Vorticity Gramian of compact Riemannian manifolds Louis Omenyi1 2 iD1 Emmanuel Nwaeze1 iD2 Friday Oyakhire1 iD3 and Monday Ekhator1 iD4

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Vorticity Gramian of compact Riemannian manifolds

Louis Omenyi1, 2 , ∗,iD1, Emmanuel Nwaeze1,iD2, Friday Oyakhire1, iD3, and Monday

Ekhator1, iD4

1Department of Mathematics and Statistics, Alex Ekwueme Federal University,

Ndufu-Alike, Nigeria (∗Corresponding author E-mail: omenyi.louis@funai.edu.ng)

Corresponding author ORCID: 0000-0002-8628-0298

2Department of Mathematical Sciences, Loughborough University, Loughborough,

Leicestershire, United Kingdom

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

arXiv:2210.06552v1 [math.GM] 17 Sep 2022

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, Rn, 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¨ar

[4] and M¨uller [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 Mis a topological space that

is locally similar to the Euclidean space and gis the Riemannian metric on M. We recall that

a Riemannian metric gon a smooth manifold Mis a symmetric, positive deﬁnite (0,2)-tensor

ﬁeld, see e.g. [16,18,1] and [13]. This means that for any point p∈M, the metric is the map

gp:TpM×TpM→Rthat is a positive deﬁnite scalar product for a tangent space TpM. 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 Mis called compact if it is compact as a topological space.

If Mis a smooth manifold, then [7] and [1] proved that there is at least one Riemannian metric

on M.

We call a function f:M→Rsmooth if for every chart (U, φ) on M, and the function

f◦φ−1:φ(U)→Ris smooth. The set of all smooth functions on the manifold Mwill be denoted

by C∞(M).Let Ωkdenote the vector space of smooth k-forms on M, and let d: Ωk→Ωk+1 be

the exterior derivative. Note that the metric which gives an inner product on the tangent space

TpMat each p∈Minduces a natural metric on each cotangent space T∗

pM, as follows. At p,

let {b1, b2,··· , bn}be an orthonormal basis for the tangent space. One obtains a metric on the

cotangent space by declaring that the dual basis {b1, b2,··· , bn}is orthonormal. Hence given

any two k-forms βand γ, we have that (β, γ) is a function on M. We call (·,·) the pointwise

inner product; see e.g. [13,6,9,18] and [20]. For a coordinate chart on M,

(x1,··· , xn) : U→Rn,

we represent gby the Gram matrix (gij ) where gij =h∂

∂xi,∂

∂xji,and h,iis the inner product on

the tangent space. The volume form dV is deﬁned be b1∧b2∧ ··· ∧ bn,and it is a well-known

fact from linear algebra that dV =p|g|dx where dx =dx1∧ ··· ∧ dxn.Using the point-wise

inner product above, one writes the L2-inner product on Ωk(M) as

hβ, γi=ZM

(β, γ)dV ∀β, γ ∈Ωk(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 Mand analyse its many

properties.

A vector ﬁeld on Mis a family {X(p)}p∈Mof tangent vectors such that X(p)∈TpMfor

any p∈M. In local coordinates chart (x1,··· , xn),

X(p) = Xi(p)∂

∂xi|x=p

The vector ﬁeld X(p) is called smooth if all functions Xiare smooth in any chart in M; see e.g.

[1,6] and [7]. We denote the set of all vector ﬁelds on Mby Γ(M).

Deﬁnition 2.1 ([9,13]).For every p∈Mthe diﬀerential map df at pis deﬁned by

dfp:TpM→Tf(p)Nwith dfp(V)(h) = V(h◦f),∀V∈TpM, ∀h∈C∞(N).

Locally, it is given by

dfp(∂

∂xj|p

) =

k=1

∂fk

∂xj|p

∂

∂yk,

where f= (f1, f 2,··· , f n).The matrix ∂fk

∂xjk,j is the Jacobian of fwith respect to the charts

(x1, x2,··· , xn)and (y1, y2,··· , yn)on Mand Nrespectively.

Deﬁnition 2.2. Let f∈C∞(M)be a smooth function. The gradient of f, denoted by gradf, is

a vector ﬁeld on Mmetrically equivalent to the diﬀerential df of f:

g(gradf, X) = df(X) = X(f),∀X∈Γ(M).

Deﬁnition 2.3. Let X∈Γ(M)on M. The divergence of Xat the point p∈Mdenoted as ∂X

is deﬁned locally as

∂X =

i=1

;i=

i=1 ∂Xi

∂xi

Γi

ij Xj;

where

Γi

jk =1

2gil∂gjl

∂xk

+∂gkl

∂xj−∂gjk

∂xl

is the Christoﬀel symbol. In local coordinates,

∂X =1

p|g|

∂

∂xj

(p|g|Xj)

with summation over j= 1,··· , n.

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VorticityGramianofcompactRiemannianmanifoldsLouisOmenyi1,2,;iD1,EmmanuelNwaeze1,iD2,FridayOyakhire1,iD3,andMondayEkhator1,iD41DepartmentofMathematicsandStatistics,AlexEkwuemeFederalUniversity,Ndufu-Alike,Nigeria(CorrespondingauthorE-mail:omenyi.louis@funai.edu.ng)CorrespondingauthorORCID:0000-0002...

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Vorticity Gramian of compact Riemannian manifolds Louis Omenyi1 2 iD1 Emmanuel Nwaeze1 iD2 Friday Oyakhire1 iD3 and Monday Ekhator1 iD4

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