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Article

# Ricci Vector Fields Revisited

by
Hanan Alohali
1,†,
Sharief Deshmukh
1,† and
Gabriel-Eduard Vîlcu
2,*,†
1
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2
Department of Mathematics and Informatics, National University of Science and Technology Politehnica Bucharest, 313 Splaiul Independenţei, 060042 Bucharest, Romania
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2024, 12(1), 144; https://doi.org/10.3390/math12010144
Submission received: 25 November 2023 / Accepted: 29 December 2023 / Published: 1 January 2024 / Corrected: 7 February 2024

## Abstract

:
We continue studying the $σ$-Ricci vector field $u$ on a Riemannian manifold $( N m , g )$, which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold $( N m , g )$, $m > 1$, of positive scalar curvature $τ$, admits a closed $σ$-Ricci vector field $u$ such that the vector $u − ∇ σ$ is an eigenvector of T with eigenvalue $τ m − 1$, if and only if it is isometric to the m-sphere $S α m$. In the second result, we show that if a compact and connected T-manifold $( N m , g )$, $m > 2$, admits a $σ$-Ricci vector field $u$ with $σ ≠ 0$ and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature $R i c u , u$ that has a suitable lower bound, then $( N m , g )$ is isometric to the m-sphere $S α m$, and the converse also holds. Finally, we show that a compact and connected Riemannian manifold $( N m , g )$ admits a $σ$-Ricci vector field $u$ with $σ$ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature $R i c u , u$ has a lower bound depending on a positive constant $α$, if and only if $( N m , g )$ is isometric to the m-sphere $S α m$.
MSC:
53C20; 53C21; 53B50

## 1. Introduction

In a recent paper, (cf. [1]), a $σ$-Ricci vector field (abbreviated as $σ$-RVF) $u$ on a m-Riemannian manifold $N m , g$ is introduced, being defined by
$1 2 £ u g = σ R i c ,$
where $£ u g$ is the Lie derivative of the metric g with respect to $u$, $σ$ is a smooth function and $R i c$ is the Ricci tensor of $N m , g$. A $σ$-RVF is a generalization of conformal vector fields (known for their utility in studying geometry and relativity), on Einstein manifolds (see [1,2,3,4,5,6,7,8,9,10,11]). Moreover, it represents a Killing vector field, which is known to have a great influence on the geometry as well as topology on which it lives (see [12,13,14,15]). Apart from these generalizations, a $σ$-RVF is a particular form of potential field of generalized solitons considered in [16,17,18]. Note that a 1-RVF $u$ on a m-Riemannian manifold $N m , g$ is a stable Ricci soliton $N m , g , u , 0$ (see [19]). Indeed, in [1], it has been observed that a $σ$-RVF on $N m , g$ is a stable solution of the generalized Ricci flow (or a $σ$-Ricci flow),
$∂ t g = 2 σ R i c , g 0 = g ,$
of the form $g t = ρ ( t ) φ t * g$, where $φ t : N m → N m$ is a 1-parameter family of diffeomorphisms generated by the vector fields $U t$ and $ρ ( t )$ is a scale factor, under the initial conditions $ρ ( 0 ) = 1$, $ρ . 0 = 0$, $U 0 = u$ and $φ 0 = i d$.
In [1], a closed $σ$-RVF $u$, with $σ ≠ 0$, on a compact and connected m-Riemannian manifold $N m , g$, $m > 2$, of nonzero scalar curvature is used with an appropriate lower bound on the integral of the Ricci curvature $R i c u , u$ to find a characterization of the m-sphere $S m ( c )$. Moreover, in [1], a closed $σ$-RVF $u$ on a complete and simply connected m-Riemannian manifold $N m , g$, $m > 2$, of positive scalar curvature, is used, where the function $σ$ is a nontrivial solution of the Fischer–Marsden equation (cf. [20]) with an appropriate upper bound on the length $∇ u$ of the covariant derivative of $u$, to find another characterization of the sphere $S m ( c )$.
The Ricci operator T of a Riemannian manifold $N m , g$ is a symmetric operator defined by
$R i c E , F = g T E , F , E , F ∈ Γ ( N m ) ,$
where $Γ ( N m )$ is a space of vector fields on $N m$. A Riemannian manifold $N m , g$ is said to be a T-manifold, if the Ricci operator T is a Codazzi tensor, i.e., it satisfies
$D E T ( F ) = D F T ( E ) , E , F ∈ Γ ( N m ) ,$
where D is the Riemannian connection on $N m , g$. It is worth noting that a T-manifold $N m , g$ has a constant scalar curvature.
In this article, we are interested in studying the geometry of $N m , g$ equipped with a $σ$-RVF $u$. In the first result, we consider a T-manifold $N m , g$ that possesses a closed $σ$-RVF $u$ and we observe that, in this case, the vector field $u − ∇ σ$ has a special role to play in shaping the geometry of the T-manifold $N m , g$. It is shown that if the scalar curvature $τ$ of a compact T-manifold $N m , g$ is positive (note that $τ$ is a constant for a T-manifold) and the vector field $u − ∇ σ$ satisfies
$T u − ∇ σ = τ m u − ∇ σ ,$
then $N m , g$ is isometric to the m-sphere $S c m$ of constant curvature c, where $τ = m ( m − 1 ) c$, and the converse also holds (cf. Theorem 1).
Then, we concentrate on a $σ$-RVF $u$ on $N m , g$ that is not necessarily closed. In this case, the 1-form $β$ dual to $u$ gives rise to a skew symmetric operator $Ψ : Γ ( N m ) → Γ ( N m )$ defined by
$g Ψ E , F = 1 2 d β E , F , E , F ∈ Γ ( N m ) ,$
and we call the operator $Ψ$ the associated operator of the $σ$-RVF $u$. In the second result of this paper, we consider a compact and connected T-manifold $N m , g$ with scalar curvature $τ = m ( m − 1 ) c$ that possesses a $σ$-RVF $u$, $σ ≠ 0$, with associated operator $Ψ$ satisfying
$Δ u = − c u , ∫ N m R i c u , u ≥ ∫ N m m − 1 m τ 2 σ 2 + Ψ 2 ,$
which necessarily implies that $N m , g$ is isometric to the m-sphere $S c m$ of constant curvature c, and the converse is also true (cf. Theorem 2), where $Δ$ is the rough Laplace operator acting on vector fields on $N m , g$.
Recall the differential equation on a Riemannian manifold $N m , g$ considered by Obata (cf. [18,21]), namely
$H e s s ( σ ) = − c σ g ,$
where $σ$ is a non-constant smooth function, c is a positive constant and $H e s s ( σ )$ is the Hessian of $σ$ defined by
$H e s s ( σ ) ( E , F ) = g D E ∇ σ , F , E , F ∈ Γ ( N m ) .$
It is known that a complete, simply connected $N m , g$ admits a nontrivial solution of (4) if and only if $N m , g$ is isometric to the sphere $S c m$ (cf. [18,21]).
There is yet another important differential equation on a Riemannian manifold $N m , g$ (cf. [7] and references therein), given by
$σ R i c − H e s s ( σ ) = 1 m τ σ − Δ σ g ,$
known as the static fluid equation, where $Δ σ$ is the Laplacian of $σ$ with respect to the metric g. A Riemannian manifold $N m , g$ that admits a nontrivial solution of the static fluid equation is called a static space. Note that under the additional assumption
$Δ σ = − τ m − 1 σ ,$
the static fluid equation reduces to the Fischer–Marsden equation (cf. [20])
$Δ σ g + σ R i c = H e s s σ .$
In the last result of this paper, we show that a compact and connected Riemannian manifold $N m , g$ with scalar curvature $τ$ possessing a $σ$-RVF $u$ with associated operator $Ψ$ and the function $σ$ is a nontrivial solution of the static perfect fluid Equation (5); furthermore, for a positive constant c, the following inequality holds:
$∫ N m R i c u , u ≥ ∫ N m m − 1 m τ 2 σ 2 + 1 m Δ σ + n c σ 2 + Ψ 2 ,$
which necessarily implies that $N m , g$ is isometric to the sphere $S c m$, and the converse is also true (cf. Theorem 3).

## 2. Preliminaries

For a $σ$-RVF $u$ on an m-dimensional Riemannian manifold $N m , g$, we let $β$ be the 1-form dual to $u$, i.e.,
$β E = g u , E , E ∈ Γ N m .$
Then, we have the associated operator $Ψ$ satisfying
$d β E , F = 1 2 g Ψ E , F , E , F ∈ Γ N m ,$
which shows that $Ψ$ is a skew symmetric operator. Using Equations (1) and (8), we obtain the following expression for the covariant derivative $∇ E u$
$D E u = σ T E + Ψ E , E ∈ Γ N m .$
where T is the Ricci operator defined by
$R i c E , F = g T E , F , E , F ∈ Γ N m .$
On employing the following expression for the curvature tensor field R of $N m , g$,
$R E , F G = D E , D F G − D [ E , F ] G , E , F , G ∈ Γ N m ,$
with Equation (9), we obtain
$R E , F u = E σ T F − F σ T E + ρ D E T ( F ) − D F T ( E ) + D E Ψ ( F ) − D F Ψ ( E ) ,$
for any $E , F ∈ Γ N m$, where
$D E T ( F ) = D E T F − T D E F .$
The scalar curvature $τ$ of $N m , g$ is given by
$τ = ∑ α = 1 m g T E α , E α ,$
where $E 1 , … , E m$ is a local frame on $N m$. The Ricci tensor is given by
$R i c E , F = ∑ α = 1 m g R E α , E F , E α ,$
and employing it in Equation (10), we conclude
$R i c F , u = R i c F , ∇ σ − τ F σ + σ g F , ∑ α = 1 m ∇ F α T ( F α ) − ρ Y τ − g F , ∑ α = 1 m ∇ F α Ψ ( F α ) ,$
where $∇ σ$ is the gradient of $σ$ and we have used the symmetry of the Ricci operator T and the skew symmetry of the associated operator $Ψ$. It is known that the gradient of scalar curvature $τ$ satisfies (cf. [22])
$1 2 ∇ τ = ∑ α = 1 m D F α T ( F α ) .$
Thus, on using Equation (12) in (11), we arrive at
$R i c F , u = R i c F , ∇ σ − τ F σ − 1 2 σ F τ − g F , ∑ α = 1 m ∇ F α Ψ ( F α )$
and, therefore,
$T u = T ∇ σ − τ ∇ σ − 1 2 ρ ∇ τ − ∑ α = 1 m ∇ F α Ψ ( F α ) .$
Lemma 1.
For a σ-RVF $u$ on a T-manifold $N m , g$, the associated operator Ψ satisfies
$D E Ψ ( F ) = R E , u F − R i c E , F ∇ σ + F σ T E , E , F ∈ Γ ( N m ) .$
Proof.
Suppose that $u$ is a $σ$-RVF on a T-manifold $N m , g$. Then, Equation (10) changes to
$D E Ψ ( F ) − D F Ψ ( E ) = R E , F u − E σ T F + F σ T E .$
Now, using the fact that the 2-form $d β$ in Equation (8) is closed and the associated operator $Ψ$ is skew symmetric, we have
$g D E Ψ ( F ) − D F Ψ ( E ) , G + g D G Ψ ( E ) , F = 0$
and employing Equation (15) in the above equation yields
$g R E , F u − E σ T F + F σ T E , G + g D G Ψ ( E ) , F = 0 .$
Thus, we have
$g D G Ψ ( E ) , F = g R G , u E , F + E σ g T G , F − R i c E , G g ∇ σ , F$
and this proves the lemma. □
On a Riemannian manifold $N m , g$ possessing a $σ$-RVF $u$, we have the second-order differential operator $∇ 2 u$ defined by
$∇ 2 u E , F = D E D F u − D D E F u , E , F ∈ Γ ( N m )$
and its trace
$Δ u = ∑ α = 1 m ∇ 2 u E α , E α$
is the rough Laplacian of the $σ$-RVF $u$.
Lemma 2.
On a connected T-manifold $N m , g$, the scalar curvature τ is a constant, and for a σ-RVF $u$ on a connected T-manifold $N m , g$ with associated operator Ψ, the rough Laplacian satisfies
$Δ u = T ∇ σ + ∑ α = 1 m D E α Ψ E α ,$
where $E 1 , … , E m$ is a local frame on $N m$.
Proof.
First, note that for a T-manifold $N m , g$, using Equation (3), we have
$E ( τ ) = E ∑ α = 1 m g T E α , E α = ∑ α = 1 m g D E T E α + T D E E α , E α + ∑ α = 1 m g T E α , D E E α = ∑ α = 1 m g D E α T E , E α + 2 ∑ α = 1 m g T E α , D E E α = ∑ α = 1 m g E , D E α T E α + 2 ∑ α = 1 m g T E α , D E E α .$
Note that
$D E E α = ∑ k ∧ α k E E k , T E α = ∑ j μ α j E j ,$
where the connection forms $∧ α k$ are skew symmetric and coefficients $μ α j$ are symmetric and, as such, we have
$∑ α = 1 m g T E α , D E E α = 0 .$
Consequently, Equation (17) yields
$∇ τ = ∑ α = 1 m D E α T E α .$
Combining it with Equation (11), we obtain $∇ τ = 0$, i.e., the scalar curvature $τ$ of a T-manifold is a constant.
Employing Equation (9), we have
$∇ 2 u E , F = E ( σ ) T ( F ) + σ D E T ( F ) + D E Ψ ( F )$
and taking the trace in the above equation, while using Equation (11) with $∇ τ = 0$, we obtain
$Δ u = T ∇ σ + ∑ α = 1 m D E α Ψ E α .$
Next, the sphere $S α m$ of constant curvature $α$ possesses a $σ$-RVF induced by a coordinate unit vector field $∂ ∂ u$ on the Euclidean space $R m + 1$. Indeed, on treating $S α m$ as an embedded surface in $R m + 1$ with unit normal $ζ$ and Weingarten operator—$α I$, we express $∂ ∂ u$ as
$∂ ∂ u = u + f ζ , f = ∂ ∂ u , ζ ,$
where $,$ is a Euclidean inner product and $u ∈ Γ S α m$. On taking g as the induced metric on $S α m$ and D as the Riemannian connection with respect to g and differentiating the above equation with respect to the vector field $E ∈ Γ S α m$, we have
$D E u = − α f E , ∇ f = α u .$
Using the first equation in (19), it follows that
$£ u g = − 2 α f g$
and for the Ricci tensor of $S α m$, we have
$R i c = ( m − 1 ) α g , τ = m ( m − 1 ) α .$
Hence, the vector field $u$ on $S α m$ obeys
$1 2 £ u g = σ R i c , σ = − 1 ( m − 1 ) α f ,$
i.e., $u$ is a $σ$-RVF on $S α m$.
Moreover, note that Equation (21) in view of Equation (19) confirms
$H e s s ( σ ) E , F = g D E ∇ σ , F = − 1 ( m − 1 ) α g D E ∇ f , F = − 1 m − 1 g D E u , F = α f m = 1 g E , F ,$
i.e.,
$H e s s ( σ ) = − α σ g , Δ σ = − m α σ .$
Combining Equations (20) and (22), we see that the function $σ$ of the $σ$-RVF $u$ on $S α m$ satisfies the static fluid equation
$σ R i c − H e s s ( σ ) = 1 m τ σ − Δ σ g .$
We investigate now whether $σ$ is a nontrivial solution. If $σ$ was a constant, by virtue of Equation (21), it would mean that f was a constant, and, in turn, by (19), it would mean that $u = 0$ and, by the same equation, would imply $f = 0$. Inserting this information in (18), we have $∂ ∂ u = 0$, a contradiction. Hence, $σ$ is a nontrivial solution of the static fluid equation on $S α m$.

## 3. $σ$-Ricci Vector Fields on $T$-Manifolds

In this section, we consider an m-dimensional T-manifold $( N m , g )$ that possesses a closed $σ$-RVF $u$. It is interesting to observe that, in this situation, the vector field $u − ∇ σ$ plays an interesting role while treating the Ricci operator T of $( N m , g )$. Note that, by Lemma 2, the scalar curvature $τ$ of a T-manifold $( N m , g )$ is a constant and we put $τ = m ( m − 1 ) α$, for a constant $α$. Here, we prove the following result.
Theorem 1.
An m-dimensional, $m > 1$, complete, and simply connected T-manifold $( N m , g )$ with positive scalar curvature τ admits a nonzero closed σ-RVF $u$, $σ ≠ 0$ satisfying
$T u − ∇ σ = τ m u − ∇ σ ,$
if and only if $( N m , g )$ is isometric to $S α m$, where $τ = m ( m − 1 ) α$.
Proof.
Suppose that the complete and simply connected T-manifold $( N m , g )$, $m > 1$, of scalar curvature $τ > 0$, admits a nonzero closed $σ$-RVF $u$, $σ ≠ 0$, which satisfies
$T u − ∇ σ = τ m u − ∇ σ .$
As the $σ$-RVF $u$ is closed, its associated operator $Ψ = 0$, and by Lemma 2, the scalar curvature $τ$ is a constant, and Equation (14) becomes
$T u = T ∇ σ − τ ∇ σ .$
Treating it with Equation (24) yields
$τ m u − ∇ σ = − τ ∇ σ$
and, as $τ > 0$, it transforms into
$u = − ( m − 1 ) ∇ σ .$
Note that, by Equation (9), we have $div u = σ τ$, and taking the divergence in Equation (26) gives
$σ τ = − ( m − 1 ) Δ σ .$
Now, inserting the value of $∇ σ$ from Equation (26) into Equation (25), we arrive at
$T u = − m − 1 m τ ∇ σ .$
Note that as $u$ is closed, Equation (9) has the form
$D E u = σ T E , E ∈ Γ ( N m ) .$
Next, we intend to compute the divergence $div T u$ and we proceed by choosing a local frame $E 1 , … , E m$ and using Equation (29)
$div T u = ∑ α = 1 m g ∇ E α T u , E α = ∑ α = 1 m g ∇ E α T ( u ) + T ∇ E α u , E α = ∑ α = 1 m g u , ∇ E α T E α + ∑ α = 1 m g ∇ E α u , T E α .$
Note that on T-manifold $( N m , g )$, by Lemma 2, $τ$ is a constant and, thus, employing Equations (12) and (29), we arrive at
$div T u = σ T 2 .$
Now, utilizing this equation in Equation (28) yields
$σ T 2 = − m − 1 m τ Δ σ .$
Inserting Equation (27) in the above equation gives
$σ T 2 = 1 m σ τ 2 ,$
i.e.,
$σ T 2 − 1 m τ 2 = 0 .$
As $N k$ is connected (being simply connected) and $σ ≠ 0$, in this situation, the above equation yields
$T 2 = 1 m τ 2 .$
However, Equation (31) is the equality in Schwartz’s inequality
$T 2 ≥ 1 m τ 2 .$
Hence, equality (31) holds if and only if
$T = τ m I$
and Equation (29) changes to
$D E u = τ m ρ E , E ∈ Γ ( N m ) .$
Thus, on employing Equation (26) in the above equation, we confirm
$D E ∇ σ = − τ m ( m − 1 ) σ E , E ∈ Γ N m .$
Note that as $u ≠ 0$ by Equation (26), the function $σ$ is a non-constant function and, also, $τ$ being a positive constant, letting $τ = m ( m − 1 ) α$, we obtain a positive constant $α$ and Equation (32) is Obata’s equation
$H e s s ( σ ) = − α ρ g ,$
proving that $( N m , g )$ is isometric to the sphere $S α m$ (cf. [18,21]).
Conversely, suppose that $( N m , g )$ is isometric to the sphere $S α m$. Then, by Equations (19)–(21), there is a nonzero $σ$-RVF $u$ on $S α m$ and, as seen earlier, the function $σ ≠ 0$ and is a non-constant function. Moreover, the Ricci operator of $S α m$ being
$T = τ m I ,$
the condition
$T u − ∇ σ = τ m u − ∇ σ$
holds, and this finishes the proof. □
In an earlier result, we considered a closed $σ$-RVF $u$ on an m-dimensional T-manifold $( N m , g )$ to find a characterization of the sphere $S α m$. Next, we consider a $σ$-RVF $u$ on an m-dimensional T-manifold $( N m , g )$ not necessarily closed and prove the following.
Theorem 2.
An m-dimensional compact and connected T-manifold $N m , g$, $m > 2$ of positive scalar curvature τ admits a σ-RVF $u$ with associated operator Ψ, $σ ≠ 0$, $Δ u = − τ m ( m − 1 ) u$ and the Ricci curvature $R i c u , u$ satisfies
$∫ N m R i c u , u ≥ ∫ N m m − 1 m τ 2 σ 2 + Ψ 2$
if and only if $N m , g$ is isometric to $S α m$, where $τ = m ( m − 1 ) α$.
Proof.
Let an m-dimensional T-manifold $( N m , g )$, $m > 2$, with scalar curvature $τ > 0$ be equipped with a $σ$-RVF $u$ with $σ ≠ 0$ and associated operator $Ψ$ such that
$Δ u = − τ m ( m − 1 ) u$
and
$∫ N m R i c u , u ≥ ∫ N m m − 1 m τ 2 σ 2 + Ψ 2 .$
Using Lemma 1, we have
$R E , u F = D E Ψ ( F ) + R i c E , F ∇ σ − F σ T E , E , F ∈ Γ ( N m )$
Employing a local frame $E 1 , … , E m$ in the above equation, we conclude
$R i c u , F = R i c ∇ σ , F − τ F σ − ∑ α = 1 m g F , D E α Ψ ( E α ) , F ∈ Γ ( N m )$
and the above equation implies
$R i c u , u = R i c ∇ σ , u − τ u σ − ∑ α = 1 m g u , D E α Ψ ( E α ) .$
Note that, by Equation (9), we have
$div u = τ σ$
and using
$div σ u = u σ + τ σ 2 ,$
in the above equation containing the expression of $R i c u , u$, we derive
$R i c u , u = R i c ∇ σ , u + τ 2 σ 2 − τ div σ u − ∑ α = 1 m g u , D E α Ψ ( E α ) .$
Next, using a local frame $E 1 , … , E m$ on $( N m , g )$, to compute the $div Ψ u$, we have, on using the skew symmetry of the associated operator $Ψ$ and Equation (9),
$div Ψ u = ∑ α = 1 m g D E α Ψ u , E α = ∑ α = 1 m g D E α Ψ ( u ) + Ψ σ T E α + Ψ E α , E α = − ∑ α = 1 m g u , D E α Ψ E α − σ ∑ α = 1 m g T E α , Ψ E α − Ψ 2$
Since T is symmetric and the associated operator $Ψ$ is skew symmetric, it follows that
$∑ α = 1 m g T E α , Ψ E α = 0$
and Equation (36) now becomes
$div Ψ u = − ∑ α = 1 m g u , D E α Ψ E α − Ψ 2$
and, inserting this equation into Equation (35), we arrive at
$R i c u , u = R i c ∇ σ , u + τ 2 σ 2 − τ div σ u + Ψ 2 + div Ψ u .$
Note that on a T-manifold $( N m , g )$, $τ$ is a constant and keeping this in mind and integrating the above equation brings us to
$∫ N m R i c u , u − R i c ∇ σ , u − τ 2 σ 2 − Ψ 2 = 0 .$
Observe that, by virtue of the symmetry of the operator T and Equations (9), (12) and (37), and the fact that $τ$ is a constant, we have
$div T u = ∑ α = 1 m g D E α T u , E α = ∑ α = 1 m g D E α T ( u ) + T σ T E α + Ψ E α , E α = σ T 2 .$
Now, using the fact that
$div σ T u = R i c ∇ σ , u + σ div T u$
in Equation (39), we arrive at
$R i c ∇ σ , u = div σ T u − σ 2 T 2 .$
Inserting the above equation in Equation (38), we confirm
$∫ N m R i c u , u + σ 2 T 2 − τ 2 σ 2 − Ψ 2 = 0$
and the above integral could be rearranged as
$∫ N m σ 2 T 2 − 1 m τ 2 = ∫ N m m − 1 m τ 2 σ 2 + Ψ 2 − ∫ N m R i c u , u .$
Treating the above equation with the inequality (34), we arrive at
$∫ N m σ 2 T 2 − 1 m τ 2 ≤ 0 .$
The integrand in the above inequality by virtue of Schwartz’s inequality is non-negative, and, therefore, we conclude
$σ 2 T 2 − 1 m τ 2 = 0 .$
As $σ ≠ 0$ and $N m$ is connected, we conclude that
$T 2 = 1 m τ 2 ,$
which, being the equality in Schwartz’s inequality, it holds if and only if
$T = τ m I .$
Consequently, as $τ$ is a constant, Equations (14) and (41) combine to arrive at
$τ m u = τ m ∇ σ − τ ∇ σ − ∑ α = 1 m ∇ F α Ψ ( F α ) ,$
for a local frame $E 1 , … , E m$ on $( N m , g )$, i.e., we have
$τ m u = − m − 1 m τ ∇ σ − ∑ α = 1 m ∇ F α Ψ ( F α ) .$
Moreover, using Equations (33) and (41) with Lemma 2, we obtain the following:
$− τ m ( m − 1 ) u = τ m ∇ σ + ∑ α = 1 m ∇ F α Ψ ( F α ) .$
Adding Equations (42) and (43), we find
$m − 2 m m − 1 τ u = − m − 2 m τ ∇ σ$
and, as $m > 2$, $τ > 0$, it confirms
$u = − ( m − 1 ) ∇ σ .$
Differentiating the above equation and using Equations (9) and (41), we have
$D E ∇ σ = − 1 m − 1 τ m σ E + Ψ E , E ∈ Γ ( N m ) ,$
which, on taking the inner product with E and noticing that $Ψ$ is a skew symmetric operator, leads to
$H e s s ( σ ) E , E = − α σ g E , E , E ∈ Γ ( N m ) ,$
where $τ = m ( m − 1 ) α$, i.e., $α$ is a positive constant. Now, polarizing the above equation confirms
$H e s s ( σ ) = − α σ g .$
Hence, $N m , g$ is isometric to $S α m$ (cf. [18,21]).
Conversely, suppose that $N m , g$ is isometric to $S α m$. Then, by Equation (21), there is a nonzero $σ$-RVF $u$ on $S α m$ with $σ ≠ 0$ and, as $u$ is is closed, the associated operator $Ψ = 0$. Moreover, it is obvious that $S α m$ is a T-manifold. Thus, using Equation (19), we have
$∇ 2 u E , F = D E D F u − D D E F u = − α E f F$
and, therefore, by treating the above equation with (16), we have
$Δ u = ∑ α = 1 m ∇ 2 u E α , E α = − α ∇ f ,$
which, by virtue of Equation (19), implies
$Δ u = − α u = − τ m ( m − 1 ) u ,$
where $τ = m ( m − 1 ) α$. Finally, using Equations (19) and (21), we have
$∇ σ = − 1 ( m − 1 ) α ∇ f = − 1 m − 1 u ,$
i.e.,
$u 2 = ( m − 1 ) 2 ∇ σ 2 .$
Now, Equation (22) implies
$σ Δ σ = − m α σ 2 ,$
which, on integrating by parts, confirms
$∫ S α m ∇ σ 2 = m α ∫ S α m σ 2 = 1 m ( m − 1 ) 2 α ∫ S α m τ 2 σ 2 .$
The Ricci curvature of $S α m$ is
$R i c u , u = ( m − 1 ) α u 2$
and, thus, using $Ψ = 0$ and Equations (44) and (45), we conclude
$∫ S α m R i c u , u = ∫ S α m ( m − 1 ) α u 2 = ∫ S α m ( m − 1 ) 3 α ∇ σ 2 = ∫ S α m m − 1 m τ 2 σ 2 + Ψ 2$
and this completes the proof. □

## 4. $σ$-Ricci Vector Fields on Static Spaces

Now, we are interested in a $σ$-RVF $u$, not necessarily closed, on a Riemannian manifold $N m , g$ with function $σ$ as a nontrivial solution of the static fluid Equation (5). Indeed, we prove the following.
Theorem 3.
If an m-dimensional compact and connected Riemannian manifold $N m , g$ admits a σ-RVF $u$ with associated operator Ψ, such that σ is a nontrivial solution of the static perfect fluid equation, for a positive constant α and the Ricci curvature $R i c u , u$, it satisfies
$∫ N m R i c u , u ≥ ∫ N m m − 1 m τ 2 σ 2 + 1 m Δ σ + m α σ 2 + Ψ 2 ,$
and $N m , g$ is isometric to $S α m$, and the converse also holds.
Proof.
Assume that $N m , g$ admits a $σ$-RVF $u$ with associated operator $Ψ$, such that $σ$ is a nontrivial solution of the static perfect fluid Equation (5) and the Ricci curvature $R i c u , u$ satisfies
$∫ N m R i c u , u ≥ ∫ N m m − 1 m τ 2 σ 2 + 1 m Δ σ + m α σ 2 + Ψ 2 .$
Then, the Hessian operator $H σ$ of the function $σ$ defined by
$g H σ E , F = H e s s ( σ ) E , F ,$
by virtue of Equation (5) satisfies
$H σ E = σ T E + 1 m Δ σ − τ σ E , E ∈ Γ N m$
Utilizing Equation (9) in the above equation, we arrive at
$H σ E = D E u − Ψ E + 1 m Δ σ − τ σ E$
and, for a positive constant $α$, the above equation could be rearranged as
$H σ + α σ I ( E ) = D E u − Ψ E + 1 m Δ σ + m α σ − τ σ E , E ∈ Γ N m .$
Choosing a local frame $E 1 , … , E m$, and using the above equation, we compute
$H σ + α σ I 2 = ∑ j = 1 m g H σ + α σ I ( E j ) , H σ + α σ I ( E j ) = ∑ j = 1 m g D E j u − Ψ E j + 1 m Δ σ + m α σ − τ σ E j , D E j u − Ψ E j + 1 m Δ σ + m α σ − τ σ E j = D u 2 + Ψ 2 + 1 m Δ σ + m α σ − τ σ 2 − 2 ∑ j = 1 m g D E j u , Ψ E j + 2 m Δ σ + m α σ − τ σ div u .$
Now, using Equation (9) and $div u = τ σ$ in the above equation, we arrive at
$H σ + α σ I 2 = D u 2 − Ψ 2 + 1 m Δ σ + m α σ − τ σ 2 + 2 τ σ m Δ σ + m α σ − τ σ ,$
i.e.,
$H σ + α σ I 2 = D u 2 − Ψ 2 + 1 m Δ σ + m α σ − τ σ Δ σ + m α σ + τ σ$
or
$H σ + α σ I 2 = D u 2 − Ψ 2 + 1 m Δ σ + m α σ 2 − 1 m τ σ 2 .$
We recall the integral formula (cf. [23])
$∫ N m D u 2 = ∫ N m R i c u , u + 1 2 £ u g 2 − div u 2 .$
Using $div u = τ σ$ and the outcome of Equation (1) in the form
$1 2 £ u g 2 = 2 σ 2 T 2$
in the above integral equation, we have
$∫ N m D u 2 = ∫ N m R i c u , u + 2 σ 2 T 2 − τ σ 2 .$
Now, integrating Equation (48) and using the above equation, we arrive at
$∫ N m H σ + α σ I 2 = ∫ N m R i c u , u + 2 σ 2 T 2 − m + 1 m τ σ 2 − Ψ 2 + 1 m Δ σ + m α σ 2$
Notice that
$2 σ 2 T 2 − m + 1 m τ σ 2 = 2 σ 2 T 2 − 1 m τ 2 − m − 1 m τ σ 2$
and, by Equation (5), we have that
$σ T E − 1 m τ σ E = H σ E − 1 m Δ σ E ,$
which implies
$σ 2 T − τ m I 2 = H σ − 1 m Δ σ I 2 .$
Combining it with Equation (50), we arrive at
$2 σ 2 T 2 − m + 1 m τ σ 2 = 2 H σ − 1 m Δ σ I 2 − m − 1 m τ σ 2 .$
Moreover, we have
$H σ − 1 m Δ σ I 2 = H σ 2 + 1 m Δ σ 2 − 2 m Δ σ ∑ j = 1 m g H σ E j , E j = H σ 2 − 1 m Δ σ 2$
Similarly, we have
$H σ + α σ I 2 = H σ 2 + 2 α σ Δ σ + m α 2 σ 2$
and utilizing it in Equation (52), we obtain
$H σ − 1 m Δ σ I 2 = H σ + α σ I 2 − 2 α σ Δ σ − m α 2 σ 2 − 1 m Δ σ 2 ,$
i.e.,
$H σ − 1 m Δ σ I 2 = H σ + α σ I 2 − 1 m Δ σ + m α σ 2 .$
Thus, in view of the above equation, (51) assumes the form
$2 σ 2 T 2 − m + 1 m τ σ 2 = 2 H σ + α σ I 2 − 2 m Δ σ + m α σ 2 − m − 1 m τ σ 2 .$
Now, inserting this value in Equation (49), we arrive at
$∫ N m H σ + α σ I 2 = ∫ N m R i c u , u + 2 H σ + α σ I 2 − m − 1 m τ σ 2 − Ψ 2 − 1 m Δ σ + m α σ 2 ,$
i.e.,
$∫ N m H σ + α σ I 2 = ∫ N m m − 1 m τ σ 2 + 1 m Δ σ + m α σ 2 + Ψ 2 − ∫ N m R i c u , u .$
Using inequality (47) in the above equation, we conclude
$∫ N m H σ + α σ I 2 ≤ 0 ,$
which proves
$H e s s ( σ ) = − α σ g ,$
where $α > 0$ is a constant and $σ$, being a nontrivial solution of a static perfect fluid, is a non-constant function. Hence, $N m , g$ is isometric to $S α m$ (cf. [18,21]).
The converse is trivial, because, by Equation (21), $S α m$ admits a $σ$-RVF $u$, and by Equation (23) and the paragraph that follows (23), $σ$ is a nontrivial solution of the static perfect fluid equation. Moreover, we have, by Equation (22), that
$Δ σ + m α σ = 0$
and, by Equation (46), we have
$∫ N m R i c u , u = ∫ N m m − 1 m τ 2 σ 2 + 1 m Δ σ + m α σ 2 + Ψ 2$
This finishes the proof. □

## Author Contributions

Conceptualization, H.A., S.D. and G.-E.V.; methodology, H.A., S.D. and G.-E.V.; software, H.A., S.D. and G.-E.V.; validation, H.A., S.D. and G.-E.V.; formal analysis, H.A., S.D. and G.-E.V.; investigation, H.A., S.D. and G.-E.V.; resources, H.A., S.D. and G.-E.V.; data curation, H.A., S.D. and G.-E.V.; writing—original draft preparation, H.A., S.D. and G.-E.V.; writing—review and editing, H.A., S.D. and G.-E.V.; visualization, H.A., S.D. and G.-E.V.; supervision, H.A., S.D. and G.-E.V.; project administration, H.A., S.D. and G.-E.V.; funding acquisition, H.A. All authors have read and agreed to the published version of the manuscript.

## Funding

The authors would like to extend their sincere appreciation to Supporting Project Number (RSPD2023R860), King Saud University, Riyadh, Saudi Arabia.

## Data Availability Statement

Data are contained within the article.

## Acknowledgments

The authors would like to extend their sincere appreciations to Supporting project number (RSPD2024R860) King Saud University, Riyadh, Saudi Arabia.

## Conflicts of Interest

The authors declare no conflicts of interest.

## References

1. Alohali, H.; Deshmukh, S. Ricci vector fields. Mathematics 2023, 11, 4622. [Google Scholar] [CrossRef]
2. Chen, B.-Y. Some results on concircular vector fields and their applications to Ricci solitons. Bull. Korean Math. Soc. 2015, 52, 1535–1547. [Google Scholar] [CrossRef]
3. Deshmukh, S.; Turki, N.B.; Vîlcu, G.-E. A note on static spaces. Results Phys. 2021, 27, 104519. [Google Scholar] [CrossRef]
4. Falcitelli, M.; Sarkar, A.; Halder, S. Conformal vector fields and conformal Ricci solitons on α-Kenmotsu manifolds. Mediterr. J. Math. 2023, 20, 127. [Google Scholar] [CrossRef]
5. Hwang, S.; Yun, G. Conformal vector fields and their applications to Einstein-type manifolds. Results Math. 2024, 79, 45. [Google Scholar] [CrossRef]
6. Khan, S.; Bukhari, M.; Alkhaldi, A.; Ali, A. Conformal vector fields of Bianchi type-I spacetimes. Mod. Phys. Lett. A 2021, 36, 2150254. [Google Scholar] [CrossRef]
7. Narmanov, A.; Rajabov, E. On the geometry of orbits of conformal vector fields. J. Geom. Symmetry Phys. 2019, 51, 29–39. [Google Scholar] [CrossRef]
8. Poddar, R.; Balasubramanian, S.; Sharma, R. Quasi-Einstein manifolds admitting conformal vector fields. Colloq. Math. 2023, 174, 81–87. [Google Scholar] [CrossRef]
9. Sharma, R. Gradient Ricci solitons with a conformal vector field. J. Geom. 2018, 109, 33. [Google Scholar] [CrossRef]
10. Sharma, R. Conformal flatness and conformal vector fields on umbilically synchronized space-times. Acta Phys. Pol. B 2023, 54, A3. [Google Scholar] [CrossRef]
11. Filho, J.F.S. Quasi-Einstein manifolds admitting a closed conformal vector field. Differ. Geom. Appl. 2024, 92, 102083. [Google Scholar] [CrossRef]
12. Nikonorov, Y. Spectral properties of Killing vector fields of constant length. J. Geom. Phys. 2019, 145, 103485. [Google Scholar] [CrossRef]
13. Lynge, W.C. Sufficient conditions for periodicity of a Killing vector field. Proc. Am. Math. Soc. 1973, 38, 614–616. [Google Scholar] [CrossRef]
14. Rong, X. Positive curvature, local and global symmetry, and fundamental groups. Am. J. Math. 1999, 121, 931–943. [Google Scholar] [CrossRef]
15. Yorozu, S. Killing vector fields on complete Riemannian manifolds. Proc. Am. Math. Soc. 1982, 84, 115–120. [Google Scholar] [CrossRef]
16. Blaga, A.M.; Chen, B.-Y. Harmonic forms and generalized solitons. Results Math. 2024, 79, 16. [Google Scholar] [CrossRef]
17. Lee, K.-H. Stability and moduli space of generalized Ricci solitons. Nonlinear Anal. 2024, 240, 113458. [Google Scholar] [CrossRef]
18. Obata, M. The conjectures about conformal transformations. J. Differ. Geom. 1971, 6, 247–258. [Google Scholar] [CrossRef]
19. Chow, B.; Lu, P.; Ni, L. Hamilton’s Ricci Flow; Graduate Studies in Mathematics; American Mathematical Society: Providence, RI, USA, 2006; Volume 77. [Google Scholar]
20. Fischer, A.E.; Marsden, J.E. Manifolds of Riemannian metrics with prescribed scalar curvature. Bull. Am. Math. Soc. 1974, 80, 479–484. [Google Scholar] [CrossRef]
21. Obata, M. Conformal transformations of Riemannian manifolds. J. Differ. Geom. 1970, 4, 311–333. [Google Scholar] [CrossRef]
22. Besse, A.L. Einstein Manifolds; Springer: Berlin/Heidelberg, Germany, 1987. [Google Scholar]
23. Yano, K. Integral Formulas in Riemannian Geometry; Marcel Dekker Inc.: New York, NY, USA, 1970. [Google Scholar]
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Alohali, H.; Deshmukh, S.; Vîlcu, G.-E. Ricci Vector Fields Revisited. Mathematics 2024, 12, 144. https://doi.org/10.3390/math12010144

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Alohali H, Deshmukh S, Vîlcu G-E. Ricci Vector Fields Revisited. Mathematics. 2024; 12(1):144. https://doi.org/10.3390/math12010144

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Alohali, Hanan, Sharief Deshmukh, and Gabriel-Eduard Vîlcu. 2024. "Ricci Vector Fields Revisited" Mathematics 12, no. 1: 144. https://doi.org/10.3390/math12010144

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