Characteristics of Sasakian Manifolds Admitting Almost ∗-Ricci Solitons

Abstract

This article presents some results of a geometric classification of Sasakian manifolds (SM) that admit an almost ∗-Ricci soliton (RS) structure $(g,\omega ,X)$. First, we show that a complete SM equipped with an almost ∗-RS with $\omega \ne$ const is a unit sphere. Then we prove that if an SM has an almost ∗-RS structure, whose potential vector is a Jacobi vector field on the integral curves of the characteristic vector field, then the manifold is a null or positive SM. Finally, we characterize those SM represented as almost ∗-RS, which are ∗-RS, ∗-Einstein or ∗-Ricci flat.

1. Introduction

Nonlinear systems that support solitons can generate self-similarity and fractals on successively smaller scales. Very often, the solitons of a given system are all self-similar to each other, see [1].

Several authors study Ricci solitons (RS) in contact metric geometry, where Sasakian manifolds (SM) are especially important. SM, which are odd-dimensional analogues of Kähler manifolds, are used to build geometrical examples, e.g., manifolds of special holonomy, Einstein manifolds (EM), Kähler manifolds and orbifolds. The geometry of SM (in particular, $\eta$-EM and Sasaki–Einstein manifolds) attracts much attention from mathematicians. We refer the reader to [2,3,4] for the latest developments in the theory of these manifolds.

Many tools for Kähler manifolds have analogues for contact manifolds, among them the ∗-Ricci tensor, introduced in [5] for almost Hermitian manifolds. In [6], the ∗-Ricci tensor was applied to a real hypersurface in a non-flat complex space form. Namely, for an almost contact manifold $M^{2n+1}(\phi ,\zeta ,\eta ,g)$ and any vector fields $Y_1,Y_2,Y_3$ on $M^{2n+1}$, we get

$$\begin{array}{c}\hfill {\mathrm{Ric}}_g^{*}(Y_1,Y_2)=\frac{1}{2}{\mathrm{trace}}_{\mathrm{g}}\left\{Y_3\to R(Y_1,\phi Y_2)\phi Y_3\right\}.\end{array}$$

(1)

If ${\mathrm{Ric}}_g^{*}$ vanishes identically then such $M^{2n+1}$ is called ∗-Ricci flat. A ∗-RS structure has been introduced in [7], using ${\mathrm{Ric}}_g^{*}$ instead of the Ricci tensor in the equation of RS,

$$\begin{array}{c}\hfill {\pounds }_{X}g+2{\mathrm{Ric}}_g^{*}=2\omega g.\end{array}$$

(2)

Here, £ is the Lie derivative. If $\omega$ in (2) belongs to $C^{\infty }\left(M\right)$ and is not a constant, then we get an almost ∗-RS, the triple $(g,\omega ,X)$. An almost ∗-RS is shrinking for $\omega >0$, steady for $\omega =0$ and expanding for $\omega <0$. Observe that a ∗-RS is ∗-EM (i.e., the ∗-Ricci tensor and the metric tensor are homothetic) if X is a Killing vector field (KVF), see [8]. Thus, it is a generalization of a ∗-EM. In particular, if $X=\nabla f$ (the gradient of $f\in C^{\infty }\left(M\right)$) in (2), then we obtain a gradient almost ∗-RS. Some generalizations of RS were studied by several authors, see [9,10,11,12,13,14,15].

Ghosh-Patra [16] first studied ∗-RS and gradient almost ∗-RS on a contact metric manifold, in particular, on an SM and $(k,\mu )$-contact manifolds. Later on, ∗-RS were studied on almost contact manifolds in [15,17,18,19,20]. Wang [20] proved that “if the metric of a Kenmotsu 3-manifold represents a ∗-RS, then the manifold is locally the hyperbolic space ${\mathbb{H}}^3(-1)$”, and it was proved in [18] that “if a non-Kenmotsu ${(k,\mu )}^{\prime }$-almost Kenmotsu manifold has a ∗-RS structure, then it is locally isometric to ${\mathbb{H}}^{n+1}(-4)\times \mathbb{R}$ under some restrictions”.

There is a natural question: “under what conditions a (gradient) almost RS is a unit sphere”? Several affirmative answers to this question on Riemannian and contact metric manifolds, are given in [9,10,12,13,14,21,22]. An almost ∗-RS is a generalization of ∗-RS and ∗-EM; thus we ask a question: “under what conditions a (gradient) almost ∗-RS is a unit sphere?”

In [16], they answered this question by showing that “if a complete SM admits a gradient almost ∗-RS, then it is a unit sphere”. Here, we consider non-gradient almost ∗-RS in the case of SM, and we are looking for conditions under which SM having an almost ∗-RS structure is a unit sphere. Our main achievement is the following.

Theorem 1.

If a complete SM $M^{2n+1}(\phi ,\zeta ,\eta ,g)$ of dimension greater than three has an almost ∗-RS structure $(g,\omega ,X)$ with $\omega \ne$ const, then it is the unit sphere ${\mathbb{S}}^{2n+1}$.

The following question arises: “under what conditions an almost RS is an RS, or, EM”? Several answers to this question can be found in [14,22,23,24]. An almost ∗-RS generalizes ∗-RS and ∗-EM; so the following question arises:

“when an almost ∗-RS is a ∗-RS, for example, an ∗-EM ?”

In [16], they found such a condition on an SM, and this question on a Kenmotsu manifold was studied in [15]. In this regard, we are interested in characterizing those SM represented as almost ∗-RS, which are ∗-RS or are ∗-EM. In the following theorem, we assume that the potential vector field is a Jacobi vector field on the integral curves of the Reeb vector field.

Theorem 2.

If an SM $M^{2n+1}(\phi ,\zeta ,\eta ,g)$ has an almost ∗-RS structure $(g,\omega ,X)$ such that X is a Jacobi field on the ζ-integral curves, then $(g,\omega ,X)$ is a ∗-RS on M.

Now, we recall some definitions, e.g., [3]. A SM $M^{2n+1}(\phi ,\zeta ,\eta ,g)$ is an η-EM if

$${\mathrm{Ric}}_{\mathrm{g}}={\alpha }_{1}g+{\alpha }_2\eta \otimes \eta ,$$

where functions ${\alpha }_1$ and ${\alpha }_2$ belong to $C^{\infty }\left(M\right)$. For K-contact manifolds (for example, SM) of dimension $\ge 4$, ${\alpha }_1,{\alpha }_2$ are real constants, see [25], thus, the scalar curvature is constant. An $\eta$-Einstein SM is a null-SM when ${\alpha }_1=-2$ and ${\alpha }_2=2n+2$, and is a positive-SM when ${\alpha }_1>-2$, see [3]. Using ([16], Theorem 8), we get the following consequence of Theorem 2.

Corollary 1.

If an SM $M^{2n+1}(\phi ,\zeta ,\eta ,g)$ has an almost ∗-RS structure $(g,\omega ,X)$ such that X is a Jacobi vector field on the ζ-integral curves, then M is positive SM and X is KVF (and g is ∗-EM), or M is null-SM and φ is invariant under X.

Remark 1.

Observe from [24] that if an SM has an almost RS structure, whose potential field is a Jacobi vector field on the$\zeta$-integral curves, then the manifold is null-SM with the expanding RS.

A vector field Y on a contact manifold $(M,\eta )$ that preserves the contact form $\eta$ is called an infinitesimal contact transformation, see [26,27], i.e.,

$$\begin{array}{c}\hfill {\pounds }_Y\eta =\nu \eta ,\end{array}$$

(3)

for some $\nu \in C^{\infty }\left(M\right)$; and if $\nu =0$, then Y is said to be strict. A vector field Y preserving $\phi ,\zeta ,\eta$ and g on a contact metric manifold is called an infinitesimal automorphism. In [16], they addressed the question by showing that “if an SM represents an almost ∗-RS such that the potential vector field is an infinitesimal contact transformation, then it is a ∗-RS". We improve this result in the following statement.

Theorem 3.

Let an SM represent an almost ∗-RS$(g,\omega ,X)$such that X is an infinitesimal contact transformation. Then X leaves φ invariant, and the manifold is η-EM of scalar curvature$(\omega +4n)n$.

Assuming compactness we get one more sufficient condition for g to be ∗-Ricci flat (e.g., ∗-EM).

Theorem 4.

Let a compact SM$M^{2n+1}(\phi ,\zeta ,\eta ,g)$admit an almost ∗-RS$(g,\omega ,X)$such that X is an infinitesimal contact transformation. Then the soliton is steady$(\omega =0)$, X is an infinitesimal automorphism, and g is ∗-Ricci flat of scalar curvature$4n^2$.

Finally, we show that the property “potential vector field and the characteristic vector field are parallel" provides “∗-Ricci flat" (e.g., ∗-EM).

Theorem 5.

Let an SM$M^{2n+1}(\phi ,\zeta ,\eta ,g)$admit an almost ∗-RS$(g,\omega ,X)$such that X is parallel to ζ. Then the soliton is steady$(\omega =0)$, V is KVF, and g is ∗-Ricci flat of scalar curvature$4n^2$.

Remark 2.

Observe from [14] that if a compact SM has an almost RS structure$(g,\omega ,X)$such that X is an infinitesimal contact transformation, then the soliton is shrinking ($\omega =2n$), X is strict and leaves$\phi$invariant, and the manifold is EM of scalar curvature$2n(2n+1)$. By Theorem 4, the soliton is steady$(\omega =0)$, X is strict and leaves$\phi$,$\zeta$and$\eta$invariant, and the manifold is ∗-EM of scalar curvature$4n^2$. In addition, note that an SM considered as an RS or an almost RS with X parallel to$\zeta$, is an EM of scalar curvature$2n(2n+1)$, and X is a KVF, see [28].

Within the framework of Theorems 3–5, we not only find sufficient conditions for ∗-EM, but also characterize the structural tensor fields and the potential vector field.

2. Preliminaries

Here, we recall some properties of SM, see [2,26]. We suppose that all manifolds are smooth and connected. The curvature tensor R on a Riemannian manifold $(M,g)$ is given by

$$\begin{array}{c}\hfill R(Y_1,Y_2)=[{\nabla }_{Y_1},{\nabla }_{Y_2}]-{\nabla }_{[Y_1,Y_2]},Y_1,Y_2\in \mathcal{X}\left(M\right),\end{array}$$

(4)

where ∇ is the Levi-Civita connection and $\mathcal{X}\left(M\right)$ is the space of vector fields on M. They define the Ricci operator Q by

$$g(Q{Y}_1,Y_2)={\mathrm{Ric}}_{\mathrm{g}}(Y_1,Y_2)={\mathrm{trace}}_{\mathrm{g}}\left\{Y_3\to R(Y_3,Y_1)Y_2\right\},Y_1,Y_2,Y_3\in \mathcal{X}\left(M\right)$$

is a symmetric $(1,1)$-tensor. Then $r={\mathrm{trace}}_{\mathrm{g}}Q$ is the scalar curvature. We get the following formula (follows from twice contracted second Bianchi identity, see, e.g., [29]):

$$\begin{array}{c}\hfill \frac{1}{2}g(Y_1,\nabla r)=(divQ)\left(Y_1\right)={\sum }_{i}g(\left({\nabla }_{E_i}Q\right)Y_1,E_i),Y_1\in \mathcal{X}\left(M\right),\end{array}$$

(5)

where $\left\{E_i\right\}$ is any local orthonormal basis on M.

For the Lie derivative of $f\in C^{\infty }\left(M\right)$ along $Y_1\in \mathcal{X}\left(M\right)$ we have

$${\pounds }_{Y_1}f=Y_1\left(f\right)=g(Y_1,\nabla f),$$

where the Hessian of f is defined by

$${\mathrm{Hess}}_f(Y_1,Y_2)={\nabla }^{2}f(Y_1,Y_2)=({\nabla }_{Y_1}\nabla f)Y_2=Y_1{Y}_2\left(f\right)-\left({\nabla }_{Y_1}{Y}_2\right)\left(f\right),Y_1,Y_2\in \mathcal{X}\left(M\right).$$

The hessian ${\mathrm{Hess}}_f$ is symmetric and $g({\nabla }_{Y_1}\nabla f,Y_2)=g({\nabla }_{Y_2}\nabla f,Y_1)$ is true, so it is a section of $T^{*}M\otimes T^{*}M$. A vector field Y on $(M,g)$ is conformal if

$$\pounds {}_{Y}g=fg$$

for some $f\in C^{\infty }\left(M\right)$. Such Y is called homothetic if $f=constant$, and Y is said to be KVF if $f=0$.

A manifold $M^{2n+1}$ is a contact manifold if $\eta \wedge {\left(d\eta \right)}^n\ne 0$ for some global 1-form $\eta$. In this case, $\zeta$ satisfying $d\eta (\zeta ,·)=0$ and $\eta \left(\zeta \right)=1$ is called a characteristic vector field. Polarization of $d\eta$ on ${\mathcal{D}}_{\equiv }=ker\eta$ allows to find a $(1,1)$-tensor $\phi$ and a Riemannian structure g satisfying

$$\begin{array}{ccc}\hfill & & {\phi }^2=-i{d}_{TM}+\eta \otimes \zeta ,\eta =g(\zeta ,·),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill & & d\eta (·,·)=g(·,\phi ·).\hfill \end{array}$$

(7)

They call such $M^{2n+1}(\phi ,\zeta ,\eta ,g)$ a contact metric manifold with associated metric g. By (6), we get

$$\phi \left(\zeta \right)=0,\eta \circ \phi =0,\mathrm{and}\mathrm{rank}\left(\phi \right)=2n.$$

Further, a contact metric manifold is called SM if the following condition is valid, see [3,26]:

$$[\phi Y_1,\phi Y_2]=-2d\eta (Y_1,Y_2)\zeta ,Y_1,Y_2\in \mathcal{X}\left(M\right).$$

The curvature tensor of an SM satisfies

$$\begin{array}{c}\hfill R(Y_1,Y_2)\zeta =\eta \left(Y_2\right)Y_1-\eta \left(Y_1\right)Y_2,Y_1,Y_2\in \mathcal{X}\left(M\right).\end{array}$$

(8)

Recall that if $\zeta$ is KVF, then M is called a K-contact manifold. A SM is K-contact, the converse is valid in dimension 3 only, see ([26], p. 87). SM satisfy the following, see ([26], p. 113):

$$\begin{array}{cc}\hfill {\nabla }_{Y_1}\zeta & =-\phi Y_1,Y_1\in \mathcal{X}\left(M\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill Q\zeta & =2n\zeta .\hfill \end{array}$$

(10)

According to ([16], Lemma 2.1):

$${\nabla }_{\zeta }Q=Q\phi -\phi Q;$$

therefore, ${\nabla }_{\zeta }Q=0$, as $\phi$ and the Ricci operator commute on an SM, see [26]. Using (9) we find the derivative of (10) in the direction of $Y_1\in \mathcal{X}\left(M\right)$,

$$\left({\nabla }_{Y_1}Q\right)\zeta =Q\phi Y_1-2n\phi Y_1.$$

(11)

The ∗-Ricci tensor ${\mathrm{Ric}}_g^{*}$ on an SM has the view (see [16], Lemma 5):

$${\mathrm{Ric}}_g^{*}(Y_1,Y_2)={\mathrm{Ric}}_{\mathrm{g}}(Y_1,Y_2)-(2n-1)g(Y_1,Y_2)-\eta \left(Y_1\right)\eta \left(Y_2\right),Y_1,Y_2\in \mathcal{X}\left(M\right).$$

(12)

Thus, we can write the ∗-RS structure (2) on an SM as

$$\begin{array}{c}\hfill \left({\pounds }_{V}g\right)(Y_1,Y_2)+2{\mathrm{Ric}}_{\mathrm{g}}(Y_1,Y_2)=2(\omega +2n-1)g(Y_1,Y_2)+2\eta \left(Y_1\right)\eta \left(Y_2\right)\end{array}$$

(13)

for all $Y_1,Y_2\in \mathcal{X}\left(M\right)$.

Remark 3.

Contracting the derivative of (8) using a local orthonormal basis${\left\{E_i\right\}}_{1\le i\le 2n+1}$on M and using (9), we get the following:

$$\left(\mathrm{div}R\right)(Y_1,Y_2)\zeta -g(R(Y_1,Y_2)\phi E_i,E_i)=-2g(\phi Y_1,Y_2).$$

Contracting the second Bianchi identity, they reduce the above equation to the following:

$$g(\left({\nabla }_{Y_1}Q\right)Y_2-\left({\nabla }_{Y_2}Q\right)Y_1,\zeta )-g(R(Y_1,Y_2)\phi E_i,E_i)=-2g(\phi Y_1,Y_2).$$

(14)

Using (11), from (14) we obtain

$$g(R(Y_1,Y_2)\phi E_i,E_i)=g(Q\phi Y_1,Y_2)+g(\phi Q{Y}_1,Y_2)-2(2n-1)g(\phi Y_1,Y_2).$$

Replacing $Y_2$ by $\phi Y_2$ in the preceding equation and using the definition (1), we represent the ∗-Ricci tensor of an SM in the form (12).

To prove Theorems 1 and 2, we need the following three results.

Proposition 1.

If an SM admits an almost ∗-RS structure, then we get the following:

$$\begin{array}{c}\hfill ({\pounds }_X\nabla )(Y_1,\zeta )+2Q\phi Y_1=2(2n-1)\phi Y_1+Y_1\left(\omega \right)\zeta +\zeta \left(\omega \right)Y_1-\eta \left(Y_1\right)\nabla \omega ,Y_1\in \mathcal{X}\left(M\right).\end{array}$$

(15)

Proof. 

Using the derivative of (13) for an arbitrary $Y_3\in \mathcal{X}\left(M\right)$ and (9), gives

$$\begin{array}{cc}\hfill & \left({\nabla }_{Y_3}{\pounds }_{X}g\right)(Y_1,Y_2)+2\left({\nabla }_{Y_3}{\mathrm{Ric}}_{\mathrm{g}}\right)(Y_1,Y_2)=2Y_3\left(\omega \right)g(Y_1,Y_2)\hfill \\ \hfill & -2\eta \left(Y_1\right)g(Y_2,\phi Y_3)-2\eta \left(Y_2\right)g(Y_1,\phi Y_3)\hfill \end{array}$$

(16)

for $Y_1,Y_2\in \mathcal{X}\left(M\right)$. Following Yano ([30] p. 23), for $Y_1,Y_2,Y_3\in \mathcal{X}\left(M\right)$ we write

$$\begin{array}{c}\hfill ({\pounds }_X{\nabla }_{Y_3}g-{\nabla }_{Y_3}{\pounds }_{X}g-{\nabla }_{[X,Y_3]}g)(Y_1,Y_2)=-g(({\pounds }_X\nabla )(Y_3,Y_1),Y_2)-g(({\pounds }_X\nabla )(Y_3,Y_2),Y_1).\end{array}$$

Inserting (16) in the above relation and using $\nabla g=0$ yields the following:

$$\begin{array}{cc}\hfill & g(({\pounds }_X\nabla )(Y_3,Y_1),Y_2)+g(({\pounds }_X\nabla )(Y_3,Y_2),Y_1)+2\left({\nabla }_{Y_3}{\mathrm{Ric}}_{\mathrm{g}}\right)(Y_1,Y_2)\hfill \\ \hfill & =2\left\{Y_3\left(\omega \right)g(Y_1,Y_2)-\eta \left(Y_1\right)g(Y_2,\phi Y_3)-\eta \left(Y_2\right)g(Y_1,\phi Y_3)\right\}.\hfill \end{array}$$

Due to the symmetry $({\pounds }_X\nabla )(Y_1,Y_2)=({\pounds }_X\nabla )(Y_2,Y_1)$, using cyclical permutations of $Y_1$, $Y_2$, $Y_3$ in the preceding equation, we get

$$\begin{array}{cc}\hfill g(({\pounds }_X\nabla )(Y_1,Y_2),Y_3)& =\left({\nabla }_{Y_3}{\mathrm{Ric}}_{\mathrm{g}}\right)(Y_1,Y_2)-\left({\nabla }_{Y_1}{\mathrm{Ric}}_{\mathrm{g}}\right)(Y_2,Y_3)-\left({\nabla }_{Y_2}{\mathrm{Ric}}_{\mathrm{g}}\right)(Y_3,Y_1)\hfill \\ & +Y_1\left(\omega \right)g(Y_2,Y_3)+Y_2\left(\omega \right)g(Y_3,Y_1)-Y_3\left(\omega \right)g(Y_1,Y_2)\hfill \\ \hfill & -2\eta \left(Y_2\right)g(\phi Y_1,Y_3)-2\eta \left(Y_1\right)g(\phi Y_2,Y_3).\hfill \end{array}$$

(17)

Since the Ricci operator Q is self-adjoint, using ${\nabla }_{\zeta }Q=0$ and (11), and replacing $Y_2$ by $\zeta$ in (17), we get (15). □

Proposition 2.

Let an SM represent an almost ∗-RS and ζ leave ω invariant. Then we get

$$\begin{array}{cc}\hfill R(Y_1,Y_2)\nabla \omega & -g(R(Y_1,Y_2)\nabla \omega ,\zeta )\zeta =4\left\{\left({\nabla }_{Y_1}Q\right)Y_2-\left({\nabla }_{Y_2}Q\right)Y_1\right\}-2\left\{Y_1\left(\omega \right)Y_2-Y_2\left(\omega \right)Y_1\right\}\hfill \\ \hfill & +2\left\{Y_1\left(\omega \right)\eta \left(Y_2\right)\zeta -Y_2\left(\omega \right)\eta \left(Y_1\right)\zeta \right\}+2(\omega -2)\{2g(Y_1,\phi Y_2)\zeta \hfill \\ \hfill & +\eta \left(Y_1\right)\phi Y_2-\eta \left(Y_2\right)\phi Y_1\}+\eta \left(Y_2\right){\nabla }_{Y_1}{\nabla }_{\zeta }\nabla \omega -\eta \left(Y_1\right){\nabla }_{Y_2}{\nabla }_{\zeta }\nabla \omega \hfill \\ \hfill & +2g(\phi Y_2,Y_1){\nabla }_{\zeta }\nabla \omega +g(\phi Y_2,{\nabla }_{Y_1}\nabla \omega )\zeta -g(\phi Y_1,{\nabla }_{Y_2}\nabla \omega )\zeta \hfill \\ \hfill & +g(\zeta ,{\nabla }_{Y_1}\nabla \omega )\phi Y_2-g(\zeta ,{\nabla }_{Y_2}\nabla \omega )\phi Y_1,Y_1,Y_2\in \mathcal{X}\left(M\right).\hfill \end{array}$$

(18)

Proof. 

Applying the Lie derivative for $R(Y_1,\zeta )\zeta =Y_1-\eta \left(Y_1\right)\zeta$ for all $Y_1\in \mathcal{X}\left(M\right)$ (follows from (8)) along X and using (8), yields

$$\begin{array}{c}\hfill \left({\pounds }_{X}R\right)(Y_1,\zeta )\zeta +R(Y_1,\zeta ){\pounds }_X\zeta +\eta \left({\pounds }_X\zeta \right)Y_1+\left({\pounds }_{X}g\right)(Y_1,\zeta )\zeta +g(Y_1,{\pounds }_X\zeta )\zeta =0.\end{array}$$

(19)

Replacing $(Y_1,Y_2)$ by $(Y_1,\zeta )$ in (13) and using (10), we acquire the following:

$$\left({\pounds }_{X}g\right)(Y_1,\zeta )=2\omega \eta \left(Y_1\right).$$

Plugging it into the Lie derivative of $\eta \left(Y_1\right)=g(Y_1,\zeta )$ and $\eta \left(\zeta \right)=1$, we find $\eta \left({\pounds }_X\zeta \right)=-\omega$ and $\left({\pounds }_X\eta \right)\left(\zeta \right)=\omega$. Thus, in view of (8), Equation (19) gives us

$$\begin{array}{c}\hfill \left({\pounds }_{X}R\right)(Y_1,\zeta )\zeta =2\omega \left\{Y_1-\eta \left(Y_1\right)\zeta \right\},Y_1\in \mathcal{X}\left(M\right).\end{array}$$

(20)

By conditions, $\zeta \left(\omega \right)=0$ is valid, thus (15) reduces to the following:

$$\begin{array}{c}\hfill ({\pounds }_X\nabla )(Y_1,\zeta )+2Q\phi Y_1=2(2n-1)\phi Y_1+Y_1\left(\omega \right)\zeta -\eta \left(Y_1\right)\nabla \omega ,Y_1\in \mathcal{X}\left(M\right).\end{array}$$

(21)

Using $\zeta$-derivative of (21), gives

$$\begin{array}{c}\hfill ({\nabla }_{\zeta }{\pounds }_X\nabla )(Y_1,\zeta )=g(Y_1,{\nabla }_{\zeta }\nabla \omega )\zeta -\eta \left(Y_1\right){\nabla }_{\zeta }\nabla \omega ,\end{array}$$

(22)

where equalities ${\nabla }_{\zeta }Q={\nabla }_{\zeta }\zeta ={\nabla }_{\zeta }\phi =0$ for an SM were used. On the other hand, using $Y_1=\zeta$ in (21), then differentiating along $Y_1$ and using (9) and the symmetry of ${\pounds }_X\nabla$, we get

$$\begin{array}{c}\hfill \left({\nabla }_{Y_1}({\pounds }_X\nabla )\right)(\zeta ,\zeta )=2({\pounds }_X\nabla )(\phi Y_1,\zeta )-{\nabla }_{Y_1}\nabla \omega .\end{array}$$

(23)

Next, differentiating $g(\zeta ,\nabla \omega )=0$ along $Y_1\in \mathcal{X}\left(M\right)$ and using (6), gives $\left(\phi Y_1\right)\left(\omega \right)=g(\zeta ,{\nabla }_{Y_1}\nabla \omega )$; therefore, it suffices to combine (6), (10), (21) and (23) to arrive at the result

$$\begin{array}{c}\hfill ({\nabla }_{Y_1}{\pounds }_X\nabla )(\zeta ,\zeta )=4Q{Y}_1-4(2n-1)Y_1-4\eta \left(Y_1\right)\zeta +2g(\zeta ,{\nabla }_{Y_1}\nabla \omega )\zeta -{\nabla }_{Y_1}\nabla \omega .\end{array}$$

(24)

We need the following commutation result, see ([30], p. 23):

$$\begin{array}{c}\hfill \left({\pounds }_{X}R\right)(Y_1,Y_2)Y_3=({\nabla }_{Y_1}{\pounds }_X\nabla )(Y_2,Y_3)-({\nabla }_{Y_2}{\pounds }_X\nabla )(Y_1,Y_3).\end{array}$$

(25)

Next, replacing both $Y_2$ and $Y_3$ by $\zeta$ in (25) and then plugging the values of $\left({\pounds }_{X}R\right)(Y_1,\zeta )\zeta$, $({\nabla }_{\zeta }{\pounds }_X\nabla )(Y_1,\zeta )$ and $({\nabla }_{Y_1}{\pounds }_X\nabla )(\zeta ,\zeta )$ from (20), (22) and (24), respectively, we get

$$\begin{array}{c}\hfill {\nabla }_{Y_1}\nabla \omega =4Q{Y}_1-2\left\{\omega +2(2n-1)\right\}{Y}_1+2(\omega -2)\eta \left(Y_1\right)\zeta +g(\zeta ,{\nabla }_X\nabla \omega )\zeta +\eta \left(Y_1\right){\nabla }_{\zeta }\nabla \omega \end{array}$$

(26)

for $Y_1\in \mathcal{X}\left(M\right)$. Using (9) in the $Y_2$-derivative of (26), we acquire

$$\begin{array}{cc}\hfill {\nabla }_{Y_2}{\nabla }_{Y_1}\nabla \omega =& 4\left\{\left({\nabla }_{Y_2}Q\right)Y_1+Q\left({\nabla }_{Y_2}{Y}_1\right)\right\}-2Y_2\left(\omega \right)\left\{Y_1-\eta \left(Y_1\right)\zeta \right\}-2\left\{\omega +2(2n-1)\right\}{\nabla }_{Y_2}{Y}_1\hfill \\ \hfill & +2(\omega -2)\left\{\eta \left({\nabla }_{Y_2}{Y}_1\right)\zeta -g(Y_1,\phi Y_2)\zeta -\eta \left(Y_1\right)\phi Y_2\right\}+\{\eta \left({\nabla }_{Y_2}{Y}_1\right)\hfill \\ \hfill & -g(Y_1,\phi Y_2)\}{\nabla }_{\zeta }\nabla \omega +\eta \left(Y_1\right){\nabla }_{Y_2}{\nabla }_{\zeta }\nabla \omega -g(\phi Y_2,{\nabla }_{Y_1}\nabla \omega )\zeta \hfill \\ \hfill & +g(\zeta ,{\nabla }_{Y_2}{\nabla }_{Y_1}\nabla \omega )\zeta -g(\zeta ,{\nabla }_{Y_1}\nabla \omega )\phi Y_2.\hfill \end{array}$$

Since ${\mathrm{Hess}}_{\omega }$ is symmetric and $\phi$ is skew-symmetric, using (26) and the above equation in (4) completes the proof of (18). □

Recall that the contact metric structure commutes with the Ricci operator, i.e., $Q\phi =\phi Q$, e.g., [26]. Its covariant derivative and (5) provide the following.

Lemma 1

(see [12]). For an SM M and all $Y_1\in \mathcal{X}\left(M\right)$, we have

$$\begin{array}{c}\hfill \left(i\right){\sum }_{i=1}^{2n+1}g(\left({\nabla }_{\phi Y_1}Q\right)\phi E_i,E_i)=0,\left(ii\right){\sum }_{i=1}^{2n+1}g(\left({\nabla }_{\phi E_i}Q\right)\phi Y_1,E_i)=-\frac{1}{2}{Y}_1\left(r\right),\end{array}$$

where ${\left\{E_i\right\}}_{1\le i\le 2n+1}$ is a local orthonormal basis on M.

3. Proof of Results

Proof of Theorem 1.

Since the characteristic vector field is KVF, ${\pounds }_{\zeta }g=0={\pounds }_{\zeta }{\mathrm{Ric}}_{\mathrm{g}}$ is valid. Applying this to the Lie derivative of (13) along $\zeta$, and using $\pounds {}_{Y_1}\pounds {}_{Y_2}g-\pounds {}_{Y_2}\pounds {}_{Y_1}g=\pounds {}_{[Y_1,Y_2]}g$, e.g., [30], we get

$$\begin{array}{c}\hfill \pounds {}_{[X,\zeta ]}g=-2\zeta \left(\omega \right)g.\end{array}$$

(27)

By (27), $[X,\zeta ]$ is a conformal vector field. This gives us the following alternatives:

(I) $[X,\zeta ]$ is non-homothetic, (II) $[X,\zeta ]$ is homothetic.

Okumura [31] proved that “if a complete SM of dimension $>3$ has a non-Killing conformal vector field, then it is a unit sphere”. Applying this theorem for (I), we conclude that $(M,g)$ is a unit sphere.

We will finish the proof by showing a contradiction for case (II). Sharma [32] proved the following: “a homothetic vector field on an SM (more generally, K-contact manifold) is necessarily a KVF”. So, (27) implies that $\zeta$ leaves $\omega$ invariant. Thus, (18) holds. Contracting it over $Y_1$ and then using (5), (8), we obtain

$$\begin{array}{cc}\hfill {\mathrm{Ric}}_{\mathrm{g}}(Y_2,\nabla \omega )& =4g(\phi Y_2,{\nabla }_{\zeta }\nabla \omega )+\eta \left(Y_2\right)div({\nabla }_{\zeta }\nabla \omega )\hfill \\ \hfill & -g(\zeta ,{\nabla }_{Y_2}{\nabla }_{\zeta }\nabla \omega )-2g(Y_2,\nabla r)+2(2n-1)g(Y_2,\nabla \omega ),\hfill \end{array}$$

(28)

where we used ${\mathrm{trace}}_{\mathrm{g}}\phi =0=\phi \zeta$, the skew-symmetry of $\phi$ and symmetry of ${\mathrm{Hess}}_{\omega }$. Differentiating the equality $g(\zeta ,{\nabla }_{\zeta }\nabla \omega )=0$ along $Y_2\in \mathcal{X}\left(M\right)$ and using (9), gives

$$g(\zeta ,{\nabla }_{Y_2}{\nabla }_{\zeta }\nabla \omega )=g(\phi Y_2,{\nabla }_{\zeta }\nabla \omega ).$$

Thus, (28) can be rewritten as

$$\begin{array}{c}\hfill {\mathrm{Ric}}_{\mathrm{g}}(Y_2,\nabla \omega )=3g(\phi Y_2,{\nabla }_{\zeta }\nabla \omega )+\eta \left(Y_2\right)div({\nabla }_{\zeta }\nabla \omega )-2Y_2\left(r\right)+2(2n-1)Y_2\left(\omega \right)\end{array}$$

(29)

for all $Y_2\in \mathcal{X}\left(M\right)$. Next, recall the following result for SM, see [26]:

$$R(\phi Y_2,\phi Y_1)Y_3=R(Y_2,Y_1)Y_3+g(Y_2,Y_3)Y_1-g(Y_1,Y_3)Y_2-g(Y_2,Y_3)Y_1+g(Y_1,Y_3)Y_2$$

for all $Y_1,Y_2,Y_3\in \mathcal{X}\left(M\right)$. Substituting $Y_1=\phi Y_1$ and $Y_2=\phi Y_2$ in (18) and using the last formula, in view of (6), $\phi \zeta =0$ and the skew-symmetry of $\phi$, we obtain

$$\begin{array}{cc}\hfill R(Y_1,Y_2)\nabla \omega & -g(R(Y_1,Y_2)\nabla \omega ,\zeta )\zeta =4\left\{\left({\nabla }_{\phi Y_1}Q\right)\phi Y_2-\left({\nabla }_{\phi Y_2}Q\right)\phi Y_1\right\}+Y_2\left(\omega \right)\left\{2Y_1-\eta \left(Y_1\right)\zeta \right\}\hfill \\ \hfill & -Y_1\left(\omega \right)\left\{2Y_2-\eta \left(Y_2\right)\zeta \right\}+4(\omega -2)g(Y_1,Y_2)\zeta +2g(Y_1,Y_2){\nabla }_{\zeta }\nabla \omega \hfill \\ \hfill & +2\eta \left(Y_2\right)g(\zeta ,{\nabla }_{Y_1}\nabla \omega )\zeta -2\eta \left(Y_1\right)g(\zeta ,{\nabla }_{Y_2}\nabla \omega )\zeta -g(Y_2,{\nabla }_{Y_1}\nabla \omega )\zeta \hfill \\ \hfill & -g(Y_1,{\nabla }_{Y_2}\nabla \omega )\zeta +g(\zeta ,{\nabla }_{Y_2}\nabla \omega )Y_1-g(\zeta ,{\nabla }_{Y_1}\nabla \omega )Y_2.\hfill \end{array}$$

(30)

On the other hand, contracting (30) over $Y_1$ and applying (8), (5), (29) and Lemma 1, we find

$$\begin{array}{c}\hfill \eta \left(Y_2\right)div({\nabla }_{\zeta }\nabla \omega )+2(n-1)\left\{Y_2\left(\omega \right)-g(\phi Y_2,{\nabla }_{\zeta }\nabla \omega )\right\}=0,\end{array}$$

(31)

where we have used the symmetry of ${\mathrm{Hess}}_{\omega }$ and that $\zeta$ leaves $\omega$ invariant. Next, replacing $Y_2$ by $\phi Y_2$ in (31), noting that (6) and using $g(\zeta ,{\nabla }_{\zeta }\nabla \omega )=0$, we acquire

$$\begin{array}{c}\hfill 2(n-1)\left\{g(\phi Y_2,\nabla \omega )+g(Y_2,{\nabla }_{\zeta }\nabla \omega )\right\}=0,Y_2\in \mathcal{X}\left(M\right).\end{array}$$

(32)

Furthermore, differentiating $g(\zeta ,\nabla \omega )=0$ along $Y_2\in \mathcal{X}\left(M\right)$ and using (9), we achieve

$$g(\zeta ,{\nabla }_{Y_2}\nabla \omega )=g(\phi Y_2,\nabla \omega ).$$

Thus, (32) for $n>1$ gives us $g(\phi Y_2,\nabla \omega )=0$; consequently, $\nabla \omega =0$. Hence, $\omega$ is constant – a contradiction with the conditions of the theorem. □

Proof of Theorem 2.

Applying Proposition 1 to the well-known formula:

$${\nabla }_{Y_1}{\nabla }_{Y_2}X-{\nabla }_{{\nabla }_{Y_1}{Y}_2}X-R(Y_1,X)Y_2=({\pounds }_X\nabla )(Y_1,Y_2),$$

see ([30], p. 23), we acquire

$$\begin{array}{c}\hfill {\nabla }_{Y_1}{\nabla }_{\zeta }X-{\nabla }_{{\nabla }_{Y_1}\zeta }X-R(Y_1,X)\zeta =-2Q\phi Y_1+2(2n-1)\phi Y_1+Y_1\left(\omega \right)\zeta +\zeta \left(\omega \right)Y_1-\eta \left(Y_1\right)\nabla \omega .\end{array}$$

(33)

By conditions, X is a Jacobi vector field on the $\zeta$-integral curves, see [33], i.e.,

$${\nabla }_{\zeta }{\nabla }_{\zeta }X+R(X,\zeta )\zeta =0.$$

Using $Y_1=\zeta$ in (33) and ${\nabla }_{\zeta }\zeta =0$ (that is a consequence of (9)), we achieve the equality $\nabla \omega =2\zeta \left(\omega \right)\zeta$, or, using the exterior derivative,

$$d\omega =2\zeta \left(\omega \right)\eta .$$

Applying exterior derivative, the Poincaré lemma ($d^2=0$), and the wedge product with $\eta$, we acquire $\zeta \left(\omega \right)\eta \wedge d\eta =0$; thus, $\zeta \left(\omega \right)=0$, as $\eta \wedge d\eta$ is nowhere zero on a contact manifold. Thus, $d\omega =0$, i.e., $\omega =constant$. □

Corollary 4 follows from ([16], Theorem 8) and our Theorem 2.

Proof of Theorem 3.

Using $d\circ {\pounds }_X={\pounds }_X\circ d$ (d commutes with the Lie derivative) and applying the operator d to (3), gives

$$\begin{array}{c}\hfill \left({\pounds }_{X}d\eta \right)(Y_1,Y_2)=\frac{1}{2}\left\{Y_1\left(\nu \right)\eta \left(Y_2\right)-Y_2\left(\nu \right)\eta \left(Y_1\right)\right\}+\nu d\eta (Y_1,Y_2)\end{array}$$

(34)

for a function $\nu \in C^{\infty }\left(M\right)$ and any $Y_1,Y_2\in \mathcal{X}\left(M\right)$. Applying the Lie derivative of (7) in the X-direction and using (2), (3) and (34), we obtain

$$\begin{array}{c}\hfill 2\left({\pounds }_X\phi \right)\left(Y_1\right)+2(2\omega -\nu +2(2n-1))\phi Y_1=4Q\phi Y_1+\eta \left(Y_1\right)\nabla \nu -Y_1\left(\nu \right)\zeta .\end{array}$$

(35)

From the first equality of (6), for $Y_1\in \mathcal{X}\left(M\right)$ we get

$$\begin{array}{c}\hfill \left({\pounds }_X\phi \right)\left(\phi Y_1\right)+\phi \left({\pounds }_X\phi \right)\left(Y_1\right)=\left({\pounds }_X\eta \right)\left(Y_1\right)\zeta +\eta \left(Y_1\right){\pounds }_X\zeta .\end{array}$$

(36)

As a result of (10), ∗-RS Equation (2) gives us

$$\left({\pounds }_{X}g\right)(Y_1,\zeta )=2\omega \eta \left(Y_1\right).$$

Taking into account this, as well as (3), it suffices to show that

$$\begin{array}{c}\hfill g({\pounds }_X\zeta ,Y_1)=(\nu -2\omega )\eta \left(Y_1\right).\end{array}$$

(37)

By direct calculation using $\phi \zeta =0$ we get $\left({\pounds }_X\phi \right)\left(\zeta \right)=0$; hence, (35) gives $\nabla \nu =\zeta \left(\nu \right)\zeta$. Thus, by (9) we get

$$\begin{array}{c}\hfill {\mathrm{Hess}}_{\nu }(Y_1,Y_2)=Y_1\left(\zeta \left(\nu \right)\right)\eta \left(Y_2\right)-\zeta \left(\nu \right)g(\phi Y_1,Y_2).\end{array}$$

(38)

Since $\phi$ is skew-symmetric and ${\mathrm{Hess}}_{\nu }$ is symmetric, by (7) and (38), for $Y_1,Y_2$ orthogonal to $\zeta$ we achieve

$$\zeta \left(\nu \right)d\eta (Y_1,Y_2)=0.$$

Thus, $\zeta \left(\nu \right)=0$, as $d\eta$ is nonzero; hence, $\nabla \nu =0$. Thus, $\nu$ is constant. Combining (2), (37) and $\eta \left(\zeta \right)=1$, we get $\nu =\omega$. Substituting this and () in (35), gives

$$\begin{array}{c}\hfill \left({\pounds }_X\phi \right)\left(\phi Y_1\right)=\phi \left({\pounds }_X\phi \right)\left(Y_1\right)=-2Q{Y}_1+(\omega +2(2n-1))Y_1-(\omega -2)\eta \left(Y_1\right)\zeta ,\end{array}$$

(39)

where the equality $Q\phi =\phi Q$ (for an SM) has been used, see ([26], p. 116). Now, by (3), (36), (37), (39) and $\nu =\omega$, we obtain the relation

$$\begin{array}{c}\hfill 2{\mathrm{Ric}}_{\mathrm{g}}=(\omega +2(2n-1\left)\right)g-(\omega -2)\eta \otimes \eta .\end{array}$$

(40)

Hence, $(M,g)$ is an $\eta$-EM of scalar curvature $n(\omega +4n)$. Substituting (40) into (35), we find ${\pounds }_X\phi =0$; thus, X leaves $\phi$ invariant. □

Proof of Theorem 4.

The volume form ${\Omega }$ on a contact metric manifold satisfies ${\Omega }=\eta \wedge {\left(d\eta \right)}^n\ne 0$; therefore, its Lie derivative in the X-direction and (3) give

$${\pounds }_X\mathsf{\Omega }=(n+1)\nu \mathsf{\Omega }.$$

Proceeding, we obtain the equality ${\pounds }_X{\Omega }=(divX){\Omega }$, from which we deduce $divX=(n+1)\nu$. Applying the Divergence theorem (for compact M), this gives $\nu =0$; consequently, $\omega =0$ by the proof of Theorem 3. Thus, (3) and (37), respectively, follows from the condition that X leaves $\eta$ and $\zeta$ invariant. Moreover, Equation (40) becomes

$$\begin{array}{c}\hfill {\mathrm{Ric}}_{\mathrm{g}}=(2n-1)g+\eta \otimes \eta .\end{array}$$

(41)

Thus, from (12) we conclude that $(M,g)$ is ∗-Ricci flat of zero ∗-scalar curvature $r^{*}$. Using (2), we find that X is KVF. Applying Theorem 3, completes the proof. □

Proof of Theorem 5.

By conditions, $X=\sigma \zeta$, where $\sigma$ is a non-zero smooth function. By the skew-symmetry of $\phi$ and (9), we obtain

$$\begin{array}{c}\hfill \left({\pounds }_{X}g\right)(Y_1,Y_2)=g({\nabla }_{Y_1}X,Y_2)+g({\nabla }_X{Y}_2,Y_1)=Y_1\left(\sigma \right)\eta \left(Y_2\right)+Y_2\left(\sigma \right)\eta \left(Y_1\right).\end{array}$$

(42)

Thus, (13) becomes

$$\begin{array}{c}\hfill Y_1\left(\sigma \right)\eta \left(Y_2\right)+Y_2\left(\sigma \right)\eta \left(Y_1\right)+2{\mathrm{Ric}}_{\mathrm{g}}(Y_1,Y_2)=2(\omega +2n-1)g(Y_1,Y_2)+2\eta \left(Y_1\right)\eta \left(Y_2\right).\end{array}$$

(43)

Using $Y_2=\zeta$ in (43) and (10), yields

$$\begin{array}{c}\hfill Y_1\left(\sigma \right)=(2\omega -\zeta \left(\sigma \right))\eta \left(Y_1\right),Y_1\in \mathcal{X}\left(M\right).\end{array}$$

(44)

Again, using $Y_1=\zeta$ and $Y_2=\zeta$ in (43) and applying (10), we get $\zeta \left(\sigma \right)=\omega$; hence, from (44) it follows that $Y_1\left(\sigma \right)=\zeta \left(\sigma \right)\eta \left(Y_1\right)$, for $Y_1\in \mathcal{X}\left(M\right)$. By the above argument, we get that $\sigma$ is constant. By (42), X is KV, and using (44) we get $\omega =0$. The above reduces (43) to (41), and therefore, g is ∗-Ricci flat (follows from (12)) and of constant scalar curvature $4n^2$. □

4. Conclusions

A modern geometrical concept of an almost ∗-RS can be important in differential geometry and theoretical physics. We study the interaction of this structure on a smooth manifold with the well-known Sasakian structure and prove some results of geometric classification. Theorem 1 contains a condition for a complete SM equipped with an almost ∗-RS to be a unit sphere. Theorems 2–5 having local character, contain conditions, ensuring that an SM equipped with an almost ∗-RS structure is a ∗-RS, e.g., a ∗-EM. Using the fact that an almost ∗-Ricci tensor on an SM can be written as (12), an almost ∗-RS reduces to the form (13), which is less general than the almost $\eta$-RS equation for an SM:

$$\begin{array}{c}\hfill \frac{1}{2}{\pounds }_{X}g+{\mathrm{Ric}}_{\mathrm{g}}+\omega g+\delta \eta \otimes \eta =0.\end{array}$$

(45)

In connection with the above, the following question arises: “are our Theorems 1–5 true under the condition (45) instead of (13), where $\omega$ and $\delta$ are arbitrary smooth functions on M?”