# Vorticity Gramian of compact Riemannian manifolds

Louis Omenyi  Department of Mathematics and Statistics, Alex Ekwueme Federal
University, Ndufu-Alike, Nigeria (∗Corresponding author E-mail:
<EMAIL_ADDRESS>
Corresponding author ORCID: 0000-0002-8628-0298  Department of Mathematical
Sciences, Loughborough University, Loughborough, Leicestershire, United
Kingdom  Emmanuel Nwaeze  Department of Mathematics and Statistics, Alex
Ekwueme Federal University, Ndufu-Alike, Nigeria (∗Corresponding author
E-mail<EMAIL_ADDRESS>
Corresponding author ORCID: 0000-0002-8628-0298  Friday Oyakhire  Department
of Mathematics and Statistics, Alex Ekwueme Federal University, Ndufu-Alike,
Nigeria (∗Corresponding author E-mail<EMAIL_ADDRESS>
Corresponding author ORCID: 0000-0002-8628-0298  Monday Ekhator  Department of
Mathematics and Statistics, Alex Ekwueme Federal University, Ndufu-Alike,
Nigeria (∗Corresponding author E-mail<EMAIL_ADDRESS>
Corresponding author ORCID: 0000-0002-8628-0298

(Preprint submitted to RGN Publications on 06/November/2021)

###### Abstract

The vorticity of a vector field on $3$-dimensional Euclidean space is usually
given by the curl of the vector field. In this paper, we extend this concept
to $n$-dimensional compact and oriented Riemannian manifold. We analyse many
properties of this operation. We prove that a vector field on a compact
Riemannian manifold admits a unique Helmholtz decomposition and establish that
every smooth vector field on an open neighbourhood of a compact Riemannian
manifold admits a Stokes’ type identity.

2010 AMS Classification: 58J65; 58J30; 53C20 Keywords and phrases: Manifold,
curl, metric tensor flow, Hodge star operator, Helmholtz decomposition

Article type: Research Article

##  1 Introduction

Vorticity is a pseudovector field describing the local spinning motion of a
flow near some point on a manifold. In classical mechanics, the dynamics of a
flow are described by its rotation and expansion. Calin and Chang [7]
expressed the rotation component by the curl vector, while the expansion is
described by the divergence function. The classical formulas involving
rotation and expansion in the case of smooth functions and vector fields on
Riemannian manifolds show that gradient vector fields do not rotate and that
the curl vector field is incompressible. Varayu, Chew et al [24] among others
showed that on Riemannian manifolds, the curl of a vector field is not a
vector field, but a tensor.

Many studies on vorticity of flows on manifolds have been ongoing for many
decades. For example, Frankel [12] in fifties established how homology of
manifolds influences vector and tensor fields on the manifolds. It showed that
the vector fields give rise to one-parameter groups of point divergence-free
transformations of the manifolds. The work of Xie and Mar [26] employed
Poisson equation for stream and vorticity equation to study $2$-dimensional
vorticity and stream function expanded in general curvilinear coordinates. It
constructed numerical algorithms of covariant, anti-covariant metric tensor
and Christoffel symbols of the first and second kinds in curvilinear
coordinates. Similarly, Perez-Garcia [23] studied exact solutions of the
vorticity equation on the sphere as a manifold. In the work of Peng and Yang
[22], the existence of the curl operator on higher dimensional Euclidean
space, $\mathbb{R}^{n},~{}~{}n>3,$ was proved.

More recently, Bauer, Kolev and Preston [5] carried out a geometric
investigations of a vorticity model equation extending the works of [25, 8,
11, 14, 15, 21] and Kim [17] on vorticity to manifold study. Besides,
Deshmukh, Pesta and Turki [10] went ahead to show that the presence of a
geodesic vector field on a Riemannian manifold influences its geometry while
B$\ddot{a}$r [4] and M$\ddot{u}$ller [19] extended the curl and divergence
operators to odd-dimensional manifolds in arbitrary basis.

In this study, we extend the concept of vorticity to $n$-dimensional compact
and oriented Riemannian manifolds and analyse many properties of this
operation. We proceed with fixing our notations and briefly explaining some
basic concepts required to follow the discussions.

Let $(M,g)$ be a Riemannian manifold. By this we mean that $M$ is a
topological space that is locally similar to the Euclidean space and $g$ is
the Riemannian metric on $M.$ We recall that a Riemannian metric $g$ on a
smooth manifold $M$ is a symmetric, positive definite $(0,2)$-tensor field,
see e.g. [16, 18, 1] and [13]. This means that for any point $p\in M,$ the
metric is the map $g_{p}:T_{p}M\times T_{p}M\rightarrow\mathbb{R}$ that is a
positive definite scalar product for a tangent space $T_{p}M.$ The Riemannian
metric enables to measure distances, angles and lengths of vectors and curves
on the manifold, see e.g. [3, 6, 9, 16] and [18], for details. We denote the
Riemannian manifold $(M,g)$ simply by $M.$ The manifold $M$ is called compact
if it is compact as a topological space. If $M$ is a smooth manifold, then [7]
and [1] proved that there is at least one Riemannian metric on $M.$

We call a function $f:M\rightarrow\mathbb{R}$ smooth if for every chart
$(U,\phi)$ on $M,$ and the function
$f\circ\phi^{-1}:\phi(U)\rightarrow\mathbb{R}$ is smooth. The set of all
smooth functions on the manifold $M$ will be denoted by $C^{\infty}(M).$ Let
$\Omega^{k}$ denote the vector space of smooth $k$-forms on $M,$ and let
$d:\Omega^{k}\rightarrow\Omega^{k+1}$ be the exterior derivative. Note that
the metric which gives an inner product on the tangent space $T_{p}M$ at each
$p\in M$ induces a natural metric on each cotangent space $T^{\ast}_{p}M,$ as
follows. At $p,$ let $\\{b_{1},b_{2},\cdots,b_{n}\\}$ be an orthonormal basis
for the tangent space. One obtains a metric on the cotangent space by
declaring that the dual basis $\\{b^{1},b^{2},\cdots,b^{n}\\}$ is orthonormal.
Hence given any two $k$-forms $\beta$ and $\gamma,$ we have that
$(\beta,\gamma)$ is a function on $M.$ We call $(\cdot,\cdot)$ the pointwise
inner product; see e.g. [13, 6, 9, 18] and [20]. For a coordinate chart on
$M,$

$(x^{1},\cdots,x^{n}):U\rightarrow\mathbb{R}^{n},$

we represent $g$ by the Gram matrix $(g_{ij})$ where
$g_{ij}=\langle\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}}\rangle,$ and $\langle,\rangle$ is the inner product on the tangent
space. The volume form $dV$ is defined be $b^{1}\wedge b^{2}\wedge\cdots\wedge
b^{n},$ and it is a well-known fact from linear algebra that $dV=\sqrt{|g|}dx$
where $dx=dx^{1}\wedge\cdots\wedge dx^{n}.$ Using the point-wise inner product
above, one writes the $L^{2}$-inner product on $\Omega^{k}(M)$ as

$\langle\beta,\gamma\rangle=\int_{M}(\beta,\gamma)dV~{}~{}\forall\beta,\gamma\in\Omega^{k}(M).$

##  2 Vector fields and differential operators

Vector fields and differential operators are the main tools used in the
analysis of vorticity in this work. We employ these tools to construct the
curl operator on $M$ and analyse its many properties.

A vector field on $M$ is a family $\\{X(p)\\}_{p\in M}$ of tangent vectors
such that $X(p)\in T_{p}M$ for any $p\in M.$ In local coordinates chart
$(x^{1},\cdots,x^{n}),$

$X(p)=X^{i}(p)\frac{\partial}{\partial x^{i}|_{x=p}}.$

The vector field $X(p)$ is called smooth if all functions $X^{i}$ are smooth
in any chart in $M;$ see e.g. [1, 6] and [7]. We denote the set of all vector
fields on $M$ by $\Gamma(M).$

######  Definition 2.1 ([9, 13]).

For every $p\in M$ the differential map $df$ at $p$ is defined by

$df_{p}:T_{p}M\rightarrow
T_{f(p)}N~{}~{}\text{with}~{}~{}df_{p}(V)(h)=V(h\circ f),~{}~{}\forall V\in
T_{p}M,~{}~{}\forall h\in C^{\infty}(N).$

Locally, it is given by

$df_{p}(\frac{\partial}{\partial x_{j|_{p}}})=\sum_{k=1}^{n}\frac{\partial
f^{k}}{\partial x_{j|_{p}}}\frac{\partial}{\partial y^{k}},$

where $f=(f^{1},f^{2},\cdots,f^{n}).$ The matrix $\Big{(}\frac{\partial
f^{k}}{\partial x_{j}}\Big{)}_{k,j}$ is the Jacobian of $f$ with respect to
the charts $(x^{1},x^{2},\cdots,x^{n})$ and $(y^{1},y^{2},\cdots,y^{n})$ on
$M$ and $N$ respectively.

######  Definition 2.2.

Let $f\in C^{\infty}(M)$ be a smooth function. The gradient of $f,$ denoted by
$\text{grad}f,$ is a vector field on $M$ metrically equivalent to the
differential $df$ of $f:$

$g(\text{grad}f,X)=df(X)=X(f),~{}~{}\forall X\in\Gamma(M).$

######  Definition 2.3.

Let $X\in\Gamma(M)$ on $M.$ The divergence of $X$ at the point $p\in M$
denoted as $\partial X$ is defined locally as

$\partial X=\sum_{i=1}^{n}X^{i}_{;i}=\sum_{i=1}^{n}\Big{(}\frac{\partial
X^{i}}{\partial x_{i}}+\sum_{j}\Gamma_{ij}^{i}X^{j}\Big{)};$

where

$\Gamma_{jk}^{i}=\frac{1}{2}g^{il}\Big{(}\frac{\partial g_{jl}}{\partial
x_{k}}+\frac{\partial g_{kl}}{\partial x_{j}}-\frac{\partial g_{jk}}{\partial
x_{l}}\Big{)}$

is the Christoffel symbol. In local coordinates,

$\partial X=\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial
x_{j}}(\sqrt{|g|}X^{j})$

with summation over $j=1,\cdots,n.$

######  Definition 2.4.

The Lie bracket $[,]:\Gamma(M)\times\Gamma(M)\rightarrow\Gamma(M)$ is defined
by

$[X,Y]=X(Y)-Y(X)~{}~{}\forall X,Y\in\Gamma(M).$

Locally,

$[X,Y]=\sum_{i,j=1}^{n}\Big{(}\frac{\partial Y^{i}}{\partial
x_{j}}X^{j}-\frac{\partial X^{i}}{\partial
x_{j}}Y^{j}\Big{)}\frac{\partial}{\partial x_{i}}.$

If $[U,V]=0,$ we say that the vector fields commute.

An extension of the usual directional derivative on the Euclidean space to
smooth manifold is linear connection.

######  Definition 2.5.

A linear connection $\nabla$ on $M$ is a map
$\nabla:\Gamma(M)\times\Gamma(M)\rightarrow\Gamma(M)$ such that $\nabla_{X}Y$
is $C^{\infty}(M)$ in $X$ and linear in $Y$ over the real field with
$\nabla_{X}(fY)=(Xf)Y+f\nabla_{X}Y,~{}~{}\forall f\in C^{\infty}(M).$

We note that $\nabla_{X}Y$ is a new vector field which, roughly speaking, is
the vector rate of change of $Y$ in the direction of $X.$ A particular
connection on Riemannian manifolds that is torsion-free is the Levi-Civita
connection. The Levi-Civita connection is defined in local coordinates as

$\nabla_{X}Y=\sum_{i,k}^{n}X^{i}\Big{(}\frac{\partial Y^{k}}{\partial
x_{i}}+\sum_{j}\Gamma_{ij}^{k}W^{j}\Big{)}\frac{\partial}{\partial x_{k}},$

where

$X=\sum_{i=1}^{n}X^{i}\frac{\partial}{\partial
x_{i}},~{}~{}Y=\sum_{k=1}^{n}Y^{k}\frac{\partial}{\partial x_{k}}$

and $\Gamma_{ij}^{k}$ are the Christoffel symbols defined by

$\Gamma_{ij}^{k}=\frac{1}{2}\sum_{m}g^{km}\Big{(}\frac{\partial
g_{jm}}{\partial x_{i}}+\frac{\partial g_{im}}{\partial x_{j}}-\frac{\partial
g_{ij}}{\partial x_{m}}\Big{)}$

where $(g^{km})$ is the inverse of $(g_{ij}).$ Note that $X$ is a Killing
vector field if $\mathcal{L}_{X}g=0.$

Let $M$ be a compact $n$-dimensional Riemannian manifold. A vector field on
$M$ which generates isometries of the Riemannian metrics is represented by a
Killing vector field $v=v^{i}e_{i},$ where $e_{i}=\frac{\partial}{\partial
x_{i}}.$ A Killing vector satisfies the differential equation
$v_{i}|_{j}=-v_{j}|_{i}$ where solidus indicates covariant differentiation. In
particular, a Killing vector is divergence free with respect to the volume
density $\sqrt{|g|},$ that is,

$\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial
x_{i}}\Big{(}\sqrt{|g|}v^{i}\Big{)}=v^{i}|_{i}=0$

since a distance preserving map conserves volume automatically. Thus, a
Killing vector, in particular, is an isometric flow. We also need to clarify
the notion of tensor on $M.$

######  Definition 2.6.

A tensor of type $(r,s)$ at $p\in M$ is a multi-linear function

$T:(T_{p}^{\ast}M)^{r}\times(T_{p}M)^{s}\rightarrow\mathbb{R}.$

A tensor field $\mathcal{T}$ of type $(r,s)$ is a smooth map, which assigns to
each point $p\in M$ an $(r,s)$-tensor $\mathcal{T}_{p}$ on $M$ at the point
$p.$ In local coordinates,

$\mathcal{T}=\mathcal{T}_{j_{1}j_{2}\cdots j_{r}}^{i_{1}i_{2}\cdots
i_{s}}dx^{j_{1}}\otimes dx^{j_{2}}\otimes\cdots\otimes
dx^{j_{r}}\otimes\frac{\partial}{\partial
x_{i_{1}}}\otimes\frac{\partial}{\partial
x_{i_{2}}}\otimes\cdots\otimes\frac{\partial}{\partial x_{i_{s}}}.$

$\mathcal{T}$ acts on $r$ one-forms and $s$ vector fields thus

$\displaystyle\mathcal{T}(\omega_{1},\omega_{2},\cdots,\omega_{r},X_{1},X_{2},\cdots,X_{s})$
$\displaystyle=$ $\displaystyle\mathcal{T}_{j_{1}j_{2}\cdots
j_{r}}^{i_{1}i_{2}\cdots i_{s}}dx_{j_{1}}(X_{1})dx_{j_{2}}(X_{2})\cdots$
$\displaystyle dx_{j_{r}}(X_{r})\frac{\partial}{\partial
x_{i_{1}}}(\omega_{1})\frac{\partial}{\partial
x_{i_{2}}}(\omega_{2})\cdots\frac{\partial}{\partial x_{i_{s}}}(\omega_{s})$
$\displaystyle=$ $\displaystyle\mathcal{T}_{j_{1}j_{2}\cdots
j_{r}}^{i_{1}i_{2}\cdots i_{s}}X_{1}^{j_{1}}X_{2}^{j_{2}}\cdots
X_{r}^{j_{r}}\omega_{1}^{i_{1}}\omega_{2}^{i_{2}}\cdots\omega_{s}^{i_{s}}.$

We say the tensor $\mathcal{T}$ is $s$ covariant and $r$ contravariant.

Let $M$ be a compact $n$-dimensional manifold and let $\rho$ be a positive
scalar density on $M.$ A $p$-tensor

$\omega=\omega^{i_{1}\cdots i_{p}}e_{i_{1}}\wedge\cdots\wedge e_{i_{p}}$

is classically a skew-symmetric contravariant tensor of order $p;$ where
$e_{i}=\frac{\partial}{\partial x_{i}}$ are the vectors of the coordinate
frame (basis vectors). Let $T_{p}$ be the linear space of all $p$-tensors on
all of the manifold, $M.$ We can now define the divergence of a tensor field.

######  Definition 2.7.

The divergence of a $p$-tensor $\omega$ written $\partial\omega$ is the
$(p-1)$-tensor

$\partial\omega=\frac{1}{\rho}\frac{\partial}{\partial
x_{j}}\Big{(}\rho\omega^{i_{1}\cdots i_{p}}e_{i_{1}}\wedge\cdots\wedge
e_{i_{p}}\Big{)}.$

Let $V_{p}=\\{\omega\in T_{p}:\partial\omega=0\\}$ be the linear space of
divergence-free $p$-tensors and let the linear space of $p$-tensor divergences
be

$D_{p}=\\{\omega\in T_{p}:\omega=\partial\omega^{\prime}~{}~{}\text{for
some}~{}~{}\omega^{\prime}\in T_{p+1}\\}.$

For $p=0$ we have that the $p$-tensor is the ordinary scalar function. For
such functions $f,$ we have that $\partial f=0.$ An easy calculation show that
$\partial^{2}-\partial\partial=0,$ hence, $D_{p}$ is a linear subspace of
$V_{p}.$ We have the following preliminary results in form of lemmas.

######  Lemma 2.8.

Let $X\in\Gamma(M),$ $T$ be an $(n,0)$-tensor field and $dV$ be the volume
form on $M.$ Let $\mathcal{L}_{X}$ be the Lie derivative of $T,$ then
$\mathcal{L}_{X}dV=(\partial X)dV.$

###### Proof.

We recall that $T=dV=\sqrt{|g|}dx_{1}\wedge dx_{2}\wedge\cdots\wedge dx_{n}$
is an $(n,0)$-tensor field on $M.$ The Lie derivative $\mathcal{L}_{X}$ of $T$
given by $\mathcal{L}_{X}(T)=T_{12\cdots n}dx_{1}\wedge
dx_{2}\wedge\cdots\wedge dx_{n}$ is also an $(n,0)$-tensor or an $n$-form
$\mathcal{L}_{X}(T)=(\mathcal{L}_{X}T)_{12\cdots n}dx_{1}\wedge
dx_{2}\wedge\cdots\wedge dx_{n}.$ We need to show that
$(\mathcal{L}_{X}T)_{12\cdots n}=(\partial X)\sqrt{g}.$ Indeed, using the
formula which gives the components of the Lie derivative of a tensor, we have

$(\mathcal{L}_{X}T)_{12\cdots n}=\frac{\partial T_{12\cdots n}}{\partial
x_{i}}X^{i}+T^{j_{1}2\cdots n}\frac{\partial X^{1}}{\partial
x_{j_{1}}}+T^{2j_{2}\cdots n}\frac{\partial X^{2}}{\partial
x_{j_{2}}}+\cdots+T^{12\cdots j_{n}}\frac{\partial X^{n}}{\partial
x_{j_{n}}}.$

As $T_{12\cdots j_{p}\cdots n}=\delta_{p,j_{p}}T_{12\cdots p\cdots n},$ we get

$\displaystyle(\mathcal{L}_{X}T)_{12\cdots n}$ $\displaystyle=$
$\displaystyle\frac{\partial T_{12\cdots n}}{\partial x_{i}}X^{i}+T_{12\cdots
n}(\frac{\partial X^{1}}{\partial x_{1}}+\cdots+\frac{\partial X^{n}}{\partial
x_{n}})$ $\displaystyle=$ $\displaystyle\frac{\partial T_{12\cdots
n}}{\partial x_{i}}X^{i}+T_{12\cdots n}\frac{\partial X^{i}}{\partial x_{i}}$
$\displaystyle=$ $\displaystyle\frac{\sqrt{g}}{\partial
x_{i}}+\sqrt{g}\frac{\partial X^{i}}{\partial x_{i}}=\frac{\partial}{\partial
x_{{}_{i}}}\Big{(}\sqrt{g}X^{i}\Big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x_{i}}\Big{(}\sqrt{g}X^{i}\Big{)}\sqrt{g}=(\partial X\sqrt{g}.$

Hence, $\mathcal{L}_{X}T=\partial X\sqrt{g}dx_{1}\wedge
dx_{1}\wedge\cdots\wedge dx_{n}=\partial XdV.$ ∎

######  Lemma 2.9.

Let $f\in C^{\infty}(M)$ and $X\in\Gamma(M).$ Then $\partial(fX)=f\partial
X+g(\text{grad}f,X).$

###### Proof.

From the definition of the $\partial$ operator, we have

$\displaystyle\partial(fX)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x_{j}}\Big{(}\sqrt{g}fX^{j}\Big{)}\sqrt{g}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{g}}\frac{\partial f}{\partial
x_{j}}\sqrt{g}X^{j}+f\frac{1}{\sqrt{g}}\frac{\partial f}{\partial
x_{j}}(\sqrt{g}X^{j})$ $\displaystyle=$ $\displaystyle\frac{\partial
f}{\partial x_{j}}X^{j}+f\partial X$ $\displaystyle=$ $\displaystyle
g_{kj}(\text{grad}f)^{k}X^{j}+f\partial X$ $\displaystyle=$ $\displaystyle
g(\text{grad}f,X)+f\partial X.$

Using that $\mathcal{L}_{X}dV=(\partial X)dV~{}~{}\forall X\in\Gamma(M),$ the
result follows. ∎

We define the adjoint operator $\delta:\Omega^{k}\rightarrow\Omega^{k+1}$ of
$d$ by requiring that $\langle\omega,\delta\beta\rangle=\langle
d\omega,\beta\rangle,~{}\forall\omega\in\Omega^{k-1},~{}~{}\text{and}~{}~{}\beta\in\Omega^{k}$
using the $L^{2}$ inner product on $\Omega^{k}(M).$ From the metric $g,$ we
can also define the Hodge star operator
$\ast:\Omega^{k}\rightarrow\Omega^{n-k}$ by requiring that for all
$\beta,\gamma\in\Omega^{k},~{}~{}\beta\wedge\ast\gamma=(\beta,\gamma)dV.$
Notice that the Hodge star operator $\ast$ is linear and point-wise.
Therefore, the inner product on $\Omega^{k}$ can be written as
$\langle\beta,\gamma\rangle=\int_{M}\beta\wedge\ast\gamma.$ So, one can find
an expression for $\delta$ acting on one-forms. Given $f\in\Omega^{0}$ and
$\omega\in\Omega^{1},$ we have

$\displaystyle\langle f,\delta\omega\rangle$ $\displaystyle=$
$\displaystyle\langle df,\omega\rangle=\int_{M}df\wedge\ast\omega$
$\displaystyle=$ $\displaystyle\int_{M}[d(f\wedge\ast\omega)-f\wedge
d\ast\omega]$ $\displaystyle=$ $\displaystyle\int_{M}-f\wedge
d\ast\omega=-\int_{M}f\wedge\ast^{2}d\ast\omega$ $\displaystyle=$
$\displaystyle\int_{M}f\wedge\ast(-\ast d\ast)\omega=\langle f,(-\ast
d\ast)\omega\rangle.$

Now we can go on to construct vorticity through the curl operator on the
compact Riemannian manifold $M$ and study their flow vorticity.

##  3 Vorticity of flows

Let $X$ be a vector field on an open subset of $U\subset M$ and $\omega_{X}$
be the associated $1$-form on $U$ dual to $X.$ So, for each $p\in U,$ the
linear functional $\omega_{X}(p)\in T^{\ast}M$ on $T_{p}M$ is $\langle
X(p),\cdot\rangle.$ It follows that the assignment $X\mapsto\omega_{X}$ is
additive in $X$ and linear with respect to multiplication by smooth function
on $U.$ Let $V$ be a finite dimensional vector space with inner product
$\langle,\rangle,$ then the dual vector space $V^{\ast}$ is naturally
isomorphic to $V$ under the map

$\alpha:V\rightarrow V^{\ast},~{}~{}\text{with}~{}~{}\alpha(v)=v^{\ast}\in
V^{\ast},\text{satisfying}~{}~{}v^{\ast}(w)=\langle v,w\rangle~{}\forall
v,w\in V.$

These lead to define the curl operator taking vector fields to vector fields
to be

$\text{curl}X=\alpha^{-1}\ast d\alpha X=g^{-1}\ast dg(X).$

In coordinate form, the curl of a vector field $X$ on a Riemannian manifold
$M$ is a $2$-covariant antisymmetric tensor $A$ with the components $A_{ij}$
given by

$A_{ij}=X_{i;j}-X_{j;i}=\frac{\partial X_{i}}{\partial x_{j}}-\frac{\partial
X_{j}}{\partial x_{i}}.$

In particular, a Riemannian metric on a manifold $M$ is an assignment of inner
product on each cotangent space $T^{\ast}_{p}M$ under the isomorphism
$\alpha.$ The inner product $g$ induces an inner product on each of the tensor
product $T_{p}M\otimes\cdots\otimes T_{p}M.$

######  Definition 3.1.

Let $X\in\Gamma(U)$ with $U$ open in $M,$ the function $\text{div}(X)\in
C^{\infty}(U)$ is characterised by

$d(\ast(\omega_{X}))=\text{div}(X)dV_{M}\Big{|}_{U}.$

To see this, let $M=\mathbb{R}^{n}$ with the standard flat Riemannian metric
and orientation in the standard linear coordinates $\\{x_{1},\cdots,x_{n}\\}.$
If $X=\sum X_{j}\partial_{x_{j}}\in\Gamma(U)$ is a vector field, since the
volume form determined by this metric and orientation is
$dx_{1}\wedge\cdots\wedge dx_{n},$ we have

$\displaystyle d(\ast(\omega_{X}))$ $\displaystyle=$ $\displaystyle
d(\ast(\sum X_{j}dX_{j}))=d(\sum X_{j}\ast(dX_{j}))$ $\displaystyle=$
$\displaystyle
d(\sum(-1)^{j-1}X_{j}dx_{1}\wedge\cdots\wedge\widehat{dx}_{n}\wedge\cdots\wedge
dx_{n})$ $\displaystyle=$ $\displaystyle\sum(-1)^{j-1}dX_{j}\wedge
dx_{1}\wedge\cdots\wedge\widehat{dx}_{n}\wedge\cdots\wedge dx_{n}$
$\displaystyle=$ $\displaystyle\sum\Big{(}\frac{\partial X_{j}}{\partial
x_{j}}\Big{)}dx_{1}\wedge\cdots\wedge dx_{n}=\sum\Big{(}\frac{\partial
X_{j}}{\partial x_{j}}\Big{)}dV_{M}=\sum_{j=1}^{n}\frac{\partial
X_{j}}{\partial x_{j}}=\text{div}(X).$

Our definition of divergence of a vector field was intrinsic to the Riemannian
structure and orientation, thus, we can likewise compute the divergence in any
oriented coordinate system on $M.$ We make the next definition.

######  Definition 3.2.

Let $M$ be a Riemannian manifold with corners. For an open $U\subseteq M$ and
$f\in C^{\infty(U)},$ the smooth vector, $\text{grad}(f),$ on $U$ is defined
by the condition: $\omega_{\text{grad}(f)}=df\in\Omega_{M}^{1}(U).$ That is,
for each $p\in U,$ we have $\langle\text{grad}(f),\cdot\rangle=df(p)$ is a
linear functional on $T_{p}M.$

Now let $M$ be a Riemannian manifold without boundary and $N$ be an oriented
submanifold with boundary inside of $M$ with constant dimension $1.$ Let $N$
be given the induced Riemannian metric of $M.$ So the boundary $\partial N$ is
assigned a collection of signs $\mathcal{E}(p)\in\\{\pm 1\\}$ for each
$p\in\partial N$ where $\mathcal{E}$ is the usual Levi-Civita symbol defined
by

$\mathcal{E}_{a_{1},\cdots,a_{n}}=\begin{cases}+1,&\text{if}~{}~{}(a_{1},\cdots,a_{n})~{}~{}\text{is
an even permutation of }~{}~{}1,2,\cdots,n,\\\
-1,&\text{if}~{}~{}(a_{1},\cdots,a_{n})~{}~{}\text{is an odd permutation of
}~{}~{}1,2,\cdots,n,\\\ 0,&\text{if otherwise};\end{cases}$

see e.g. [3] and [6]. Let $dl$ be the length form on $N$ and $T$ be the
tangent field dual to $dl.$ It can be proved that for any $f\in C^{\infty}(M)$
for which $f\Big{|}_{N}\in C^{\infty}(N)$ is compactly supported, the smooth
inner product function $\langle\text{grad}(f)\Big{|}_{N},T\rangle$ is
compactly supported on $N$ and

$\int_{N}\langle\text{grad}(f)\Big{|}_{N},T\rangle dl=\sum_{p\in\partial
N}\mathcal{E}(p)f(p).$

In this way, we see that vector fields give rise to one-parameter groups of
point transformations of the the manifold and one may be interested in those
point transformations that are divergence-free. We call such vectors and their
associated transformations simply “flows”. The next lemma ensures the
existence of vector fields generating flows on $M.$

######  Lemma 3.3.

Let $V$ be a nonzero vector field at a point $p$ on the manifold $M.$ Then
there exists a system of coordinates
$(\bar{x}^{1},\bar{x}^{2},\cdots,\bar{x}^{n})$ about $p$ such that there is
$j\in\\{1,\cdots,n\\}$ for which $V=\frac{\partial}{\partial\bar{x}_{j}}.$ We
call this Rectification lemma.

###### Proof.

This lemma follows from the fact that in a compact Riemannian manifold, $M,$
if $p\in M$ there is an open neighbourhood $V$ of $p$ in the ambient manifold
$\mathbb{R}^{n+1}$ such that if $U\subset\mathbb{R}^{n}$ is open and
$\phi:U\rightarrow\mathbb{R}^{n+1}$ is smooth, then $\phi(U)$ is a
homomorphism. Besides, any transition Jacobian on $U$ for change of
coordinates has full rank for every $p\in U.$ This proves the lemma. ∎

Given a vector field $X,$ consider the system

$\frac{dc^{k}}{dt}=X^{k}(c(t)),~{}~{}k=1,2,\cdots,n,$ (3.1)

where $c(t)$ is the integral curve associated with $X.$ The next result shows
that the system (3.1) can be solved locally around the point $x_{0}=c(0),$ for
$0<t<\epsilon.$

######  Proposition 3.4 (Existence and uniqueness).

Given $x_{0}\in M$ and let $X$ be a nonzero vector field on an open set
$U\subset M$ of $x_{0},$ then there is $\epsilon>0$ such that the system (3.1)
has a unique solution $c:[0,\epsilon)\rightarrow U$ such that $c(0)=x_{0}.$

###### Proof.

By the rectification lemma, there is a local change of coordinates
$\bar{x}=\phi(x)$ such that the system (3.1) becomes

$\frac{dc^{k}}{dt}=\delta_{kn},~{}~{}k=1,2,\cdots,n~{}~{}\text{where}~{}~{}\bar{c}=\phi(c).$

This system has a unique solution through the point $\bar{x}_{0}=\phi(x_{0})$
given by

$\bar{c}^{k}(t)=\bar{x}^{k}_{0},~{}~{}k=1,2,\cdots,n-1~{}~{}\text{and}~{}~{}\bar{c}^{n}(t)=t+\bar{x}^{n}_{0}.$

Hence this will hold also for the system (3.1) in a neighbourhood of
$x_{0}=\phi^{-1}(\bar{x}_{0}).$ ∎

Let $f_{j}:=\frac{\partial f}{\partial x_{j}},$ and $f^{i}:=g^{ij}f_{j}$ so
that $\nabla f=f^{i}\frac{\partial}{\partial x}=g^{ij}\frac{\partial
f}{\partial x_{j}}\frac{\partial}{\partial x},$ we have more propositions that
will lead us to the main results of this work.

######  Proposition 3.5.

If $X\in\Gamma(M)$ then $X=\text{grad}\phi\iff\text{curl}X=0$ and
$tr(\text{curl}X)=0.$

###### Proof.

Let $X=\text{grad}\phi,$ then $X^{k}=g^{kj}\frac{\partial\phi}{\partial
x_{j}}$ and $X_{i}=\frac{\partial\phi}{\partial x_{i}}.$ So we have

$(\text{curl}X)_{ij}=\frac{\partial X_{i}}{\partial x_{j}}-\frac{\partial
X_{j}}{\partial x_{i}}=\frac{\partial^{2}\phi}{\partial x_{j}\partial
x_{i}}-\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}=0.$

Conversely, let $X\in\Gamma(M)$ such that $\text{curl}X=0.$ Then,
$\frac{\partial X_{i}}{\partial x_{j}}=\frac{\partial X_{j}}{\partial x_{i}}.$
Thus the $1$-form $\omega=\sum X_{k}dx_{k}$ is exact. This means there is a
locally defined function such that $\omega=df=\sum\frac{\partial f}{\partial
x_{k}}dx_{k}.$ Thus, $X_{k}=X^{j}.$ Besides,
$tr(\text{curl}X)=g^{ij}(X_{i;j}-X_{j;i})=X_{;j}^{j}-X_{;i}^{i}=0;$ which
completes the proof. ∎

Let $\langle,\rangle$ be the Riemannian metric with associated Levi-Civita
connection $\nabla.$ We prove global and invariant properties of the curl
operator on $M.$

######  Proposition 3.6.

If $A=\text{curl}X,$ then
$A(U,V)=\langle\nabla_{V}X,U\rangle-\langle\nabla_{U}X,V\rangle~{}~{}\forall
U,V\in\Gamma(M).$

###### Proof.

For every $U,V\in\Gamma(M),$ we have

$\displaystyle A(U,V)$ $\displaystyle=$ $\displaystyle
A_{ij}U^{i}V^{j}=(X_{i;j}-X_{j;i})U^{i}V^{j}=(\nabla_{\partial_{j}}X)_{i}U^{i}V^{j}-(\nabla_{\partial_{i}}X)_{j}U^{i}V^{j}$
$\displaystyle=$ $\displaystyle\langle\nabla_{\partial_{j}}X,U\rangle
V^{j}-\langle\nabla_{\partial_{i}}X,U\rangle
U^{i}=\langle\nabla_{V^{j}\partial_{j}}X,U\rangle-\langle\nabla_{U^{i}\partial_{j}}X,V\rangle$
$\displaystyle=$
$\displaystyle\langle\nabla_{V}X,U\rangle-\langle\nabla_{U}X,V\rangle.$

∎

######  Proposition 3.7.

Let $A=\text{curl}X,$ where $X\in\Gamma(M).$ Then

$A(U,V)=V\langle X,U\rangle-U\langle X,V\rangle+\langle X,[U,V]\rangle.$

###### Proof.

Since $\nabla$ is a metric connection

$V\langle X,U\rangle=\langle\nabla_{V}X,U\rangle+\langle
X,\nabla_{V}U\rangle~{}~{}\text{and}~{}~{}U\langle
X,V\rangle=\langle\nabla_{U}X,V\rangle+\langle X,\nabla_{U}V\rangle.$

Using the symmetry of $\nabla,$ subtracting we obtain

$V\langle X,U\rangle-U\langle X,V\rangle=A(U,V)+\langle X,[V,U]\rangle,$

which proves the claim. ∎

The following result shows the relation between the curl, Levi-Civita
connection and the Lie derivative.

######  Proposition 3.8.

If $A=\text{curl}X$ and $\nabla$ is the Levi-Civita connection on $M,$ then

$A(U,V)=2\langle\nabla_{V}X,U\rangle-(L_{X}g)(U,V).$

###### Proof.

We have

$2\langle\nabla_{V}X,U\rangle=V\langle X,U\rangle+X\langle U,V\rangle-U\langle
V,X\rangle-\langle V,[X,U]\rangle+\langle X,[U,V]\rangle+\langle
U,[V,X]\rangle.$

That is,

$\displaystyle 2\langle\nabla_{V}X,U\rangle$ $\displaystyle=$ $\displaystyle
A(U,V)+X\langle U,V\rangle-\langle V,[X,U]\rangle+\langle U,[V,X]\rangle$
$\displaystyle=$ $\displaystyle A(U,V)+X\langle U,V\rangle-\langle
V,L_{X}U\rangle-\langle U,L_{X}V\rangle.$

Using that $(L_{X}g)(U,V)=X\langle U,V\rangle-\langle L_{X}U,V\rangle-\langle
U,L_{X}V\rangle,$ we obtain

$2\langle\nabla_{V}X,U\rangle=A(U,V)+(L_{X}g)(U,V).$

∎

We can now show a Helmholtz decomposition of vector fields on $M.$ This is the
theorem that follows. That is, we prove that a vector field $X$ on a compact
Riemannian manifold can be uniquely decomposed as a sum of two vectors $Y$ and
$Z,$ where $Y$ is the rotation component and $Z$ the expansion component.

######  Theorem 3.9.

If $X\in\Gamma(M),$ there are two vector fields $Y$ and $Z$ on $M$ such that
$X=Y+Z,$ with $\text{div}Y=0$ and $\text{curl}Z=0.$ Moreover, the
decomposition is unique.

###### Proof.

Let $\eta=\text{div}X$ and let $\phi$ solve the elliptic equation
$\nabla\circ\nabla\phi=\Delta\phi=\eta.$ Take $Z=\nabla\phi$ and
$Y=X-\nabla\phi.$ Then $\text{curl}Z=\text{curl}\nabla\phi=0$ and
$\text{div}Y=\eta-\nabla\phi=0.$ This proves the existence of $Y$ and $Z.$

Now suppose two decompositions of $X$ so that $X=Y_{1}+Z_{1}=Y_{2}+Z_{2}.$ As
$\text{curl}Z_{1}=0,$ it follows that there are two functions
$\phi_{i},~{}i=1,2$ such that $Z_{i}=\nabla\phi_{i},~{}i=1,2.$
So, $Y_{2}-Y_{1}=\nabla(\phi_{2}-\phi_{1}).$

Denote $W=Y_{2}-Y_{1}$ and $\phi=\phi_{2}-\phi_{1},$ then
$\text{div}W=\text{div}\nabla\phi.$ Since $\text{div}Y_{2}-\text{div}Y_{1}=0$
we get $\Delta\phi=0,$ thus, $\phi_{2}-\phi_{1}$ must be constant. Taking the
gradient yields $Z_{2}-Z_{1}=0.$ Then we have also that $Y_{1}=Y_{2},$ hence
the decomposition is unique. ∎

For example, let
$X=(x_{1}-x_{2})\partial_{x_{1}}+(x_{1}+x_{2})\partial_{x_{2}}.$ Then the
Helmholtz decomposition is

$X=Y+Z~{}~{}\text{with}~{}~{}Y=x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}~{}~{}\text{and}~{}~{}Z=-x_{2}\partial_{x_{1}}+x_{1}\partial_{x_{2}}.$

To visualise this, we consider the $2$-dimensional unit sphere with a local
parametrisation

$\Phi:(-\frac{\pi}{2},\frac{\pi}{2})\rightarrow\mathbb{R}^{3}~{}~{}\text{with}~{}~{}\Phi(\theta,\phi)=\begin{pmatrix}\cos\theta\cos\phi\\\
\cos\theta\sin\phi\\\ \sin\theta\end{pmatrix}.$

The differentials $\frac{\partial\Phi}{\partial\theta}$ and
$\frac{\partial\Phi}{\partial\phi}$ give the Gramian
$\displaystyle{g=\Big{(}g_{ij}\Big{)}_{ij}=\begin{pmatrix}1&0\\\
0&\cos^{2}\theta\end{pmatrix}}$ which has the associated vector field flow
$X_{S^{2}}=(1,\cos^{2}\theta)$ and stream plot as Figure 1. Observe that
$X_{S^{2}}$ is divergence-free and that
$\text{curl}X_{S^{2}}=-2\cos\theta\sin\theta.$

Figure 1: Stream plot of the vector field $X_{S^{2}}$ on $S^{2}.$

By the Helmholtz decomposition,

$X_{S^{2}}=Y_{S^{2}}+Z_{S^{2}}~{}~{}\text{with}~{}~{}Y_{S^{2}}=X_{S^{2}}~{}~{}\text{and}~{}~{}Z_{S^{2}}=(-\cos^{2}\theta,1)~{}~{}\text{with}~{}~{}\text{curl}Z_{S^{2}}=0.$

The stream plot of the Helmholtz decomposed $X_{S^{2}}$ flow is Figure 2.

Figure 2: Stream plot of the decomposed $X_{S^{2}}.$

Finally, another main result of this work is the Stokes’ type result:

######  Theorem 3.10.

Let $X$ be smooth vector field on an open neighbourhood $U$ of $Z\subseteq M$
with $X\Big{|}_{Z}$ compactly supported and let $dA$ be the area form on $Z.$
Then the smooth function $\langle\text{curl}(X)\Big{|}_{Z},\hat{N}\rangle$ on
$Z,$ where $\hat{N}$ is the outward unit normal field along $Z$ in $M,$ is
compactly supported and satisfies the Stokes’ type identity

$\int_{Z}\langle\text{curl}(X)\Big{|}_{Z},\hat{N}\rangle dA=\int_{\partial
Z}\langle X\Big{|}_{\partial Z},T\rangle dl.$ (3.2)

###### Proof.

Let $\eta=\omega_{X}\in\Omega_{M}^{1}(U)$ be dual to $X.$ Thus
$\eta\Big{|}_{Z}\in\Omega_{Z}^{1}(Z)$ is compactly supported as it vanishes at
points where $X\Big{|}_{Z}$ vanishes. By the definition of the curl, we have
$\omega_{\text{curl}(X)}=\ast(d\eta).$ Since $\ast\circ\ast=(-1)^{r(n-r)}$ on
$r$-forms on an $n$\- dimensional manifold, for $n=3$ and $r=1,$ we have
$\ast\circ\ast=1$ on $1$-forms on $U,$ so
$d\eta=\ast(\omega_{\text{curl}(X)}).$ Similarly,

$\langle\text{curl}(X)\Big{|}_{Z},\hat{N}\rangle dA=(d\eta)\Big{|}_{Z}$

inside of $\Omega_{Z}^{2}(Z).$ But $(d\eta)\Big{|}_{Z}=d(\eta\Big{|}_{Z})$
since $d$ and pullback commute along the closed embedding of $Z$ into $M.$
This is compactly supported on $Z.$ For compactly supported $1$-form
$\eta\Big{|}_{Z}$ on the $2$\- dimensional manifold $Z$ with boundary, we get

$\int_{Z}\langle\text{curl}(X)\Big{|}_{Z},\hat{N}\rangle
dA=\int_{Z}(d\eta)\Big{|}_{Z}=\int_{Z}d(\eta\Big{|}_{Z})=\int_{\partial
Z}\eta\Big{|}_{\partial Z}$

showing (3.2).

Thus our problem comes to proving the identity $\eta\Big{|}_{\partial
Z}=\langle X\Big{|}_{\partial Z},T\rangle dl\in\Omega_{\partial
Z}^{1}(\partial Z).$ But since $\eta|_{Z}=\omega_{X|_{Z}}\in\Omega_{Z}^{1}(Z)$
it follows that $\omega_{X|_{Z}}=\omega_{X}Z$ pointwise.

Let $X|_{Z}=G$ and $G|_{\partial Z}=H,$ we want to prove that for any smooth
vector field $H$ along $\partial Z,$

$\langle H,T\rangle dl=\omega_{H}\in\Omega_{\partial Z}^{1}(\partial Z)$ (3.3)

generally for any $1$-dimensional Riemannian manifold $C$ with length form
$dl$ dual to the tangent field $T.$ Evaluating both sides of (3.3) at a point
$p\in C,$ we obtain a $1$-dimensional real vector space $V$ endowed with an
inner product. So, let $V=T_{p}C,$ if $v\in V$ with length for $\phi,$ and for
$t\in V$ the vector dual to $\phi$ we have $\langle v,t\rangle\phi=\langle
v,\cdot\rangle$ is in the dual space. It suffices to check this equality when
evaluating both sides on the basis $\\{t\\}.$ But since $\phi(t)=1$ by the
definition of $t,$ the result follows. ∎

##  4 Conclusion

We have constructed vorticity of vector field flows on compact smooth
Riemannian manifolds through differential operators mainly the curl and the
divergence operators. Many properties of vorticity on the manifolds were
established and we proved that a vector field on a compact Riemannian manifold
admits a unique Helmholtz decomposition. We also proved a Stokes’ type
identity for the curl operator on smooth tensor fields on $M.$

This study can be extended to applications of the central ideas of the paper
to physical flow problems in engineering and industry. One can as well study
the vorticity of climate variabilities on the earth surface using the
machinery developed in this paper. This will throw more light on the
mathematical analysis of climate change.

## Acknowledgement

The authors are grateful to the members of Research and Development Committee
of the Department of Mathematics and Statistics of the Alex Ekwueme Federal
University, Ndufu-Alike, Nigeria for their comments and suggestions that have
improved the readability of this paper. We also thank members of our families
for their cooperation while this research lasted. All errors are our
responsibility.

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11institutetext: Thüringer Landessternwarte, Sternwarte 5, D-07778 Tautenburg,
Germany 11email<EMAIL_ADDRESS>22institutetext: Leibniz-Institut für
Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany

# Gaia-supported re-discovery of a remarkable weak line quasar from a
variability and proper motion survey††thanks: Based on observations with the
2.2 m telescope of the German-Spanish Astronomical Center, Calar Alto, jointly
operated by the Max-Planck-Institut für Astronomie Heidelberg and the
Instituto de Astrofísica de Andalucía (CSIC).

Helmut Meusinger and Ralf-Dieter Scholz 1122

(Received June 28, 2022; accepted July 28, 2022)

We demonstrate that VPMS J170850.95+433223.7 is a weak line quasar (WLQ) which
is remarkable in several respects. It was already classified as a probable
quasar two decades ago, but with considerable uncertainty. The non-significant
proper motion and parallax from the Gaia early data release 3 have solidified
this assumption. Based on previously unpublished spectra, we show that VPMS
J170850.95+433223.7 is a WLQ at $z=2.345$ with immeasurably faint broad
emission lines in the rest-frame ultraviolet. A preliminary estimate suggests
that it hosts a supermassive black hole of $\sim 10^{9}\,M_{\odot}$ accreting
close to the Eddington limit, perhaps at the super-Eddington level. We
identify two absorber systems with blueward velocity offsets of $0.05c$ and
$0.1c$, which could represent high-velocity outflows, which are perhaps
related to the high accretion state of the quasar.

###### Key Words.:

Proper motions - Quasars: emission lines - Quasars: absorption lines -
Quasars: individual: VPMS J170850.95+433223.7

##  1 Introduction

The weakness of broad emission lines is the defining characteristics of a rare
class of high-luminosity active galactic nuclei (AGN) called weak-line quasars
(WLQs). They were first discovered about a quarter of a century ago (McDowell
et al., 1995; Fan et al., 1999). The Sloan Digital Sky Survey (SDSS; York et
al., 2000) has revealed the existence of a (still relatively small) population
of WLQs (e.g. Diamond-Stanic et al., 2009; Plotkin et al., 2010; Meusinger &
Balafkan, 2014). 111Following Diamond-Stanic et al. (2009), WLQs are defined
as quasars having equivalent widths (EWs) of either the Ly$\alpha$+N v
emission line complex or the C iv line below the $3\sigma$ threshold of the EW
distribution in the parent quasar sample, that is $\mbox{EW(Ly$\alpha$+{N
{v}})}<15.4$ Å (Diamond-Stanic et al., 2009) and $\mbox{EW({C {iv}})}<4.8$ Å
(Meusinger & Balafkan, 2014). A number of different scenarios have been
proposed, including, in particular, extremely high accretion rates, anemic
broad emission line regions, and a gravitational lensed accretion disk (AD).
Currently, the idea of a shielding gas component between the central X-ray
source and the broad emission line region appears particularly attractive
(e.g. Paul et al., 2022, and references therein).

The Tautenburg – Calar Alto Variability and (zero) Proper Motion Survey (VPMS)
is a quasar search project that is based on optical long-term variability and
non-detectable proper motions measured on a large number of imaging
observations with the Schmidt camera of the Tautenburg 2-m telescope (Scholz
et al., 1997; Meusinger & Brunzendorf, 2001; Meusinger et al., 2002). One of
the goals of VPMS was to search for quasars with odd spectra that might not be
picked out in colour-based quasar surveys. An example of this is the unusual
broad-absorption line (BAL) quasar VPMS J134246.24+284027.5 (Meusinger et al.,
2005). Here, we present another remarkable quasar discovered from this survey,
the WLQ VPMS J170850.95+433223.7 (hereafter VPMS J1708+4332). Both sources are
located in the SDSS footprint area, but were not targeted for spectroscopy by
SDSS.

VPMS J1708+4332 was classified as a high-priority quasar candidate in the VPMS
field around the globular cluster M 92. Follow-up observations yielded a
spectrum similar to that of a WLQ, but at first it could not be assigned with
reasonable certainty to a quasar. It was not detected as a radio source at the
flux level of the FIRST survey (Becker et al., 1995). Recently, the Gaia early
data release 3 (EDR3; Gaia Collaboration et al., 2021a) has provided data that
make a significant contribution to distinguishing between quasars and
foreground stars in the case of unclear spectra.

We present and interpret the spectra and the Gaia data in Sect. 2. In Sect. 3,
we discuss some properties of the quasar, Sect. 4 gives the conclusions. We
assume Lambda Cold Dark Matter ($\Lambda$CDM) cosmology with $H_{0}=73$ km s-1
Mpc-1, $\Omega_{\rm\Lambda}=0.73$, and $\Omega_{\rm M}=0.27$.

##  2 Observations and analysis of the spectra

VPMS J1708+4332 was observed with the focal reducer and faint object
spectrograph CAFOS at the 2.2-m telescope of the German-Spanish Astronomical
Centre (DSAZ) on Calar Alto, Spain, in July 1998. The grism B-400 was used
with a wavelength coverage of $3200-8000$ Å and a dispersion of 10 Å px-1 on
the SITe1d CCD. In a subsequent campaign in July 2004, VPMS J1708+4332 was
observed at higher resolution using the grisms B-200 and B-100, which have a
wavelength coverage of $3200-7000$ Å and $3200-5800$ Å and a dispersion of
$\sim 5$ Å px-1 and $\sim 2$ Å px-1, respectively. One B-200 spectrum and two
B-100 spectra were recorded, each with an exposure time of 1200 seconds. The
spectra were reduced using the ESO MIDAS data reduction package with standard
procedures including bias correction, flat-fielding, cosmic ray removal, sky
subtraction, wavelength calibration, and a rough flux calibration. The
wavelength calibration was done by means of calibration lamp spectra. An exact
flux calibration was not carried out because it was not considered absolutely
necessary at that time. Galactic foreground reddening was corrected adopting
$E(B-V)=0.014$ from Schlafly & Finkbeiner (2011) and the Milky Way reddening
curve from Pei (1992).

In a first attempt (Meusinger & Brunzendorf, 2001), we had identified the peak
in the B-400 spectrum of VPMS J1708+4332 with the Ly $\alpha$ \+ N v line
complex and thus estimated a redshift of $z\approx 2.39$ from the best fit of
the SDSS quasar composite spectrum template (Vanden Berk et al., 2001) to the
continuum (Fig. 1). Later, Richards et al. (2009) listed this source as a
quasar candidate with a photometric redshift $z_{\rm ph}=2.215$ based on the
SDSS magnitudes. The mid-infrared (mid-IR) colour indices $W1-W2=0.8$ and
$W2-W3=3.7$ from the Wide-field Infrared Survey Explorer (WISE; Wright et al.,
2010) are typical for quasars (Jarrett et al., 2017) and thus also seem to
support this assessment. But, nevertheless, the classification of VPMS
J1708+4332 as a quasar could not be considered to be certain especially
because the absorption features in our spectrum could not be convincingly
explained.

Figure 1: CAFOS B-400 spectrum of VPMS J1708+4332 (black) in arbitrary units.
For comparison, the arbitrarily scaled SDSS quasar composite spectrum (Vanden
Berk et al., 2001) is shown (magenta) redshifted into the observer frame with
$z=2.39$ (Meusinger & Brunzendorf, 2001).

We therefore also considered the alternative possibility that VPMS J1708+4332
might be a rare star type. If so, the spectrum most closely resembles that of
an O-type subdwarf (Jeffery et al., 2021): A relatively strong line is seen
close to the position of He ii $\lambda\,4686$, which, however, turns out to
be a double line in the B-100 spectrum. There are no hydrogen lines, perhaps
with the exception of a weak feature at 4863 Å, where hot subdwarfs show a
blend of H$\beta$ with a He i line. According to the scheme of Jeffery et al.
(2021), which is based on line ratios, we would have to classify the spectrum
as sdO3-4. Such stars are very hot ($T_{\rm eff}\ga 30\,000$ K). In the
optical and near-IR, the spectral energy distribution (SED) can be fitted by a
black body of such a high temperature only if strong reddening is invoked, for
instance $E(B-V)=0.75$ for a 35 000 K black body. The source of the reddening
could be related to the dusty component which is indicated by the above-
mentioned mid-IR excess. The distance to such a strongly reddened sdO star is
estimated to be $\sim 3$ kpc, which raises the question of whether this is
compatible with the zero-proper motion from VPMS.

With the availability of the data from the Gaia satellite, the question of
zero-proper motion (and parallax) can be answered on a more secure basis. Gaia
EDR3 (Gaia Collaboration et al., 2021a) lists a non-significant, slightly
negative parallax $Plx=-0.0648\pm 0.0725$ mas and a non-significant, zero
proper motion $pmRA=-0.076\pm 0.082$ mas/yr, $pmDE=+0.023\pm 0.090$ mas/yr.
Zero parallaxes and proper motions are exceptional among the objects with
$Plx<0.5$ mas and parallax errors $<0.1$ mas in the sky region around VPMS
J1708+4332 and only shown by known quasars or quasar candidates (see Fig. 2).
Considering objects with similar magnitudes ($Gmag=17.555\pm 0.5$ mag) within
20 arcmin from VPMS J1708+4332, we compared its various EDR3 astrometric
quality parameters, for example parallax and proper motion errors,
astrometric_sigma5d_max, astrometric_excess_noise, ipd_frac_multi_peak, ruwe,
astrometric_gof_al, and visibility_periods_used (cf. Table A.1 in Gaia
Collaboration et al., 2021b), with their typical regional values and found no
obvious outliers.

Figure 2: Proper motion (PM) vs. parallax (Plx) diagram from Gaia EDR3 for the
objects with parallax errors $<0.1$ mas within two degrees around VPMS
J1708+4332. Red frames, quasars from SDSS DR16 (Ahumada et al., 2020); blue
frames, quasar candidates from Richards et al. (2009); and blue asterisk, VPMS
J1708+4332. For clarity, error bars have only been plotted for VPMS
J1708+4332. The concentration of sources at log PM $\sim 0.75$ is due to the
globular cluster M 92.  Figure 3: CAFOS B-100 spectrum of VPMS J1708+4332
(arbitrary units). Identified absorption lines, from absorber systems at three
different redshifts $z$, are marked by vertical lines with colours
corresponding to $z$. Dashed vertical lines indicate common transitions, and
dotted lines mark unusual ones. NV and CIV denote the line doublets N v
$\lambda\lambda\,1238.8,1242.8$ and C iv $\lambda\lambda\,1548.2,1550.8$,
respectively. For comparison, the redshifted ($z=2.345$) and arbitrarily
scaled SDSS quasar composite spectrum (Vanden Berk et al., 2001) has been over
plotted (magenta).

Nevertheless, we decided to check the EDR3 proper motion of VPMS J1708+4332
with its previous position measurements made over a longer time baseline
before Gaia started to work. The software of Gudehus (2001) allows for a
weighted proper motion solution (with the parallax set to zero), using
positional data of different quality. Using it, we combined the very precise
positions from three Gaia data releases, DR1 (Gaia Collaboration et al.,
2016), DR2 (Gaia Collaboration et al., 2018), and EDR3 at epochs 2015.0,
2015.5, and 2016.0, with the less precise positions from the best optical
catalogues of ground-based observations. Two earlier epochs, from 2004 and
2005, are given in SDSS DR12 (Alam et al., 2015), whereas two intermediate
epochs (both from 2013) are listed in the First U.S. Naval Observatory Robotic
Astrometric Telescope Catalog (URAT1; Zacharias et al., 2015) and in Pan-
STARRS release 1 (PS1; Chambers & et al., 2017). Compared to the given Gaia
catalogue errors of our target ($<$0.3 mas in DR1 and $<$0.1 mas in DR2 and
EDR3), its URAT1 errors (11 mas) appeared to be realistic, whereas PS1
(1.9-3.5 mas) and especially SDSS (2-3 mas) errors seemed to be too small.
According to Tian et al. (2017), the typical positional precision in PS1 and
SDSS is 10 mas and 25 mas, respectively. Therefore, we assumed these larger
PS1 and SDSS positional errors in our weighted proper motion solution. The
resulting proper motion, $pmRA=-0.04\pm 0.18$ mas/yr, $pmDE=+0.24\pm 0.23$
mas/yr, was not significant and also did not change if even less precise
(errors of several 100 mas) positions measured on old Schmidt plates, with
epochs between 1954 and 1992, were included. We considered our confirmed zero
proper motion to support the zero parallax measured in EDR3 and hence
extragalactic distance of VPMS J1708+4332. In Gaia DR3 (Gaia Collaboration et
al., 2022), our target is listed as a quasar candidate with a probability of
being a quasar of $>0.999$ and with $z=2.445\pm 0.014$.

Figure 4: Magnitude-redshift diagram of WLQs for $1.5\leq z\leq 3.5$. Red
symbols: WLQ sample from Diamond-Stanic et al. (2009) (framed crosses); EW-
selected sub-sample ($\mbox{EW({C {iv}}})<4.8$ Å) from Meusinger & Balafkan
(2014) (squares); and VPMS J1708+4332 (asterisk). For comparison, the
distribution of all SDSS DR7 quasars from the Shen et al. (2011) catalogue is
marked by the median relation (thick blue curve) and the standard deviation
(hatched green area) .

Figure 3 shows the B-100 spectrum of VPMS J1708+4332 in the wavelength
interval in which narrow absorption lines were found.222Apart from the
slightly lower resolution, the B-200 spectrum (not shown here) looks very
similar. For the line identification, we used the table of transitions with
$\lambda\geq 1215$ Å seen in BAL quasars (Hall et al., 2002). We were able to
assign essentially all lines only after we dropped the assumption that they
all belong to the same $z$. The key to the line identification are the two
observed double lines at $\sim 4685$ Å and 4930 Å. The wavelength ratio of
their two line components is close to that of the C iv
$\lambda\lambda\,1548.2,1550.8$ doublet. The identification of these two
double lines with C iv results in $z_{\rm abs,1}=2.023$ and $z_{\rm
abs,2}=2.128$. At these redshifts, several other observed lines could be
assigned to common lines from the input table, particularly Ly $\alpha$,
VPMSJ1708 N v $\lambda\lambda\,1239,1243$, Si iv $\lambda\lambda\,1394,1403$,
and Al iii $\lambda\lambda\,1855,1863$. Finally, we assume a third absorber at
$z_{\rm abs,3}=2.345$, which explains the strong absorption feature at
$\lambda\sim 4150$ Å as being due to the N v doublet and the unidentified
lines at $\lambda\la 4050$ Å as being due to the Ly $\alpha$ forest. The
redshift $z_{\rm abs,3}$ is close to $z$ from the continuum fit (Fig. 1). We
simply set $z_{\rm abs,3}$ equal to the systemic redshift $z$, whose correct
value can probably best be determined by measuring the
$[{O\textsc{III}}]\lambda\lambda 4959,5007$ lines by IR spectroscopy in the
H-band.

We would like to note that WLQs tend to be relatively bright compared to the
typical SDSS quasars at the same redshift (Fig. 4; see also Meusinger &
Balafkan, 2014; Luo et al., 2015). With an i-band apparent magnitude of
$i=17.54$, VPMS J1708+4332 is one of the WLQs with the strongest excess
compared to the median relation.

##  3 Discussion

The spectrum of VPMS J1708+4332 is comparable to those of the two WLQs SDSS
J114153.34+021924.3 and SDSS J123743.08+630144.9 discussed by Shemmer et al.
(2010). With their exceptionally weak, and actually undetectable, UV emission
lines, these objects are placed at the low-end tail of the EW distribution of
broad emission lines, not only of the type 1 quasars, but also of the WLQs.

VPMS J1708+4332 is also noteworthy in relation to the slope of the SED. In
general, WLQs exhibit SEDs that are broadly consistent with the continuum of
normal quasars (e.g. Diamond-Stanic et al., 2009). Figure 5 shows the observed
SED of VPMS J1708+4332 based on photometric data from UV to mid-IR. In
addition to the magnitudes $U,B,V$ from VPMS, $u,g,r,i,z$ from SDSS, and
$W1,W2,W3,W4$ from WISE, the magnitudes $J$ and $H$ from the Two-Micron All-
Sky Survey (2MASS; Skrutskie et al., 2006) were used and an upper limit in the
near-UV band of the Galaxy Evolution Explorer (Galex; Morrissey et al., 2007).
A trend towards a steeper continuum of WLQs at rest-frame wavelengths $\ga
1500$ Å was noticed by Meusinger & Balafkan (2014). VPMS J1708+4332 appears
extreme in this respect (Fig. 5). The spectral slope between 2200 Å and 4000 Å
(rest frame) is $\alpha=-2.31$ ($F_{\lambda}\propto\lambda^{\alpha}$). For
comparison, we estimated a mean slope $\bar{\alpha}=-1.74\pm 0.35$ for the
same wavelength interval from the SDSS spectra of the 45 WLQs from Meusinger &
Balafkan (2014) with $0.7\leq z\leq 2.2$ and an equivalent width of the broad
Mg ii $\lambda\,2800$ line $<15$ Å. We note that the fluxes from the SDSS
magnitudes agree well with those from VPMS. Because the VPMS fluxes are
averaged over five decades, it is unlikely that the steep increase from near-
IR to optical (observer frame) is due to flux variations in the rest-frame UV.

Figure 5: The SED of VPMS J1708+4332 based on reddening corrected photometric
data from VPMS, SDSS, 2MASS, and WISE (symbols). The error bars are usually
smaller than the symbol size. The downward error is an upper limit from Galex
in the near-UV. The SWIRE template for bright quasars (Polletta et al., 2007)
has been over plotted (blue), redshifted into the observer frame, and scaled
to the observed SED in the near-IR.

In the standard model (Shakura & Sunyaev, 1973), the radial temperature
profile of the AD is $T(r)=T_{\ast}\,f(r)$, where $f(r)$ describes the radial
dependence. The temperature parameter $T_{\ast}$ (in K) is given by
$T_{\ast}=2.2\times 10^{9}\,M_{\bullet}^{-1/2}\,\dot{M}_{\bullet}^{1/4}$,
where $M_{\bullet}$ is the black hole (BH) mass in $M_{\odot}$ and
$\dot{M}_{\bullet}$ is the mass accretion rate in $M_{\odot}$ yr-1 (e.g.
Pereyra et al., 2006). A steep slope of the SED in the UV indicates a high
$T_{\ast}$ and thus, at a given mass, a high accretion rate. Because it is not
possible to determine the virial BH mass from the available spectra, we
tentatively estimated $M_{\bullet}$ and $\dot{M}_{\bullet}$ from $T_{\ast}$
and the luminosity $L$. The combination of the above equation for $T_{\ast}$
with the relation between $\dot{M}_{\bullet}$, $M_{\bullet}$, and the optical
luminosity from Davis & Laor (2011, their Eq. 8) results in
$M_{\bullet}=2.9\times 10^{9}\,(L_{\rm
opt,45}/\cos\theta)^{0.52}\,T_{\ast}^{-0.25}$ and
$\dot{M}_{\bullet}=0.13\,(L_{\rm opt,45}/\cos\theta)^{1.04}\,T_{\ast}^{0.22}$,
where $L_{\rm opt,45}$ is the optical luminosity $\lambda L_{\lambda}$ at 4681
Å (rest frame) in units of $10^{45}$ erg s-1 and $\theta$ is the AD
inclination angle. The effect of the latter is primarily due to the
$\cos\theta$ dependence of the projected AD area. We have applied the Davis &
Laor (2011) relation here for two reasons. Firstly, because the optical
luminosity is relatively independent of the inner regions of the disk affected
by strong relativistic effects. Secondly, it allows us to compare the results
with the WLQ sample of Meusinger & Balafkan (2014). We estimate
$T_{\ast}=(1.3\pm 0.1)\times 10^{5}$ K from the $T_{\ast}-\alpha$ relation for
the multi-temperature black-body model of the AD and $L_{\rm opt}=(2.4\pm
0.6)\times 10^{46}$ erg s-1 from the observed flux density (Fig. 5). This
yields $M_{\bullet}=9.1\times 10^{8}\,M_{\odot}$ and
$\dot{M}_{\bullet}=57\,M_{\odot}\,\mbox{yr}^{-1}$ for a mean disk inclination
$\cos\theta=0.8$. For the range $\theta=0\degr...60\degr$ usually assumed for
quasars, one finds $M_{\bullet}=(8.1...11.6)\times 10^{8}\,M_{\odot}$ and
$\dot{M}_{\bullet}=45...94\,M_{\odot}\,\mbox{yr}^{-1}$ (higher values at
larger $\theta$). The uncertainties in $L_{\rm opt},T_{\ast},$ and $z$ result
in uncertainties of 14% for $M_{\bullet}$ and 26% for $\dot{M}_{\bullet}$.

The accretion state of quasars is usually expressed by the Eddington ratio
$\varepsilon=L_{\rm bol}/L_{\rm Edd}$, where the Eddington luminosity $L_{\rm
Edd}$ is determined by $M_{\bullet}$. We calculated the bolometric luminosity
$L_{\rm bol}$ from the monochromatic luminosities $L_{\lambda}$ at 1450 Å and
3000 Å, respectively, using bolometric corrections $\zeta_{1450}$ and
$\zeta_{\rm 3000}$ listed by Runnoe et al. (2012). The mean value of the
resulting Eddington ratio for $\cos\theta=0.8$ (Table 3) is $\epsilon=2.0$,
with an estimated uncertainty of 70%. For $\theta=60\degr$ and $0\degr$, we
find $\epsilon=1.66$ and 2.25, respectively. VPMS J1708+4332 thus belongs to
the upper end of the Eddington ratio distribution of both WLQs (Shemmer &
Lieber, 2015; Meusinger & Balafkan, 2014) and type 1 quasars (Kelly & Shen,
2013).

Table 1: Eddington ratio $\varepsilon$ for $\cos\theta=0.8$ and bolometric
corrections $\zeta$ at 1450 Å and 3000 Å from different sources.

Reference | $\varepsilon\,(\zeta_{1450})$ | $\varepsilon\,(\zeta_{3000})$
---|---|---
Elvis et al. (1994)aa$a$Taken from Runnoe et al. (2012), their table 2.bb$b$Recalculated by Runnoe et al. (2012), | 1.92 | 1.54
Richards et al. (2006)aa$a$Taken from Runnoe et al. (2012), their table 2.bb$b$Recalculated by Runnoe et al. (2012), | 1.42 | 1.25
Nemmen & Brotherton (2010)aa$a$Taken from Runnoe et al. (2012), their table 2. | 1.83 | 2.38
Runnoe et al. (2012)aa$a$Taken from Runnoe et al. (2012), their table 2. | 2.56 | 2.09
Runnoe et al. (2012)ccfootnotemark: $c$ | 2.06 | 3.32
333 $b$$b$footnotetext: and from their eq. 9.

A more general approach also involves the BH spin parameter $a$. Campitiello
et al. (2018) presented analytic approximations of the AD emission for
rotating BHs. Their $L(a,\theta)-\dot{M}_{\bullet}$ relation (their Eq. B3)
confirms a high accretion rate ($\sim 30M_{\odot}\,\mbox{yr}^{-1}$ for $a\ga
0.9$ and $60...110M_{\odot}\,\mbox{yr}^{-1}$ for $a\approx 0.1$) for all
$\theta\leq 60\degr$. In particular, they describe the dependence of the peak
frequency and the peak luminosity of the AD spectrum on
$M_{\bullet},\dot{M}_{\bullet},\theta$, and $a$ for $\theta=0\degr$. We used
their Eqs. 12-14 to calculate $M_{\bullet}$ and $\dot{M}_{\bullet}$ as a
function of $a$, assuming that the observed peak of the SED (Fig. 5) is not
primarily caused by the Ly$\alpha$ forest. The result is
$(M_{\bullet}[10^{9}M_{\odot}],\dot{M}_{\bullet}[M_{\odot}\,\mbox{yr}^{-1}])=(4.0,36)$
for $a=0$ and $(4.6,12)$ for $a=0.998$. There is no evidence for a pole-on
view of VPMS J1708+4332, but the effect of the inclination is only moderate
for $\theta\la 60\degr$ (e.g. a factor $<2$ for $L$; see Campitiello et al.,
2018, their Figs. B.1 and B.2). Therefore, we can thus assume that this result
reflects the correct order of magnitude and does not contradict the above
conclusion of a high Eddington ratio. It should be mentioned that $\epsilon\ga
0.3$ violates the limit for a thin disk and could thus indicate a slim or
thick AD (Abramowicz et al., 1988), whose vertical structure must be taken
into account by numerical models. Such an inner AD structure has been
suggested to be related to a gas component that may shield the broad line
region from the ionising continuum (Luo et al., 2015).

Another remarkable property of VPMS J1708+4332 is the blueshift of the two
narrow absorption line systems at $z_{\rm abs,1}$ and $z_{\rm abs,2}$. If they
are intrinsic, that is not from intervening systems, the wavelength shifts
relative to the adopted systemic redshift indicate outflows from the central
engine with velocities of $v\ga 0.05c$ and $\ga 0.1c$,
respectively.444$v/c=(R^{2}-1)/(R^{2}+1)$, where $R=(1+z)/(1+z_{\rm abs})$

##  4 Conclusions

Supported by the data from Gaia DR3, we have identified VPMS J1708+4332 as a
WLQ at $z=2.345$ with undetectable weak UV emission lines. We tentatively
estimate a large BH mass and a high accretion rate close to the Eddington
limit, which probably indicates a rapid BH growth phase. We identified two
narrow absorption line systems with blueshifts which, if intrinsic, represent
outflows with moderately high velocities of $\sim 0.05c$ and $\sim 0.1c$. This
outflow may be related to the high accretion state. Near-IR spectroscopy could
be helpful to estimate the virial BH mass based on the H$\beta$ line (e.g.
McDowell et al., 1995; Shemmer et al., 2010). VPMS J1708+4332 provides another
example of unusual AGN that may be under-represented in colour-based quasar
surveys.

###### Acknowledgements.

We thank the referee for useful comments and suggestions. We are grateful to
the staff of the Calar Alto observatory for their kind support. H. M.
acknowledges financial support from the Deutsche Forschungsgemeinschaft under
grants Me1359/3 and Me1350/8.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of data products from the Sloan Digital Sky Survey (SDSS). Funding for the
SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions (see below), the National Science Foundation, the
National Aeronautics and Space Administration, the U.S. Department of Energy,
the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education
Funding Council for England. The SDSS Web site is http://www.sdss.org/. The
SDSS is managed by the Astrophysical Research Consortium (ARC) for the
Participating Institutions. The Participating Institutions are: the American
Museum of Natural History, Astrophysical Institute Potsdam, University of
Basel, University of Cambridge (Cambridge University), Case Western Reserve
University, the University of Chicago, the Fermi National Accelerator
Laboratory (Fermilab), the Institute for Advanced Study, the Japan
Participation Group, the Johns Hopkins University, the Joint Institute for
Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and
Cosmology, the Korean Scientist Group, the Los Alamos National Laboratory, the
Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for
Astrophysics (MPA), the New Mexico State University, the Ohio State
University, the University of Pittsburgh, University of Portsmouth, Princeton
University, the United States Naval Observatory, and the University of
Washington.
This publication has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France, and of the NASA/IPAC Infrared Science Archive (IRSA),
operated by the Jet Propulsion Laboratories/California Institute of
Technology, founded by the National Aeronautic and Space Administration. In
particular, this publication makes use of data products from the Wide-field
Infrared Survey Explorer, which is a joint project of the University of
California, Los Angeles, and the Jet Propulsion Laboratory/California
Institute of Technology, funded by the National Aeronautics and Space
Administration. In addition, we used data products from the Two Micron All Sky
Survey, which is a joint project of the University of Massachusetts and the
Infrared Processing and Analysis Center/California Institute of Technology,
funded by the National Aeronautics and Space Administration and the National
Science Foundation. We also used observations made with the NASA Galaxy
Evolution Explorer, GALEX, which is operated for NASA by the California
Institute of Technology under NASA contract NAS5-98034.

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