$N(\kappa)$-paracontact metric manifolds admitting the Fischer-Marsden conjecture

1.   Introduction

Paracontact geometry equipped with the nullity distribution contributes a crucial part in the development of modern paracontact geometry. The pioneering work of Kaneyuki and Williams ^[1] opened the door to the study of paracontact geometry for researchers. The para-Kähler manifolds with its applications in pseudo-Riemannian geometry and mathematical physics have motivated researchers to concentrate on paracontact geometry. In ^[2], Zamkovoy presented a systematic research of paracontact metric manifolds (PMMs). The geometrical and physical properties of PMMs have been studied by many researchers. Calvaruso et al. ^[3] have investigated paracontact metric structures on the unit tangent bundle. The properties of bi-paracontact structure and Legendre foliations have been explored in ^[4]. Blaga ^[5] has studied the properties of Lorentzian para-Sasakian manifolds endowed with $\eta$-Ricci solitons. Three-dimensional paracontact metric manifolds have studied in ^[6,7,8,9]. Cappelletti-Montano et al. ^[10] introduced the notion of paracontact $(\kappa, \mu)$-spaces and obtained their various properties, where $\kappa$ and $\mu$ are real constants. After that, the properties of paracontact metric $(\kappa, \mu)$-spaces have been studied in ^{[11,12,13,14]}. The classification of paracontact metric $(\kappa, \mu)$-spaces with non-trivial examples were given in ^[15,16].

Let $M^{2n+1}$ be a $(2n+1)$-dimensional PMM, and $g$ is a pseudo-Riemannian metric of $M^{2n+1}$. If the set of all pseudo-Riemannian metrics of unit volume on $M^{2n+1}$ is represented by $\mathcal{G}$, then we have

where $g^{*}$ represents the $(0, 2)$-type symmetric bilinear tensor, $\triangle_{g}$ is the negative Laplacian of the pseudo-Riemannian metric $g$, $S_{g}$ is the Ricci tensor corresponding to $g$ and $'div'$ and $'tr'$ are used for divergence and trace, respectively. Here, $\mathcal{L}_{g}$ is the linearized scalar curvature operator. Let $\mathcal{L}^{*}_{g}$ represent the formal $L^{2}$-adjoint of $\mathcal{L}_{g}$. Then, the aforementioned equation assumes the form

where $Hess_{g}\lambda$ is the Hessian of the smooth function $\lambda$ corresponding to $g$ and is defined by the relation $Hess_{g} \lambda ({\mathfrak{U}_1}, {\mathfrak{U}_2}) = g(\nabla_{{\mathfrak{U}_1}} D \lambda, {\mathfrak{U}_2})$, $\forall \, \, {\mathfrak{U}_1}, {\mathfrak{U}_2} \in \mathfrak{X}(M^{2n+1})$. Here, $D$ denotes the gradient operator and $\mathfrak{X}(M^{2n+1})$ the collection of all smooth vector fields of $M^{2n+1}$. In the present paper, we represent $\mathcal{L}^{*}_{g}(\lambda) = 0$ as the Fischer-Marsden equation (briefly, by $FME$). The doublet $(g, \lambda)$ satisfying the equation $\mathcal{L}^{*}_{g}(\lambda) = 0$, for $\lambda \ne 0$, is known as the non-trivial solution of the $FME$. Bourguignon ^[17], Fischer and Marsden ^[18] considered a Riemannian manifold satisfying the $FME$ and proved that the scalar curvature of the manifold is constant. In $2000$, Corvino ^[19] showed that a complete Riemannian manifold with the warped product metric $g^{*} = g-\lambda^{2}dt^{2}$ is Einstein if and only if the doublet $(g, \lambda)$ is a non-trivial solution of the $FME$. Recently, Al-Dayal et al. ^[20] have considered a semi-Riemannian manifold satisfying the Fischer-Marsden equation and proved that the manifold under consideration is a quasi-Einstein manifold. In $1974$, Fischer and Marsden ^[18] conjectured that, if a compact Riemannian $n$-manifold concedes $(g, \lambda)$ with $\lambda \ne 0$, we attain an Einstein manifold. The first counter example of the Fischer-Marsden conjecture (in short, $FMC$) was given by Kobayashi ^[21]. In ^[22], Cernea and Guan established that a closed homogeneous Riemannian manifold $(M, g)$ satisfying the equation $\mathcal{L}^{*}_{g}(\lambda) = 0$ is locally isometric to $E \times S^{m}$, where $E$ and $S^{m}$ denote the Einstein manifold and the Euclidean sphere, respectively. Recently, Patra and Ghosh ^[23] proved that if a $K$-contact (or a $(\kappa, \mu)$-contact) metric manifold has $\mathcal{L}^{*}_{g}(\lambda) = 0$, the manifold is Einstein (or locally isometric to the sphere $S^{2n+1}$). Prakasha et al. ^[24] considered a $(2n+1)$-dimensional ${(\kappa, \mu)'}$-almost Kenmotsu manifold $(M, g)$ admits $(g, \lambda \ne 0)$ and proved that $(M, g)$ is locally isometric to $\mathbb{H}^{n+1}(\alpha) \times_{f} \mathbb{R}^{n}$ or $\mathbb{B}^{n+1}(\alpha') \times_{f'} \mathbb{R}^{n}$. In ^[25,26,27], Chaubey et al. studied the properties of Kenmotsu manifolds, generalized Sasakian-space-forms and cosymplectic manifolds satisfying the Fischer-Marsden conjecture. Deshmukh et al. ^[28] explored the Fischer-Marsden conjecture in Riemannian manifolds. Very recently, Suh et al. ^[29,30,31] and Venkatesha et al. ^[32] have explored the properties of FMC on the hypersurfaces of space-forms.

The above studies inspire us to characterize $N(\kappa)$-paracontact metric manifolds (in brief, NKPMMs) satisfying the $FME$, that is, $\mathcal{L}^{*}_{g}(\lambda) = 0$. Following an overview in Section $1$, in Section $2$ we gather the basic known results and definitions of PMMs. Section $3$ deals with the study of three-dimensional NKPMMs satisfying the Fischer-Marsden equation and prove that the manifold under consideration is Einstein. In Section $4$, we characterize an NKPMM satisfying the equation $\mathcal{L}^{*}_{g}(\lambda) = 0$ for $n > 1$. It is proved that either there does not exist an Einstein NKPMM with $\kappa > -1$, or the manifold $M^{2n+1}$ is locally isometric to the product of a hyperbolic space $\mathbb{H}^{n}(-4)$ and a Euclidean space $\mathbb{E}^{n+1}$.

2.   Paracontact metric manifolds

Let $M^{2n+1}$ be a $(2n+1)$-dimensional differentiable manifold of class $C^{\infty}$. Then, a triplet $(\phi, \xi, \eta)$ defined on $M^{2n+1}$ and satisfying the relations

where $\phi$ is a tensor field of type $(1, 1)$, $\eta$ a tensor field of type $(0, 1)$ and the Reeb vector field $\xi$, is known as an almost paracontact structure on $M^{2n+1}$. The manifold $M^{2n+1}$ equipped with the structure $(\phi, \xi, \eta)$ is called an almost paracontact manifold. It is noticed that the structure tensor $\phi$ induces an almost paracomplex structure $J$ on the horizontal distribution $\mathcal{D} = ker (\eta)$, that is, the eigensubbundles $\mathcal{D^{+}}$ and $\mathcal{D^{-}}$ have equal dimension $n$ corresponding to the eigenvalues $+1$ and $-1$ of $J$, respectively. If $M^{2n+1}$ admits a pseudo-Riemannian metric $g$ of type $(0, 2)$ such that the relations

hold for all ${\mathfrak{U}_1}, \, {\mathfrak{U}_2} \, \, \in \mathfrak{X}(M^{2n+1})$, then $(M^{2n+1}, g)$ is known as an almost PMM. From (2.1) and (2.2), it follows that the following relations

hold for all ${\mathfrak{U}_1}, \, {\mathfrak{U}_2} \, \, \in \mathfrak{X}(M^{2n+1})$. An almost PMM $M^{2n+1}$ with $g({\mathfrak{U}_1}, \phi{{\mathfrak{U}_2}}) = d \eta({\mathfrak{U}_1}, {\mathfrak{U}_2})$ becomes a PMM. Here $d$ represents the exterior derivative operator.

An almost paracontact metric structure with $[\phi, \phi ]-2d\eta \otimes \xi = 0$ is said to be normal, where $[\phi, \phi ]$ represents the Nijenhuis tensor corresponding to the structure tensor $\phi$. In ^[2], Zamkovoy proved that an almost PMM $M^{2n+1}$ possesses at least a (locally) $\phi$-basis, that is, the set $\{E_{1}, E_{2}, E_{3}, ..., E_{n}, \; \phi {E_{1}}, \phi {E_{2}}, \phi {E_{3}}, ..., \phi {E_{n}}, \xi \}$ represents a (locally) pseudo-orthonormal basis of the vector fields, where $E_{1}, E_{2}, E_{3}, \;..., E_{n}, \xi$ and $\phi { E_{1}}, \phi {E_{2}}, \phi {E_{3}}, \;..., \; \phi {E_{n}}$ are space-like and time-like vector fields, respectively. In $M^{2n+1}$, the $\phi$-basis is determined by a (locally) pseudo-orthonormal basis of $ker(\eta)$. If possible, we suppose that $e_{3}$ is time-like and $\{e_{2}, e_{3}\}$ a pseudo-orthonormal basis of $ker(\eta)$. Then, from Eq (2.2) we conclude that $\phi {e_{2}}\in ker{(\eta)}$ is time-like and orthonormal to $e_{2}$. Thus $\phi {e_{2}} = \pm e_{3}$ and hence we consider $\{e_{2}, \pm e_{3}, \xi \}$ to be a $\phi$-basis on $M^{3}$. For more details, we refer to ^[7]. On a PMM $M^{2n+1}$, a symmetric and trace-free $(1, 1)$-type tensor $h$, defined by $h = \frac{1}{2} \mathcal{L}_{\xi }\phi$, satisfies

for all ${\mathfrak{U}_1}\in \mathfrak{X}(M^{2n+1})$, where $\nabla$ and $tr{h}$ denote the Levi-Civita connection and the trace of the operator $h$, respectively. An almost paracontact metric structure is said to be a $K$-paracontact structure if $\xi$ is Killing, that is, $h = 0$. An almost PMM is said to be a para-Sasakian manifold if and only if

for all ${\mathfrak{U}_1}, \, {\mathfrak{U}_2} \, \, \in \mathfrak{X}(M^{2n+1})$. A normal PMM is para-Sasakian and satisfies

The converse is not true. Here $R$ denotes the curvature tensor corresponding to $\nabla$.

Next, we consider that the Reeb vector field $\xi$ of a $(2n+1)$-dimensional PMM $M^{2n+1}$ belongs to the $(\kappa, \mu)$-nullity distribution.

Definition 2.1. A PMM $M^{2n+1}$ is said to be a paracontact $(\kappa, \mu)$-manifold if

for all ${\mathfrak{U}_1}, \, {\mathfrak{U}_2} \, \, \in \mathfrak{X}(M^{2n+1})$, where $\kappa$ and $\mu$ are real constants ^[12].

As a particular case, the paracontact metric $(\kappa, \mu)$-manifold with $\mu = 0$ reduces to an NKPMM. Hence, the above equation becomes

In light of Eqs (2.2), (2.5) and (2.6), we have

where $S$ denotes the Ricci tensor of $M^{2n+1}$. For $dim \, M = 3$, the NKPMM $M^3$ satisfies the following relations

for each ${\mathfrak{U}_1}, \, {\mathfrak{U}_2} \in \mathfrak{X}(M^{2n+1})$, where the Ricci operator associated with the Ricci tensor $S$ is $Q$, that is, $S(\cdot, \cdot) = g(Q\cdot, \cdot)$ and $r$ denotes the scalar curvature of $M^{2n+1}$ ^[9].

3.   Three-dimensional NKPMM satisfying the Fischer-Marsden conjecture

This section deals with the study of the Fischer-Marsden conjecture within the framework of a three-dimensional NKPMM. In this section, we represent $M^3$ as a three-dimensional NKPMM. We recall the following results.

Lemma 3.1. A three-dimensional paracontact metric $(\kappa, \mu)$-manifold is Einsteinian if and only if $\kappa = \mu = 0$ (see Corollary $4.14$, ^[10]).

De et al. ^[33] showed that the following results hold on a three-dimensional NKPMM.

Lemma 3.2. i) If and only if the manifold is an Einstein manifold, an $M^3$ is Ricci semisymmetric.

ii) If and only if the manifold has constant curvature $\kappa$, an $M^3$ is Ricci semisymmetric.

iii) An $M^3$ is Riccisymmetric if and only if the manifold is of constant curvature $\kappa$.

Before proving our main results, we prove the following propositions.

Proposition 3.1. If a PMM $M^{2n+1}$ satisfies the FMC, then we have

for all vector fields ${\mathfrak{U}_1}$ and ${\mathfrak{U}_2}$ of $M^{2n+1}$, where $f = -\frac{r \lambda}{2n}$.

Proof. Assume that there is a non-trivial solution $(g, \lambda)$ to the equation $\mathcal{L}^{*}_g (\lambda) = 0$. Then, from Eq (1.1), we have

where ${\triangle_{g}}\lambda = -\frac{r \lambda}{2n}$. Thus, the $FME$ can be written as

Equation (3.2)'s covariant derivative along the vector field ${\mathfrak{U}_2}$ results in

Interchanging ${\mathfrak{U}_1}$ and ${\mathfrak{U}_2}$ in (3.3) and using the obtained equation, (3.2) and (3.3) in $R({\mathfrak{U}_1}, {\mathfrak{U}_2})D \lambda = [\nabla_{{\mathfrak{U}_1}}, \nabla_{{\mathfrak{U}_2}}] D \lambda-\nabla_{[{\mathfrak{U}_1}, {\mathfrak{U}_2}]} D \lambda$, we immediately get the required result. □

Proposition 3.2. On $M^3$, we have

Proof. From Eq (2.10), we have $Q\xi = 2\kappa \xi$, where $\kappa$ is a real constant. Taking the covariant derivative of this equation along the vector field ${\mathfrak{U}_2}$, and using Eqs (2.3), (2.5) and (2.9), we obtain

Differentiating Eq (2.9) once more along the Reeb vector field $\xi$, we have

In light of Eqs (2.2)–(2.5), (2.9) and (2.10), the above equation becomes

Thus, in view of (3.5) and (3.6), we get the statement of Proposition 3.2. □

From Eq (3.5), we have

where $div\, Q$ denotes the divergence of Ricci operator $Q$. This equation shows that the scalar curvature $r$ is locally constant along the vector field $\xi$.

Now, we are going to prove the main result of this section. Changing ${\mathfrak{U}_1}$ by $\xi$ in (3.1), we obtain

Taking the inner product of (3.8) with ${\mathfrak{U}_1}$ and then calling Eqs (2.2), (2.3), (2.10) and (3.4), we obtain

In view of Eqs (2.1) and (2.8), we get

This result also holds well for the $(2n+1)$-dimensional NKPMM. Eq (3.9) along with Eq (3.10) gives

Let $\{e_i, \, i = 1, \, 2, \, 3\}$ be a local orthonormal basis on $M^3$. Setting ${\mathfrak{U}_1} = {\mathfrak{U}_2} = e_i$ in (3.11) and summing for $i$, $i = 1, \, 2, \, 3$, we conclude that

which becomes

It is obvious that, on $M^3$, $2f = -r \lambda$, and hence it gives us

From Eqs (3.7) and (3.11), if $(div\, Q)(\xi) = 0$, then we have

Setting ${\mathfrak{U}_1} = \xi$ in (3.12) and using the Eqs (2.1)–(2.4) and (2.10), we get

By replacing ${\mathfrak{U}_1}$ in Eq (3.12) with $\phi{\mathfrak{U}_1}$ and then using Eq (2.3), we discover

Interchanging ${\mathfrak{U}_1}$ and ${\mathfrak{U}_2}$ in (3.13), we get

Adding (3.13) and (3.14), we find

Therefore, $\lambda \ne 0$ as we are interested in the Fischer-Marsden equation's non-trivial solution. Now we divide our study into two cases as:

Case I. We suppose that $r \ne 6 \kappa$. Then, the above equation along with (2.5) reflects that $\nabla \xi = 0.$ This result together with Eq (2.6) shows that $R({\mathfrak{U}_1}, {\mathfrak{U}_2})\xi = 0$, and hence we have $\mu = 0$ and $\kappa = 0$. These results and Lemma 3.1 infer that the three-dimensional NKPMM obeying the $FME$ is Einstein.

Case II. Let us assume that $r = 6 \kappa$. That is, the scalar curvature of the three-dimensional NKPMM satisfying $\mathcal{L}^{*}_g (\lambda) = 0$ is constant. This shows that

In consequence of $r = 6\kappa$ and (3.15), Eq (3.12) reduces to

This shows that either $S = 2 \kappa g$ or $(\xi \lambda) = 0$. If possible, we consider that $(\xi \lambda) = 0$, and hence $g(\xi, D \lambda) = 0$. Differentiating $g(\xi, D \lambda) = 0$ covariantly along ${\mathfrak{U}_1}$, we find

The above equation along with Eqs (2.2), (2.5), (2.10) and (3.2) give

Substituting ${\mathfrak{U}_1} = \xi$ in (3.17) and using Eqs (2.3) and (2.4), we get

From Eqs (3.2) and (3.18), we conclude that $r = 4 \kappa$, which contradicts our hypothesis. Hence, $(\xi \lambda) \ne 0$, and thus Eq (3.16) gives $S = 2 \kappa g$. By considering the above discussions and Lemma 3.1, we state:

Theorem 3.1. An $M^3$ satisfying the Fischer-Marsden conjecture is Einstein.

In light of Lemma 3.2 and Theorem 3.1, we can state the following:

Corollary 3.1. Let the Fischer-Marsden conjecture hold on an $M^3$. Then, the following conditions are equivalent:

$(i) \; M^3$ is Einstein,

$(ii) \; M^3$ is Ricci semisymmetric,

$(iii) \; M^3$ is a space of constant curvature,

$(iv) \; M^3$ is Ricci symmetric.

4.   The Fischer-Marsden conjecture on an NKPMM

The aim of this section is to study the properties of a $(2n+1)$-dimensional NKPMM satisfying the Fischer-Marsden conjecture. We will utilize the following result to support our main finding.

Lemma 4.1. ^[34] Let $M^{2n+1}$ be a PMM and suppose that $R({\mathfrak{U}_1}, {\mathfrak{U}_2})\xi = 0$ for all vector fields ${\mathfrak{U}_1}$, ${\mathfrak{U}_2}.$ Then, locally, $M^{2n+1}$ is the product of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of negative curvature equal to $-4$ for $n > 1$.

In ^[10], Cappelletti-Montano et al. characterized $(2n+1)$-dimensional paracontact metric $(\kappa, \mu)$-manifolds and proved many interesting results. It is observed that the Ricci operator $Q$ of a $(2n+1)$-dimensional NKPMM satisfies the following relation

for $\kappa \ne -1$, where $I$ is an identity operator on $M^{2n+1}$. Throughout this section, we suppose that $\kappa \ne -1$. The symmetric tensor field $h$ also satisfies

for all ${\mathfrak{U}_1}, \, {\mathfrak{U}_2} \, \, \in \mathfrak{X}(M^{2n+1})$. From (4.1), we have

which gives

In consequence of Eqs (2.1)–(2.3), (2.5), (3.1), (4.2) and (4.3), we get

From Eqs (3.10) and (4.4), we have

Contracting Eq (4.1), we obtain

and hence Eq (3.2) becomes

which infers

Using Eq (4.7) in (4.5), we obtain

Setting ${\mathfrak{U}_1} = \xi$ in (4.8) and using Eqs (2.1)–(2.4) and (2.7), we get

This shows that either $n \kappa +n-1 = 0$ or $({\mathfrak{U}_2} \lambda)-(\xi \lambda)\eta({\mathfrak{U}_2}) = 0$.

Case I. We suppose that $n \kappa +n-1 = 0$. Thus, $\kappa = -1+\frac{1}{n} > -1$ for $n > 1$.

Cappelletti-Montano et al. ^[10] investigated several results of $(2n+1)$-dimensional paracontact metric $(\kappa, \mu)$-manifolds for $\kappa > -1$. They proved that a three-dimensional NKPMM with $\kappa > -1$ is an $\eta$-Einstein manifold. They also showed that there is no paracontact $(\kappa, \mu)$-manifold for $\kappa > -1$ and $n > 1$ that can be Einstein.

Case II. If possible, we suppose that $({\mathfrak{U}_2} \lambda)-(\xi \lambda)\eta({\mathfrak{U}_2}) = 0$ for $n > 1$ on $M^{2n+1}$. Thus we have $D \lambda = (\xi \lambda)\xi$. By covariantly differentiating this outcome along the vector field ${\mathfrak{U}_1}$, we discover

According to Eq (3.2), the previous equation has the following form:

Taking a local frame field and contracting Eq (4.9), we get

Again replacing ${\mathfrak{U}_1}$ by $\xi$ in Eq (3.2) and taking the inner product with $\xi$, we find

Equations (4.10) and (4.11) along with Eqs (3.2) and (4.6) give us

By the hypothesis $\lambda \ne0$, we have $\kappa = 0$ for $n > 1$ and, from Eq (2.6), we get $R({\mathfrak{U}_1}, {\mathfrak{U}_2})\xi = 0$. This result along with the Lemma 4.1 tell us that $M^{2n+1}$, $n > 1$, is locally isometric to the product of the Euclidean space $\mathbb{E}^{n+1}$ and a hyperbolic space $\mathbb{H}^{n}(-4)$ of constant curvature $-4$. Thus, we are in a position to state the following:

Theorem 4.1. Let a $(2n+1)$-dimensional $N(k)$-paracontact metric manifold $M^{2n+1}$ with $n > 1$ satisfy the equation $\mathcal{L}^{*}_{g}(\lambda) = 0$. Then, either $M^{2n+1}$ is locally isometric to $\mathbb{E}^{n+1} \times \mathbb{H}^n$ or there does not exist an Einstein NKPMM with $\kappa > -1$ for $M^{2n+1}$.

5.   Conclusions

The notion of the Fischer-Marsden conjecture on Riemannian manifolds was introduced by Fischer and Marsden ^[18], and it has been further extended by Bourguignon ^[17]. This conjecture on some classes of almost contact metric manifolds has been explored by many researchers. In this manuscript, we defined the Fischer-Marsden conjecture on semi-Riemannian manifolds, and in particular, we studied the non-trivial solutions of the Fischer-Marsden equation on $N(k)$-paracontact metric manifolds. This manuscript may open a door for researchers to explore the non-trivial solutions of the Fischer-Marsden equation on the classes of semi-Riemannian manifolds.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in writing the paper.

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RP23105).

Conflict of interest

The authors declare that there is no conflict of interest in this paper.