g\wedge A$ be the Kulkarni-Nomizu product of $g$ and $A$. Then the relation

[TABLE]

is satisfied at $x$ if and only if the condition

[TABLE]

holds at $x$.

With the help of Theorem 3.6 and Lemma 3.1 we have the following result:

Corollary 3.8**.**

Let $M$ be a (2n+1)-dimensional $N(\kappa)$ contact metric manifold $M$ $(n>1)$, satisfying the condition $C\cdot S=0$. Then $T\cdot T=\alpha Q(g,T)$, where $T=g\wedge S$ and $\alpha=\left((2n-1)\kappa-\frac{(2n-1)r}{2n(2n+1)}\right)$.

Corollary 3.9**.**

Let $M$ be an $(2n+1)$-dimensional $\eta$-Einstein $N(\kappa)$-contact metric manifold $M$ $(n>1)$. Then the condition $C\cdot S=0$ holds on $M$.

Proof. We suppose that $M(n>1)$, be an $\eta$-Einstein $N(\kappa)$-contact metric manifold. It is well-known that Weyl tensor $C$ has all symmetries of a curvature tensor. In view of (1.10) and (2.18) we have

[TABLE]

for all vector fields $X,Y,Z,W\in\chi(M)$. By using (1.6), (2.10), (2.12) and (2.18), by a straightforward calculation, we get $(C(X,Y)\cdot S)(Z,W)=0$, so we get required result. This completes the proof.

Theorem 3.7**.**

If a $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold $M$ is Weyl-pseudosymmetric then $M$ is either locally isometric to the Riemannian product $E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat for $n=1$ or $\eta$-Einstein manifold.

Proof. Let $M$ $(n>1)$, be a Weyl-pseudosymmetric $N(\kappa)$-contact metric manifold. Then from (1.9) we have

[TABLE]

In view of (1.7) and (1.8), equation (3.20) yields

[TABLE]

[TABLE]

[TABLE]

Taking $X=\xi$ in (3.21), using (1.2) and (2.6) we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Taking the inner product of (3.22) with $\xi$ and then putting $Y=Z$, we get

[TABLE]

[TABLE]

On contracting (3.23) along $Z$, we obtain

[TABLE]

If $f_{C}=0$, that is, the $M$ is Weyl-semisymmetric. Then from (3.24), either $\kappa=0$, or

[TABLE]

which gives

[TABLE]

where $\alpha=-\left\{\kappa+\frac{(2n-1)r}{2n(2n+1)}\right\}$, $\beta=\left\{(2n+1)\kappa+\frac{(2n-1)r}{2n(2n+1)}\right\}$.

If $\kappa=0$, then by Proposition 2.2, the manifold is locally isometric to the Riemannian product $E^{n+1}(0)\times S^{n}(4)$ for $n>1$ and flat for $n=1$.

On the other hand if $f_{C}\neq 0$ and $\eta(C(\xi,U)V)\neq 0$ then (3.24) gives $f_{C}=-\kappa$. So we have the following result.

Corollary 3.10**.**

Every Weyl-pseudosymmetric $(2n+1)$-dimensional $N(\kappa)$-contact metric manifold, $(n>1)$, is of the form $R\cdot C=-\kappa Q(g,C)$.

Corollary 3.11**.**

Every Weyl-pseudosymmetric $(2n+1)$-dimensional Sasakian manifold, $(n>1)$, is of the form $R\cdot C=-Q(g,C)$.

Using the Proposition 2.3, we have the following

Theorem 3.8**.**

A $(2n+1)$-dimensional Weyl-pseudosymmetric $N(\kappa)$-contact metric manifold, $(n>1)$, is a Sasakian manifold.

4. Example

Example.4.1 We consider a $3$-dimensional manifold $M=\{(x,y,z)\in\Re^{3},(x,y,z)\neq(0,0,0)\},$ where $(x,y,z)$ is the standard coordinate in $\Re^{3}$. Let $(e_{1},e_{2},e_{3})$ be linearly independent vector fields in $\Re^{3}$ defined by

[TABLE]

and

[TABLE]

Let $g$ be the Riemannian metric defined by

[TABLE]

Let $\eta$ be the $1$-form such that

[TABLE]

for any $U\in\chi(M)$. Let $\phi$ be the $(1,1)$-tensor field defined by

[TABLE]

Making use of the linearity of $\phi$ and $g$, we have

[TABLE]

[TABLE]

and

[TABLE]

for any $U,W\in\chi(M).$ Moreover

[TABLE]

The Riemannian connection $\nabla$ of metric tensor $g$ is given by Koszul’s formula as

[TABLE]

Using the Koszul’s formula, we get

[TABLE]

Consequently, the manifold satisfies the relation

[TABLE]

and

[TABLE]

Thus we have

[TABLE]

for any vector field $X$. Therefore, the manifold is a contact metric manifold for $e_{3}=\xi$.

Now, we find the components of curvature tensor as

[TABLE]

From the expression of above, we conclude the manifold is $N(0)$-contact metric manifold.

Example.4.2 We consider a $5$-dimensional differentiable manifold

[TABLE]

where $(x,y,z,u,v)$ denote the standard coordinate in $\mathbb{R}^{5}$. Let $(e_{1},e_{2},e_{3},e_{4},e_{5})$ are five vector fields in $\mathbb{R}^{5}$ which satisfies

[TABLE]

[TABLE]

We also define the Riemannian metric $g$ by

[TABLE]

[TABLE]

Let the $1-$form $\eta$ be $\eta(Z)=g(Z,e_{1})$ for any $Z\in\chi(M)$.

Let $\phi$ be the $(1,1)$-tensor field defined by

[TABLE]

By the linearity properties of $\phi$ and $g$, we have

[TABLE]

for arbitrary vector fields $X,Y\in\chi(M)$. Moreover,

[TABLE]

We recall the Koszul’s formula as

[TABLE]

for arbitrary vector fields $X,Y,Z\in\chi(M)$. Using Koszul’s formula we get

[TABLE]

With the help of above relation, it is notice that $\nabla_{X}\xi=-\phi X-\phi hX$ for $\xi=e_{1}$. Therefore, the manifold is a contact metric manifold with the contact structure $(\phi,\eta,\xi,g)$.

Now, we find the curvature tensors as follows

[TABLE]

In view of the expressions of the curvature tensors we conclude that the manifold is a $N(-\lambda^{2})$-contact metric manifold.