Projective Vector Fields on Semi-Riemannian Manifolds

Abstract

This paper explores the properties of projective vector fields on semi-Riemannian manifolds. The main result establishes that if a projective vector field P on such a manifold is also a conformal vector field with potential function $\psi$ and the vector field $\zeta$ dual to $d\psi$ does not change its causal character, then P is homothetic, or $\zeta$ is a light-like vector field. Additionally, it is shown that a complete Riemannian manifold admits a projective vector field that is also conformal and non-Killing if and only if it is locally Euclidean. The paper also presents other results related to the characterization of Killing and parallel vector fields using the Ricci curvature and the Hessian of the function given by the inner product of the vector field.

1. Introduction

Consider an n-dimensional semi-Riemannian manifold $(N,h)$, $n\ge 2$, and denoted by $\mathfrak{X}\left(N\right)$ the collection of all smooth vector fields on N.

At a point p in N, the tangent vector X is called space-like (respectively, time-like or light-like) if $h_p(X,X)\ge 0$ (respectively, $\le 0$ or $=0$). The zero vector is classified as space-like. The norm $\left|X\right|$ is defined as ${\left|h(X,X)\right|}^{\frac{1}{2}}$. Analogously, a vector field X on N is called space-like (respectively, time-like or light-like) if it is this at each point. The zero vector field is also considered space-like.

A vector field P is called a projective if its local flow preserves the geodesics of $(N,h)$ in the set-theoretic sense. If the flow of P preserves geodesics in the mapping sense, then it is called affine. It is not difficult to see that a vector field P on N is projective if there exists a differential 1-form $\mu$ on N (called the associated differential 1-form to P), such that

$$({\pounds }_P\nabla )(U,V)=\mu \left(U\right)V+\mu \left(V\right)U,$$

(1)

for all $U,V\in \mathfrak{X}\left(N\right)$, where ${\pounds }_P$ is the Lie derivative along P, and ∇ is the Levi-Civita connection of $(N,h)$. Here the Lie derivative ${\pounds }_P$ acts on ∇ as follows:

$$({\pounds }_P\nabla )(U,V)=[P,{\nabla }_{U}V]-{\nabla }_{[P,U]}V-{\nabla }_U[P,V],$$

for all $U,V\in \mathfrak{X}\left(N\right)$.

Of course, when ${\pounds }_P\nabla =0$, P is an affine vector field. A projective vector field satisfies

$$\mu \left(U\right)={\displaystyle \frac{1}{n+1}}{\nabla }_U\left(div\left(P\right)\right),$$

(2)

for all $U\in \mathfrak{X}\left(N\right)$. See Lemma 3 below (see also [1]).

According to [2], a complete Riemannian manifold N with a parallel Ricci tensor, which admits a non-affine projective vector fields, has a positive constant curvature. In [3], it has been shown that if N is a compact Riemannian manifold with non-positive constant scalar curvature, any projective vector field on N is Killing. Furthermore, in [4], it is proven that if a compact simply connected Riemannian manifold with constant scalar curvature admits a projective vector field which is not Killing, then N must be isometric to sphere.

In [5], a set of integral inequalities within a compact, orientable Riemannian manifold with constant scalar curvature that allows for a projective vector field, subsequently deriving the necessary and sufficient conditions for such a Riemannian manifold to be isometric to a sphere.

In addition, Section 4 will explore conformal projective vector fields. Conformal vector fields are crucial. They are significant elements in the study of the geometry of various types of manifolds. A smooth vector field P on a semi-Riemannian manifold $(N,h)$ is termed a conformal vector field if its flow results in conformal transformations or, equivalently, if the Lie derivative ${\pounds }_{P}h$ with respect to the metric h along the vector field P satisfies the condition [6] (see also [7]):

$${\pounds }_{P}h=2\psi h,$$

(3)

where $\psi$ is a smooth function on N (called the potential function of P). In this case, it is straightforward to see that:

$$\psi ={\displaystyle \frac{div\left(P\right)}{n}}.$$

(4)

Examples of conformal vector fields include homothetic vector fields, where $\psi$ remains constant, and Killing vector fields, where $\psi =0$.

A notable question in the study of Riemannian manifold geometry is identifying spheres within the category of compact connected Riemannian manifolds. Obata provided one such identification [8,9]. Many authors extensively studied Riemannian manifolds with constant scalar curvature allowing for non-isometric conformal vector fields. They aimed to prove a conjecture about the Euclidean sphere as the unique compact orientable Riemannian manifold admitting a metric of constant scalar curvature R carrying a conformal vector field X. Notable researchers include Goldberg and Kobayashi [10], Nagano [11], Obata [12], and Yano and Hagano [13]. Interested readers can find a summary of these results in Yano [14]. We also reference the following works for recent studies on conformal vector fields in semi-Riemannian manifolds: [15,16,17,18,19].

This paper examines the properties of projective vector fields in semi-Riemannian manifolds. Initially, we demonstrate that a projective field, which is also a conformal vector field within a semi-Riemannian manifold, is inherently homothetic. This paper is structured as follows. Section 2 provides some preliminaries and Section 3 focuses on validating various theorems related to projective vector fields within a semi-Riemannian manifold. This includes multiple characterization results and confirms certain theorems on projective vector fields in such manifolds. We demonstrate that any projective vector field P with a non-negative $\mu \left(P\right)$ on a Riemannian compact manifold must be a Killing vector field. Also, we establish the impossibility of a non-parallel projective vector field P with a non-negative $\mu \left(P\right)$ on a Riemannian compact manifold with non-positive Ricci curvature. For non-compact manifolds where the metric h is not necessarily positive definite (i.e., $(N,h)$ is semi-Riemannian), we show that a projective vector field P on N with constant length and fulfilling $Ric(P,P)\le 0$ must be parallel. Furthermore, we prove that any projective vector field P with a non-negative $\mu \left(P\right)$ on a Riemannian manifold, where the Hessian of the function $h(P,P)$ is non-positive, is necessarily a geodesic vector field. We also identify several necessary and sufficient conditions for a projective vector field on a semi-Riemannian manifold to be Killing.

Additionally, in Theorems 7 and 9, we establish the necessary and sufficient conditions for a projective vector field on a semi-Riemannian manifold to be parallel.

In Section 4, we explore projective vector fields on semi-Riemannian manifolds that also serve as conformal vector fields. First, we show that if P is a projective vector field which is also a conformal vector field on a semi-Riemannian manifold such that ${\pounds }_{P}h=2\psi h$, and if the vector field $\zeta$ dual to $d\psi$ does not change its causal character, then P is homothetic or $\zeta$ is a light-like vector field. Then, we prove that a complete Riemannian manifold has a non-Killing projective vector field that is also conformal if and only if it is locally Euclidean. We also generalize two results in [20,21] in two directions: We focus on semi-Riemannian manifolds rather than Riemannian manifolds, and we examine conformal vector fields instead of affine vector fields (referred to as Jacobi-type vector fields in [20]).

2. Perliminaries

For the concepts and formulas discussed in this section, we suggest referring to the following books [22,23].

On a semi-Riemannian manifold of dimension $n\ge 2$, denoted as $(N,h)$, with a Levi-Civita connection ∇ and a local orthonormal frame $\{E_1,\dots ,E_n\}$. The Ricci curvature tensor is a symmetric tensor defined as follows:

$$Ric(U,V)=\sum _{i=1}^n{ϵ}_{i}h(R(U,E_i)V,E_i),$$

(5)

where U and V are vector fields on N, and ${ϵ}_i=h(E_i,E_i)$. Here, the curvature tensor of N is given by

$$R(U,V)W={\nabla }_{[U,V]}W-{\nabla }_U{\nabla }_{V}W+{\nabla }_V{\nabla }_{U}W,$$

for all $U,V,W\in \mathfrak{X}\left(N\right)$. The divergence of a vector field U is defined by

$$div\left(U\right)=\sum _{i=1}^n{ϵ}_{i}h({\nabla }_{E_i}U,E_i).$$

(6)

where ${ϵ}_i=h(E_i,E_i)$. The vector field U is called incomperssible if $div\left(U\right)=0$. That means that the flow of U preserves the volume of $(N,h)$. For a smooth function f on N, the Hessian, denoted $Hess\left(f\right)$, is a symmetric tensor of type $(0,2)$. It is defined by the equation

$$Hess\left(f\right)(U,V)=h({\nabla }_U\nabla f,V),$$

(7)

for all $U,V\in \mathfrak{X}\left(N\right)$, where the symbol $\nabla f$ represents the gradient of f.

The second covariant derivative of the vector field P in the direction of the vector fields U and V is defined by

$${\nabla }_{U,V}^{2}P={\nabla }_U{\nabla }_{V}P-{\nabla }_{{\nabla }_{U}V}P.$$

For operators A and B on N, the inner product between A and B is given by

$$<A,B>=tr\left(A{B}^t\right),$$

where $tr$ denoted the trace. The norm of the operator A is determined as

$$\left|\right|A\left|\right|=\sqrt{<A,A>}.$$

The following lemma characterizes projective vector fields in terms of the second covariant derivative and the curvature tensor.

Lemma 1. 

Let P be a projective vector field on a semi-Riemannian manifold $(N,h)$. Then, P satisfies the following equation:

$${\nabla }_{U,V}^{2}P+R(U,P)V=\mu \left(U\right)V+\mu \left(V\right)U,$$

for all $U,V\in \mathfrak{X}\left(N\right)$, where μ the differential 1-form associated to P.

Proof. 

For $U,V\in \mathfrak{X}\left(N\right)$, we have

$$\begin{array}{cc}\hfill ({\pounds }_P\nabla )(U,V)& ={\pounds }_P{\nabla }_{U}V-{\nabla }_{{\pounds }_{P}U}V-{\nabla }_U{\pounds }_{P}V\hfill \\ & =[P,{\nabla }_{U}V]-{\nabla }_{[P,U]}V-{\nabla }_U[P,V]\hfill \\ & ={\nabla }_P{\nabla }_{U}V-{\nabla }_{{\nabla }_{U}V}P-{\nabla }_{[P,U]}V-{\nabla }_U{\nabla }_{P}V+{\nabla }_U{\nabla }_{V}P\hfill \\ & ={\nabla }_{U,V}^{2}P+[{\nabla }_P,{\nabla }_U]V-{\nabla }_{[P,U]}V\hfill \\ & ={\nabla }_{U,V}^{2}P+R(U,P)V.\hfill \end{array}$$

So, P is a projective vector field if and only if ${\nabla }_{U,V}^{2}P+R(P,U)V=\mu \left(U\right)V+\mu \left(V\right)U$ for all $U,V\in \mathfrak{X}\left(N\right)$. □

For any vector field P on $(N,h)$, let ${\omega }_P$ denote the 1- form dual to P, that is, ${\omega }_P\left(U\right)=h(U,P)$, for all $U\in \mathfrak{X}\left(N\right)$. We associate the (1, 1)-tensor $A_P$ defined by

$$A_P\left(U\right)={\nabla }_{U}P,$$

(8)

for all $U\in \mathfrak{X}\left(N\right)$.

We write

$$A_P=B+\theta ,$$

(9)

where B and $\theta$ are the symmetric and anti-symmetric components of $A_P$, respectively.

The assertion presented here is an alternative form of Lemma 1 presented in terms of the operator of $A_P$.

Lemma 2. 

A vector field P on a semi-Riemannian manifold $(N,h)$ with an associated differential 1-form μ is projective if and only if it satisfies the following equation.

$$\left({\nabla }_U{A}_P+R(U,P)\right)V=\mu \left(U\right)V+\mu \left(V\right)U,$$

(10)

for all $U,V\in \mathfrak{X}\left(N\right)$.

Lemma 3. 

If P is a projective vector field on an n-dimensional semi-Riemannian manifold $(N,h)$ with associated differential 1-form μ, then

$$\mu \left(U\right)={\displaystyle \frac{1}{n+1}}{\nabla }_U\left(div\left(P\right)\right),$$

for all $U\in \mathfrak{X}\left(N\right)$.

Proof. 

Let $\{E_1,\dots ,E_n\}$ be a local orthonormal frame on N, and set ${ϵ}_i=h(E_i,E_i)=\pm 1$. By tracing (10) in Lemma 2, we obtain

$$\begin{array}{cc}\hfill tr\left({\nabla }_U\left(A_P\right)\right)+tr\left(R(U,P)\right)& =\mu \left(U\right)\sum _{i=1}^n{ϵ}_{i}h(E_i,E_i)+\sum _{i=1}^n{ϵ}_{i}h(\mu \left(E_i\right)U,E_i)\hfill \\ & =n\mu \left(U\right)+\mu \left(\sum _{i=1}^n{ϵ}_{i}h(U,E_i)E_i\right)\hfill \\ & =n\mu \left(U\right)+\mu \left(U\right)\hfill \\ & =(n+1)\mu \left(U\right),\hfill \end{array}$$

for all $U\in \mathfrak{X}\left(N\right)$.

Since $tr\left({\nabla }_U\left(A_P\right)\right)={\nabla }_U\left(tr\left(A_P\right)\right)={\nabla }_U\left(div\left(P\right)\right)$, we get $\mu \left(U\right)={\displaystyle \frac{1}{n+1}}{\nabla }_U\left(div\left(P\right)\right)$. □

Now, we present a generalized formulation of the Bochner formula, which will be employed in the forthcoming sections. (cf. [24]).

Theorem 1. 

Let $(N,h)$ be a semi-Riemannian manifold. Then

$${\nabla }_U\left(div\left(U\right)\right)+Ric(U,U)-div\left({\nabla }_{U}U\right)+tr\left(A_U^2\right)=0,$$

(11)

for all $U\in \mathfrak{X}\left(N\right)$.

Proof. 

Let $\{E_1,\dots ,E_n\}$ be a local orthonormal frame on N that we assume to be parallel, where n is the dimension of N, and let $U\in \mathfrak{X}\left(N\right)$. It is straightforward to see that

$$({\pounds }_U\nabla )(U,E_i)=({\pounds }_U\nabla )(E_i,U),$$

for all $i=1,...,n$.

Then, by (5) and (6), we get

$$\begin{array}{cc}\hfill {\nabla }_U\left(div\left(U\right)\right)+Ric(U,U)-div\left({\nabla }_{U}U\right)+tr\left(A_U^2\right)& =\sum _{i=1}^n{ϵ}_{i}h({\nabla }_U{\nabla }_{E_i}U+R(E_i,U)U\hfill \\ & -{\nabla }_{E_i}{\nabla }_{U}U+{\nabla }_{{\nabla }_{E_i}U}U,E_i)\hfill \\ & =\sum _{i=1}^n{ϵ}_{i}h(({\pounds }_U\nabla )(U,E_i),E_i))\hfill \\ & -h(({\pounds }_U\nabla )(E_i,U),E_i)=0,\hfill \end{array}$$

where ${ϵ}_i=h(E_i,E_i)=\pm 1$. □

3. Characterizations of Projective Vector Fields on Semi-Riemannian Manifolds

In this section, we provide several results on projective vector fields in a semi-Riemannian manifold, including various useful formulas. These formulas help us derive significant results, allowing projective vector fields to be identified as either Killing or parallel vector fields.

Theorem 2. 

Let $(N,h)$ be a semi-Riemannian manifold. For a projective vector field P on N, the equation below holds for P

$$\Delta f=-Ric(P,P)+2\mu \left(P\right)+\left|\right|A_P{\left|\right|}^2,$$

(12)

where $f={\displaystyle \frac{1}{2}}h(P,P)$.

Proof. 

Let P be a projective vector field on a semi-Riemannian manifold $(N,h)$. By Lemma 1, it follows that

$${\nabla }_{U,V}^{2}P+R(U,P)V=\mu \left(U\right)V+\mu \left(V\right)U,$$

(13)

for all $U,V\in \mathfrak{X}\left(N\right)$. Furthermore, we obtain

$$\begin{array}{c}\hfill h(\nabla f,V)={\nabla }_{V}f=h(A_P\left(V\right),P),\end{array}$$

(14)

for all $V\in \mathfrak{X}\left(N\right)$. It follows that, for any $U\in \mathfrak{X}\left(N\right)$, we have

$${\nabla }_{U}h(\nabla f,V)={\nabla }_{U}h(A_P\left(V\right),P),$$

which implies that

$$h({\nabla }_U\nabla f,V)+h(\nabla f,{\nabla }_{U}V)=h({\nabla }_U{\nabla }_{V}P,P)+h({\nabla }_{V}P,{\nabla }_{U}P).$$

Thus, according to (7) and (14), we conclude that

$$Hessf(U,V)=h({\nabla }_{U,V}^{2}P,P)+h({\nabla }_{V}P,{\nabla }_{U}P),$$

(15)

By substituting (13) into (15), we get

$$Hessf(U,V)=-h(R(U,P)P,V)+\mu \left(U\right)h(V,P)+\mu \left(V\right)h(U,P)+h({\nabla }_{V}P,{\nabla }_{U}P)$$

(16)

By computing the trace of Equation (16) with respect to a local orthonormal frame $\{E_1,\dots ,E_n\}$, and considering both the symmetry of B and the anti-symmetry of $\theta$, together with the fact that $\Delta f=tr\left(Hessf\right)$, we can obtain

$$\begin{array}{cc}\hfill \Delta f& =-\sum _{i=1}^n{ϵ}_{i}h(R(P,E_i)P,E_i)+2\sum _{i=1}^n{ϵ}_{E_i}\mu \left(E_i\right)h(P,E_i)+\sum _{i=1}^n{ϵ}_{i}h({\nabla }_{E_i}P,{\nabla }_{E_i}P)\hfill \\ & =-Ric(P,P)+2\mu \left(\sum _{i=1}^n{ϵ}_{i}h(P,E_i)E_i\right)+\sum _{i=1}^n{ϵ}_{i}h\left(A_P\left(E_i\right),A_P\left(E_i\right)\right)\hfill \\ & =-Ric(P,P)+2\mu \left(P\right)+\sum _{i=1}^n{ϵ}_{i}h\left((B+\theta )\left(E_i\right),(B+\theta )\left(E_i\right)\right)\hfill \\ & =-Ric(P,P)+2\mu \left(P\right)+\sum _{i=1}^n{ϵ}_{i}h\left(B^2\left(E_i\right)-{\theta }^2\left(E_i\right),E_i\right)\hfill \\ & =-Ric(P,P)+2\mu \left(P\right)+tr\left(B^2\right)-tr\left({\theta }^2\right)\hfill \\ & =-Ric(P,P)+2\mu \left(P\right)+tr\left(B^{t}B\right)+tr\left({\theta }^t\theta \right)\hfill \\ & =-Ric(P,P)+2\mu \left(P\right)+{\left|\right|B\left|\right|}^2+{\left|\right|\theta \left|\right|}^2\hfill \\ & =-Ric(P,P)+2\mu \left(P\right)+\left|\right|A_P{\left|\right|}^2.\hfill \end{array}$$

□

We return to the decomposition (9), from which we deduce that

$$tr\left(A_P^2\right)={\left|\right|B\left|\right|}^2-{\left|\right|\theta \left|\right|}^2,$$

(17)

and

$$\left|\right|A_P{\left|\right|}^2={\left|\right|B\left|\right|}^2+{\left|\right|\theta \left|\right|}^2.$$

(18)

Thus, $tr\left(A_P^2\right)=\left|\right|A_P{\left|\right|}^2$ if $A_P$ is symmetric, and $tr\left(A_P^2\right)=-\left|\right|A_P{\left|\right|}^2$ if $A_P$ is anti-symmetric (that is, P is a Killing vector field). Also, from (9) and (14), we get

$$\nabla f=B\left(P\right)-\theta \left(P\right).$$

(19)

Now, for a projective vector field P on semi-Riemannian manfold $(N,h)$, we give a very useful formula.

Theorem 3. 

Let $(N,h)$ be a semi-Riemannian manifold. For a projective vector field P on N, the equation below holds

$$div\left({\nabla }_{P}P\right)+\Delta f={2\left|\right|B\left|\right|}^2+(n+3)\mu \left(P\right),$$

(20)

where B is the symmetric part of $A_P$, and $f={\displaystyle \frac{1}{2}}h(P,P)$.

Proof. 

From (11) and (12), we have

$$div\left({\nabla }_{P}P\right)={\nabla }_P\left(div\left(P\right)\right)+Ric(P,P)+tr\left(A_P^2\right),$$

and

$$\Delta f=-Ric(P,P)+2\mu \left(P\right)+\left|\right|A_P{\left|\right|}^2,$$

respectively.

By adding those two equations, and using (17) and (18), we obtain

$$\begin{array}{cc}\hfill div\left({\nabla }_{P}P\right)+{\displaystyle \frac{1}{2}}\Delta h(P,P)& =2\mu \left(P\right)+\left|\right|A_P{\left|\right|}^2+{\nabla }_P\left(div\left(P\right)\right)+tr\left(A_P^2\right)\hfill \\ & =2\mu \left(P\right)+{\left|\right|B\left|\right|}^2+{\left|\right|\theta \left|\right|}^2+(n+1)\mu \left(P\right)+{\left|\right|B\left|\right|}^2-{\left|\right|\theta \left|\right|}^2\hfill \\ & ={2\left|\right|B\left|\right|}^2+(n+3)\mu \left(P\right).\hfill \end{array}$$

□

We can derive several consequences from (20). The first one is a characterization of Killing vector fields on compact Riemannian manifolds among projective ones.

Theorem 4. 

Let $(N,h)$ be an n-dimensional compact Riemannian manifold, and let P be a projective vector field on N. If $\mu \left(P\right)\ge 0$, then P is a Killing vector field.

Proof. 

Given that P is a projective vector field on the compact Riemannian manifold $(N,h)$, by integrating Equation (20), we obtain

$${\int }_N\left({2\left|\right|B\left|\right|}^2+(n+3)\mu \left(P\right)\right)dV=0.$$

(21)

This leads to the deduction that $B=0$, as $\mu \left(P\right)\ge 0$, which implies that $A_P$ is anti-symmetric, and meaning that P is a Killing vector field. □

When considering a semi-Riemannian manifold $(N,h)$ which may not be compact, an interesting problem arises: What conditions need to be satisfied for a projective vector field to become a Killing vector field? The following two corollaries can be derived directly from (12) and the important formula (20).

Corollary 1. 

Let $(N,h)$ be an n-dimensional semi-Riemannian manifold, with a projective geodesic vector field P where $\mu \left(P\right)\ge 0$. Then, P has a constant length if and only if it is a Killing vector field. In this case, $\mu \left(P\right)=0$.

Corollary 2. 

Let $(N,h)$ be an n-dimensional semi-Riemannian manifold, with a projective vector field P of constant length and $\mu \left(P\right)\ge 0$. Then, ${\nabla }_{P}P$ is an incompressible vector field if and only if P is a Killing vector field. In this case, $\mu \left(P\right)=0$.

The result below guarantees that a non-parallel projective vector field cannot exist on a compact Riemannian manifold with non-positive Ricci curvature. This is a consequence of Formula (12).

Corollary 3. 

Let P be a projective vector field on a compact Riemannian manifold $(N,h)$, with $\mu \left(P\right)\ge 0$. If $Ric(P,P)\le 0$, then P is a parallel vector field.

Proof. 

By integrating both sides of (12), we obtain

$${\int }_N\left(\left|\right|A_P{\left|\right|}^2+2\mu \left(P\right)\right)dV={\int }_{N}Ric(P,P)dV.$$

Considering that $\mu \left(P\right)\ge 0$ and $Ric(P,P)\le 0$, we deduce that $\left|\right|A_P{\left|\right|}^2$ is zero. Thus, $A_P=0$ and P must be a parallel vector field. □

When N is not necessarily compact, the following holds. This is also a consequence of Formula (12).

Corollary 4. 

Let P be a projective vector field of constant length on the semi-Riemannian manifold $(N,h)$ such that $\mu \left(P\right)\ge 0$. If $Ric(P,P)\le 0$, then P is a parallel vector field.

Proof. 

Given that $h(P,P)$ is constant, (12) reduces to

$$\left|\right|A_P{\left|\right|}^2+2\mu \left(P\right)=Ric(P,P).$$

Since $\mu \left(P\right)\ge 0$ and $Ric(P,P)\le 0$, it follows that $A_P=0$, which means that P is parallel. □

Corollary 5. 

If the Ricci curvature of a semi-Riemannian manifold $(N,h)$ is non-positive, then $(N,h)$ admits no non-zero parallel projective vector field P with $\mu \left(P\right)\ge 0$.

The subsequent result characterizes projective vector fields on a Riemannian manifold in terms of the Hessian of the length of these vector fields.

Theorem 5. 

Let P be a projective vector field on a Riemannian manifold $(N,h)$ with $\mu \left(P\right)\ge 0$, and let $f={\displaystyle \frac{1}{2}}{\left|P\right|}^2$. If $Hessf(P,P)\le 0$, then P is a geodesic vector field.

Proof. 

Taking $U=V=P$ into (16), it follows that

$$Hessf(P,P)={2\mu \left(P\right)\left|P\right|}^2+{\left|{\nabla }_{P}P\right|}^2.$$

(22)

Since $\mu \left(P\right)\ge 0$ and $Hessf(P,P)\le 0$, it follows that $|{\nabla }_P{P|}^2=0$. Thus, P is a geodesic vector field. □

From this result, we obtain an important consequence.

Corollary 6. 

Consider a Riemannian manifold $(N,h)$. There does not exist any nonzero geodesic projective vector field P such that $\mu \left(P\right)\ge 0$ and $Hessf\le 0$, where $f={\displaystyle \frac{1}{2}}{\left|P\right|}^2$.

We generalize Theorem 2 in [20] to projective vector fields on semi-Riemannian manifolds.

Theorem 6. 

Let P be a projective vector field on a semi-Riemannian manifold. Then, P is a Killing vector field if and only if the following holds

$$\Delta f\le {\left|\right|\theta \left|\right|}^2+2\mu \left(P\right)-Ric(P,P),$$

(23)

where θ is the anti-symmetric part of $A_P$, and $f={\displaystyle \frac{1}{2}}h(P,P)$.

Proof. 

Assuming (23) holds, then by (12), we have

$$\left|\right|A_P{\left|\right|}^2\le {\left|\right|\theta \left|\right|}^2.$$

By (18), we obtain $B=0$, and P is a Killing vector field. The converse is trivial. □

In the following result, we prove that under a simple condition in terms of Ricci curvature, a geodesic projective vector field must be parallel.

Theorem 7. 

Let $(N,h)$ be an n-dimensional connected semi-Riemannian manifold, admitting a geodesic projective vector field P with $\mu \left(P\right)\ge 0$. Then, P is parallel field if and only if the following holds

$$Ric(P,P)+{\left|\right|B\left|\right|}^2+(n+1)\mu \left(P\right)\le 0.$$

(24)

In particular, if P is a geodesic vector field, then $Ric(P,P)\le 0$.

Proof. 

Let P be a geodesic projective vector field. Then, by applying the generalized Bochner formula (11) and referring to (2), we obtain

$$Ric(P,P)=-tr\left(A_P^2\right)-Pdiv\left(P\right)={\left|\right|\theta \left|\right|}^2-{\left|\right|B\left|\right|}^2-(n+1)\mu \left(P\right).$$

(25)

Assuming that $Ric(P,P)+{\left|\right|B\left|\right|}^2+(n+1)\mu \left(P\right)\le 0$, we deduce from (25) that $\theta =0$. Since P is geodsic, it follows from (9) that $B\left(P\right)=0$.

By substituting these quantities into (19), we get $\nabla f=0$. So, we deduce that $h(P,P)$ is constant. Substituting this into (12), we obtain

$$Ric(P,P)={\left|\right|B\left|\right|}^2+2\mu \left(P\right).$$

Given that $Ric(P,P)+{\left|\right|B\left|\right|}^2+(n+1)\mu \left(P\right)\le 0$ and $\mu \left(P\right)\ge 0$, it follows that $B=0$. This, with the fact $\theta =0$ implies that $A_P=0$, which means that P is parallel. □

Next, we generalize Theorem 4 in [20] to semi-Riemannian manifolds admitting a projective vector field.

Theorem 8. 

Let P be a projective vector field on a connected semi-Riemannian manifold with $\mu \left(P\right)\ge 0$. Assume that ${\nabla }_{P}P$ is space-like, and define $f={\displaystyle \frac{1}{2}}h(P,P)$. Then P is a Killing vector field with constant length if and only if $Hessf(P,P)\le 4\mu \left(P\right)f$ and $\Delta f\le 0$. In this case, $\mu =0$ and $Ric(P,P)\ge 0$, where the equality is valid if and only if P is a parallel vector field.

Proof. 

Assume that P is a Killing vector field with constant length. This means that $B=0$ and f is constant. It follows that $\Delta f=0$ and $A_P$ is anti-symmetric. In particular, we deduce from (2) that $\mu =0$. Also, we deduce from (19) that $\theta \left(P\right)=0$.

By using (9), we deduce that

$${\nabla }_{P}P=A_P\left(P\right)=B\left(P\right)+\theta \left(P\right)=0,$$

meaning that P is a geodesic vector field. Since $\mu =0$, $\Delta f=0$, and $\left|\right|A_P{\left|\right|}^2={\left|\right|\theta \left|\right|}^2$, we get by (12) that $Ric(P,P)={\left|\right|\theta \left|\right|}^2\ge 0$, where the equality is valid if and only if $\theta =0$, that is $A_P=0$, meaning that P is parallel.

Conversely, if $Hessf(P,P)\le 4\mu \left(P\right)f$ and $\Delta f\le 0$. By taking $U=V=P$ in (16), we obtain

$$Hessf(P,P)=2\mu \left(P\right)h(P,P)+h({\nabla }_{P}P,{\nabla }_{P}P)$$

It follows that $h({\nabla }_{P}P,{\nabla }_{P}P)\le 0$. Since ${\nabla }_{P}P$ is spacelike, we deduce that ${\nabla }_{P}P=0$, meaning that P is a geodesic vector field.

By (20), we have

$$\Delta f={2\left|\right|B\left|\right|}^2+(n+3)\mu \left(P\right)$$

Since we are assuming $\Delta f\le 0$ and $\mu \left(P\right)\ge 0$, we deduce from the last equation that $B=0$ and $\mu \left(P\right)=0$. Therefore, $A_P$ is anti-symmetric, indicating that P is a Killing vector field. In particular, $div\left(P\right)=0$, which leads to $\mu =0$ according to (2).

On the other hand, since P is a geodesic vector field and $B=0$, we deduce from (9) that $\theta \left(P\right)=0$. It follows from (19) that $\nabla f=0$, meaning that f is constant.

By taking $U=P$ in (11) and substituting the values $div\left(P\right)=tr\left(A_P\right)=0$, ${\nabla }_{P}P=0$, and $tr\left(A_P^2\right)=-{\left|\right|\theta \left|\right|}^2$, we deduce that $Ric(P,P)={\left|\right|\theta \left|\right|}^2\ge 0$, where the equality is valid if and only if $\theta =0$, that is $A_P=0$, meaning that P is a parallel vector field. □

When the projective vector field P is light-like, we drive the following consequence.

Corollary 7. 

Let P be a light-like projective vector field on a connected semi-Riemannian manifold with $\mu \left(P\right)\ge 0$. Assume that ${\nabla }_{P}P$ is space-like, and define $f={\displaystyle \frac{1}{2}}h(P,P)$. Then P is a Killing vector field if and only if $Hessf(P,P)\le 0$.

Next, we give a characterization of Killing vector fields on semi-Riemannian manifolds in terms of the Ricci curvature and the Hessian of the length of such a vector field.

Theorem 9. 

Let P be a projective vector field on a connected semi-Riemannian manifold with $\mu \left(P\right)\ge 0$. Assume that ${\nabla }_{P}P$ is space-like, and define $f={\displaystyle \frac{1}{2}}h(P,P)$. Then, P is a Killing vector field with constant length if and only if $Hessf(P,P)\le 4\mu \left(P\right)f$ and $Ric(P,P)+(n+1)\mu \left(P\right)\ge 0$.

Proof. 

Assume that P is a Killing vector field. This means that $B=0$. Since P has a constant length, by (19), we get $\theta \left(P\right)=0$. It follows that P is geodesic. Referring to Equation (22), we observe $Hessf(P,P)=4\mu \left(P\right)f$. Thus, Equation (25) shows $Ric(P,P)+(n+1)\mu \left(P\right)={\left|\right|\theta \left|\right|}^2\ge 0$.

Conversely, assume that $Hessf(P,P)-4\mu \left(P\right)f\le 0$. Since ${\nabla }_{P}P$ is space-like, then (22) implies that P is geodesic. Since $Ric(P,P)+(n+1)\mu \left(P\right)\ge 0$, and by (20), we see that $B=0$. Hence, P is a Killing vector field. Using Equation (14), we see that $\theta \left(P\right)=0$. Thus, f is constant. □

Remark 1. 

The existence of a parallel vector field on an n-dimensional Riemannian manifold suggests that the metric locally splits into a product of a one-dimensional Riemannian manifold and an $(n-1)$-dimensional Riemannian manifold. However, in this paper, the siuation is more complex because the manifold is semi-Riemannian (i.e. the metric is indefinite), meaning the vector field P can have a non-constant causal character, being time-like at some points, light-like at others, and space-like at different points.

4. Conformal Projective Vector Fields Are Homothtic

The main objective of this section is to investigate whether a complete semi-Riemannian manifold, which admits a projective vector field that is also a conformal vector field, can be characterized as a Euclidean space. Initially, we show that a projective vector field, which is also a conformal vector field on a semi-Riemannian manifold is homothetic.

If P is a conformal vector field on a semi-Riemannian manifold $(N,h)$ such that ${\pounds }_{P}h=2\psi h$, then the following equation holds

$$({\pounds }_P\nabla )(U,V)=d\psi \left(U\right)V+d\psi \left(V\right)U-h(U,V)\zeta ,$$

(26)

where $\zeta$ is the vector field associated to the 1-form $d\psi$, i.e. $d\psi \left(U\right)=h(\zeta ,U)$, for all $U\in \mathfrak{X}\left(N\right)$. See, for example [14].

Theorem 10. 

Let P be a projective vector field on a n-dimensional semi-Riemannian manifold $(N,h)$, $n\ge 2$. If P is conformal, such that ${\pounds }_{P}h=2\psi h$, and the vector field ζ dual to $d\psi$ does not change its causal character, then P is homothetic or ζ is a light-like vector field.

Proof. 

Using (1) and (26), we have

$$h(U,V)\zeta =(d\psi -\mu )\left(U\right)V+(d\psi -\mu )\left(V\right)U,$$

(27)

for all $U,V\in \mathfrak{X}\left(N\right)$.

On the other hand, by (2) and (4), we have

$$\mu \left(U\right)={\displaystyle \frac{n}{n+1}}d\psi \left(U\right)={\displaystyle \frac{n}{n+1}}h(U,\zeta ),$$

for all $U\in \mathfrak{X}\left(N\right)$.

Substituting this into (27), it becomes

$$h(U,V)\zeta ={\displaystyle \frac{1}{n}}(\mu \left(U\right)V+\mu \left(V\right)U),$$

for all $U,V\in \mathfrak{X}\left(N\right)$.

By setting $U=V=\zeta$ in the above equation, we get

$$h(\zeta ,\zeta )\zeta ={\displaystyle \frac{2n}{n+1}}h(\zeta ,\zeta )\zeta ,$$

which implies

$$\left({\displaystyle \frac{n-1}{n+1}}\right)h(\zeta ,\zeta )\zeta =0.$$

Given that $n\ge 2$, and since $\zeta$ does not change its causal character, we deduce that $\zeta =0$ or $\zeta$ is ligth-like. However, if $\zeta =0$, then $\mu =0$ and $\psi$ is a constant. Thus, P is homothetic. □

In the next theorem, we show that a complete Riemannian manifold possesses a non-Killing projective vector field which is also conformal if and only if it is locally Euclidean.

Theorem 11. 

If $(N,h)$ is an n-dimensional complete Riemannian manifold, $n\ge 2$, that admits a non-Killing projective vector field that is also conformal, then $(N,h)$ is locally Euclidean.

Proof. 

According to Theorem 10, such a vector field must be homothetic. By Lemma 2, page 242, in [21], and by [25], $(N,h)$ is necessarily a locally Euclidean space. □

Remark 2. 

In [21], Lemma 2, page 244 (see also [20], Theorem 6), it has been proved that if $(N,h)$ is a complete Riemannian manifold that admits an affine vector field that is also a non-Killing gradient conformal vector field, then $(N,h)$ is isometric to a Euclidean space. Furthermore, it was proved in [20], that a complete Riemannian manifold $(N,h)$ admits an affine vector field P that is also a non-Killing conformal vector field that annihilates the operator ϕ (the anti-symmetric part of $A_P$) if and only if $(N,h)$ is locally Euclidean. It is clear that these results evidently represent particular cases of our Theorem 11.

5. Conclusions

This paper has presented a detailed investigation of projective vector fields on semi-Riemannian manifolds, focusing on their geometric properties and interactions with the curvature tensor. We derived essential results that relate the curvature tensor and the second covariant derivative of a projective vector field. Our results show that on a semi-Riemannian manifold with non-positive Ricci curvature, there is no non-zero parallel projective vector field P with the corresponding differential 1-form $\mu$ satisfies the condition $\mu \left(P\right)\ge 0$. Furthermore, we showed that on a compact Riemannian manifold, a projective vector field with constant length and non-positive Ricci curvature must be parallel. Additionally, we explored the conditions under which a projective vector field becomes a Killing vector field, specifically in the case of light-like vector fields. We also showed that a non-Killing, conformal projective vector field exists on a complete Riemannian manifold if and only if the manifold is locally Euclidean.

These results contribute to the broader understanding of the interplay between projective, conformal, and Killing vector fields, with potential implications for future studies in differential geometry and general relativity.