Ricci Vector Fields

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Ricci Vector Fields

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Submitted:

26 October 2023

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27 October 2023

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Abstract

We introduce a special vector field ω on a Riemannian manifold (N^m, g), such that the Lie derivative of the metric g with respect to ω is equal to ρRic, where Ric is the Ricci curvature of (N^m, g) and ρ is a smooth function on N^{m} and call this vector field a ρ-Ricci vector field. We use ρ-Ricci vector field on a Riemannian manifold (N^m, g) and find two characterizations of m-sphere S^m(α). In first result, we show that an m-dimensional compact and connected Riemannian manifold (N^m, g) with nonzero scalar curvature admits a ρ-Ricci vector field ω such that ρ is nonconstant function and the integral of Ric(ω,ω) has a suitable lower bound is necessary and sufficient for (N^m, g) to be isometric to m-sphere S^m(α). In second result, we show that an m-dimensional complete and simply connected Riemannian manifold (N^m, g) of positive scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nontrivial solution of Fischer-Marsden equation and the squared length of the covariant derivative of ω has an appropriate upper bound, if and only if, (N^m, g) to be isometric to m-sphere S^m(α).

Keywords:

Subject: Computer Science and Mathematics - Geometry and Topology

1. Introduction

An m-dimensional complete simply connected Riemannian manifold of constant curvature

α

is isometric to one of the spaces the m-sphere

S^{m} (α)

, the Euclidean space

R^{m}

or the hyperbolic space

H^{m} (α)

according as

α > 0

α = 0

α < 0

respectively (cf. [2]). Since, this classification, there has been an interest in obtaining necessary and sufficient conditions on complete Riemannian manifolds so that they are isometric to one of the three model spaces

S^{m} (α)

R^{m}

and

H^{m} (α)

respectively. In that one of most sought questions is in obtaining different characterizations of spheres

S^{m} (α)

among complete Riemannian manifolds. In obtaining these characterizations most of the times conformal and Killing vector fields are used on an m-dimensional complete Riemannian manifold

(N^{m}, g)

(cf. [1,4,5,6,7,8,9,10,11,14,15]). A vector field

u

on m-Riemannian manifold

(N^{m}, g)

is a conformal vector field if the Lie derivative

£_{u} g

has expression

£_{u} g = 2 f g,

where f is a smooth function called the conformal factor. If

f = 0

in above definition, then

u

is called a Killing vector field.

In this paper, we are interested in a vector field

ω

on an m-dimensional Riemannian manifold

(N^{m}, g)

that satisfies

\frac{1}{2} £_{ω} g = ρ R i c,

(1.1)

where

£_{ω} g

is the Lie-derivative of the metric g with respect to

ω

ρ

is a smooth function and

R i c

is the Ricci tensor of

(N^{m}, g)

. We call

ω

satisfying equation (1.1) a

ρ

-Ricci vector field on

(N^{m}, g)

. Naturally, if

(N^{m}, g)

is an Einstein manifold, then a

ρ

-Ricci vector field

ω

is a conformal vector field on

(N^{m}, g)

(cf. [4,5,6,7,8,9]). If in the equation (1.1), we take

ρ = 0

, then 0-Ricci vector field

ω

(N^{m}, g)

is a Killing vector field on

(N^{m}, g)

(cf. [10]). A

ρ

-Ricci vector field on

(N^{m}, g)

is also a particular form of potential field of a generalized soliton (cf. [12]), with

α = - ρ

and

β = γ = 0

We could also approach to equation (1.1) in other context (cf. [3]). On the m-dimensional Riemannian manifold

(N^{m}, g)

, take a smooth function

ρ

and consider 1-parameter family of metrics

g (t)

satisfying generalized Ricci flow (or

ρ

-Ricci flow) equation

\partial_{t} g = 2 ρ R i c, g (0) = g .

(1.2)

To reach a solution of above flow, we take a 1-parameter family of diffeormorphisms

φ_{t} : N^{m} \to N^{m}

generated by the family of vector fields

W (t)

and

σ (t)

be a scale factor. Then we are interested in a solution of flow (1.2) of the form

g (t) = σ (t) φ_{t}^{*} (g) .

Differentiating above equation with respect to t and substituting

t = 0

, while assuming

σ (0) = 1

\dot{σ} (0) = 0

W (0) = ω

and using

φ_{0} = i d

, we get

£_{ω} g - 2 ρ R i c = 0,

which is equation (1.1). Thus, a

ρ

-Ricci vector field

ω

(N^{m}, g)

can be considered as stable solution of the flow (1.2).

We see that as a trivial example on the Euclidean space

R^{m}

, a constant vector field

a

is a

ρ

-Ricci vector field for any smooth function

ρ

R^{m}

. Similarly on the complex Euclidean space

C^{m}

with complex structure J and the vector field

ξ = \sum_{i = 1}^{m} z^{i} \frac{\partial}{\partial z^{i}},

where

z^{1}, . .,, z^{m}

are Euclidean coordinates, the vector field

ω = J ξ

is a

ρ

-Ricci vector field for any smooth function

ρ

C^{m}

Next, we show that on the sphere

S^{m} (α)

of constant curvature

α

, there are many

ρ

-Ricci vector fields. With the imbedding

i : S^{m} (α) \to R^{m + 1}

and unit normal

ξ

and shape operator

- \sqrt{α} I

, on taking a nonzero constant vector field

b

on the Euclidean space

R^{m + 1}

, we have

b = ω + f ξ

, where

f = 〈b, ξ〉

and

ω

is the tangential component of

b

to the sphere

S^{m} (α)

. Denote the induced metric on the sphere

S^{m} (α)

by g and the Riemannian connection by D. Then differentiating above equation with respect to the vector field X on

S^{m} (α)

, we have

D_{X} ω = - \sqrt{α} f X, \nabla f = \sqrt{α} ω,

(1.3)

where

\nabla f

is the gradient of f. Using the first equation in (1.3), it follows that

£_{ω} g = - 2 \sqrt{α} f g

and the Ricci tensor of the sphere

S^{m} (α)

is given by

R i c = (m - 1) α g .

Thus, we see that the vector field

ω

on the sphere

S^{m} (α)

satisfies

\frac{1}{2} £_{ω} g = ρ R i c, ρ = - \frac{1}{(m - 1) \sqrt{α}} f,

(1.4)

that is,

ω

is a

ρ

-Ricci vector field on the sphere

S^{m} (α)

. Indeed. for each nonzero constant vector field on the Euclidean space

R^{m + 1}

, there is a

ρ

-Ricci vector field on the sphere

S^{m} (α)

Above example naturally leads to a question: Under what conditions a compact and connected m-dimensional Riemannian manifold

(N^{m}, g)

admitting a

ρ

-Ricci vector field

ω

is isometric to a m-sphere

S^{m} (α)

There are two well known differential equations on a Riemannian manifold

(N^{m}, g)

, the first is Obata’s differential equation namely (cf. [14,15]),

H e s s (σ) = - α σ g,

(1.5)

where

σ

is a non-constant smooth function,

α

is a positive constant and

H e s s (σ)

is the Hessian of

σ

defined by

H e s s (σ) (X, Y) = g (D_{X} \nabla σ, Y),

for smooth vector fields

X, Y

N^{m}

. Obata proved that a necessary and sufficient condition for a complete and simply connected Riemannian manifold

(N^{m}, g)

to admit a nontrivial solution of differential equation (1.5) is that

(N^{m}, g)

is isometric to the sphere

S^{m} (α)

(cf. [14,15]). The other differential equation on

(N^{m}, g)

is Fischer-Marsden equation (cf. [13])

(Δ σ) g + σ R i c = H e s s (σ),

(1.6)

where

σ

is a smooth function on

N^{m}

and

Δ σ = d i v (\nabla σ)

is the Laplacian of

σ

. We shall use the abbreviation for the above Fischer-Marsden equation as FM-equation. Taking trace in the FM-equation (1.6), we get

Δ σ = - \frac{τ}{m - 1} σ,

(1.7)

where

τ = T r R i c

is the scalar curvature of the Riemannian manifold

(N^{m}, g)

. It is known that if

(N^{m}, g)

admits a nontrivial solution of FM-equation, then the scalar curvature

τ

is necessarily constant (cf. [13]).

Note that by equation (1.3), the smooth function f on the sphere

S^{m} (α)

has Hessian

H e s s (f) (X, Y) = g (D_{X} \nabla f, Y) = \sqrt{α} g (D_{X} ω, Y) = - α f g (X, Y),

the Laplacian

Δ f = d i v (\sqrt{α} ω) = - m α f

and

R i c = (m - 1) α g

. Consequently, on

S^{m} (α)

, we see that

(Δ f) g + f R i c = H e s s (f),

(1.7)

that is, f is a solution of FM-equation on the sphere

S^{m} (α)

. If we combine the two, namely a Riemannian manifold

(N^{m}, g)

admits a

ρ

-Ricci vector field

ω

such that

ρ

is a nontrivial solution of the FM-equation on

(N^{m}, g)

and seek additional condition under which

(N^{m}, g)

is isometric to

S^{m} (α)

? Notice that the

ρ

-Ricci vector field

ω

on the sphere

S^{m} (α)

is a closed vector field. Therefore, in this paper, we use the closed

ρ

-Ricci vector field

ω

on a Riemannian manifold

(N^{m}, g)

and answer these two question in section-3, where we find two characterizations of the sphere

S^{m} (α)

2. Preliminaries

Let

ω

be a closed

ρ

-Ricci vector field on an m-dimensional Riemannian manifold

(N^{m}, g)

. If

β

is the 1-form dual to

ω

, that is,

β (X) = g (ω, X), X \in Θ (T N^{m}),

(2.1)

where

Θ (T N^{m})

is the space of smooth sections of the tangent bundle

T N^{m}

, then we have

d β = 0

. We denote by

\nabla_{X}

the covariant derivative operator with respect to the Riemannian connection on

(N^{m}, g)

and notice that for the closed

ρ

-Ricci vector field

ω

, we have

\begin{matrix} 2 g (\nabla_{X} ω, Y) & = & g (\nabla_{X} ω, Y) + g (\nabla_{Y} ω, X) + g (\nabla_{X} ω, Y) - g (\nabla_{Y} ω, X) \\ = & (£_{ω} g) (X, Y) + d β (X, Y) = 2 ρ R i c (X, Y) . \end{matrix}

Thus, for a closed

ρ

-Ricci vector field

ω

, we have

\nabla_{X} ω = ρ T X, X \in Θ (T N^{m}),

(2.2)

where T is a symmetric operator called Ricci operator given by

R i c (X, Y) = g (T X, Y) .

Using the expression for the curvature tensor field R of

(N^{m}, g)

R (X, Y) Z = [\nabla_{X}, \nabla_{Y}] Z - \nabla_{[X . Y]} Z, X, Y, Z \in Θ (T N^{m}),

and equation (2.2), we get

R (X, Y) ω = X (ρ) T Y - Y (ρ) T X + ρ ((\nabla_{X} T) (Y) - (\nabla_{Y} T) (X)),

(2.3)

X, Y \in Θ (T N^{m})

, where

(\nabla_{X} T) (Y) = \nabla_{X} T Y - T (\nabla_{X} Y)

. The scalar curvature

τ

(N^{m}, g)

is given by

τ = T r T

, where

T r T

is the trace of the symmetric operator T. Choosing a local frame

\{F_{1}, . ., F_{m}\}

and using the definition of the Ricci tensor

R i c

R i c (X, Y) = \sum_{j = 1}^{m} g (R (F_{j}, X) Y, F_{j}),

together with equation (1.3), we conclude that

R i c (Y, ω) = R i c (Y, \nabla ρ) - τ Y (ρ) + ρ g (Y, \sum_{j = 1}^{m} (\nabla_{F_{j}} T) (F_{j})) - ρ Y (τ),

(2.4)

where

\nabla ρ

is the gradient of

ρ

. It is known that the gradient of scalar curvature

τ

satisfies (cf. [2])

\frac{1}{2} \nabla τ = \sum_{j = 1}^{m} (\nabla_{F_{j}} T) (F_{j}) .

(2.5)

Consequently, equation (2.4) takes the form

R i c (Y, ω) = R i c (Y, \nabla ρ) - τ Y (ρ) - \frac{1}{2} ρ Y (τ)

(2.6)

and we have

T (ω) = T (\nabla ρ) - τ \nabla ρ - \frac{1}{2} ρ \nabla τ .

(2.7)

3. Characterizing spheres via $ρ$ -Ricci fields

Let

ω

be a closed

ρ

-Ricci vector field on an m-dimensional Riemannian manifold

(N^{m}, g)

. We shall use

ρ

-Ricci vector field and find two characterizations of m-sphere

S^{m} (α)

. In our first result, we prove the following result:

Theorem 1.

A closed ρ-Ricci vector field ω on an m-dimensional compact and connected Riemannian manifold

(N^{m}, g)

m > 2

with scalar curvature

τ \neq 0

and nonzero nonconstant function ρ satisfies

\int_{M} R i c (ω, ω) \geq \frac{m - 1}{m} \int_{M} {(d i v ω)}^{2},

if and only if, τ is a positive constant

m (m - 1) α

, and

(N^{m}, g)

is isometric to

S^{m} (α)

Proof.

Let

(N^{m}, g)

be an m-dimensional compact and connected Riemannian manifold,

m > 2

with scalar scalar curvature

τ \neq 0

and

ω

be a closed

ρ

-Ricci vector field defined on

(N^{m}, g)

with nonzero and nonconstant function

ρ

satisfying

\int_{M} R i c (ω, ω) \geq \frac{m - 1}{m} \int_{M} {(d i v ω)}^{2} .

(3.1)

Then using equation (2.2), we have

d i v ω = ρ τ .

(3.2)

Choosing a local orthonormal frame

\{F_{1}, . ., F_{m}\}

and using

{∥T∥}^{2} = \sum_{j = 1}^{m} g (T F_{j}, T F_{j})

and an outcome of equation (2.2) as

(£_{ω} g) (X, Y) = 2 ρ g (T X, Y), X, Y \in Θ (T N^{m}),

we conclude

\frac{1}{2} {|£_{ω} g|}^{2} = 2 ρ^{2} {∥T∥}^{2} .

(3.3)

Note that, we have

\begin{matrix} {∥T - \frac{τ}{m} I∥}^{2} & = & \sum_{j = 1}^{m} g ((T E_{j} - \frac{τ}{m} E_{j}), (T E_{j} - \frac{τ}{m} E_{j})) \\ = & {∥T∥}^{2} + \frac{1}{m} τ^{2} - 2 \sum_{j = 1}^{m} g (T E_{j}, \frac{τ}{m} E_{j}), \end{matrix}

that is,

{∥T - \frac{τ}{m} I∥}^{2} = {∥T∥}^{2} - \frac{1}{m} τ^{2} .

(3.4)

Now, using equation (2.2), we have

ρ (T X - \frac{τ}{m} X) = (\nabla_{X} ω - \frac{τ}{m} ρ X),

which in view of a local frame

\{F_{1}, . ., F_{m}\}

(N^{m}, g)

implies

\begin{matrix} ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} & = & \sum_{j = 1}^{m} g (ρ (T E_{j} - \frac{τ}{m} E_{j}), ρ (T E_{j} - \frac{τ}{m} E_{j})) \\ = & \sum_{j = 1}^{m} g (\nabla_{E_{j}} ω - \frac{τ}{m} ρ E_{j}, \nabla_{E_{j}} ω - \frac{τ}{m} ρ E_{j}) \\ = & {∥\nabla ω∥}^{2} + \frac{1}{m} τ^{2} ρ^{2} - \frac{2}{m} τ ρ d i v ω . \end{matrix}

Using (3.2), in above equation, yields

ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} = {∥\nabla ω∥}^{2} - \frac{1}{m} τ^{2} ρ^{2},

which on integration gives

\int_{N^{m}} ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} = \int_{N^{m}} ({∥\nabla ω∥}^{2} - \frac{1}{m} τ^{2} ρ^{2}) .

(3.5)

Next, we recall the following integral formula (cf. [16])

\int_{N^{m}} (R i c (ω, ω) + \frac{1}{2} {|£_{ω} g|}^{2} - {∥\nabla ω∥}^{2} - {(d i v ω)}^{2}) = 0,

and employing it in equation (3.5), we conclude

\int_{N^{m}} ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} = \int_{N^{m}} (R i c (ω, ω) + \frac{1}{2} {|£_{ω} g|}^{2} - {(d i v ω)}^{2} - \frac{1}{m} τ^{2} ρ^{2}) .

Using equations (3.2) and (3.3) in above equation, yields

\int_{N^{m}} ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} = \int_{N^{m}} (R i c (ω, ω) + 2 ρ^{2} {∥T∥}^{2} - τ^{2} ρ^{2} - \frac{1}{m} τ^{2} ρ^{2}),

that is,

\int_{N^{m}} ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} = \int_{N^{m}} (R i c (ω, ω) + 2 ρ^{2} ({∥T∥}^{2} - \frac{1}{m} τ^{2} ρ^{2}) - τ^{2} ρ^{2} + \frac{1}{m} τ^{2} ρ^{2}) .

In view of equation (3.4), above equation implies

\int_{N^{m}} ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} = \int_{N^{m}} (\frac{m - 1}{m} τ^{2} ρ^{2} - R i c (ω, ω))

and substituting from equation (3.2), it yields

\int_{N^{m}} ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} = \frac{m - 1}{m} \int_{N^{m}} {(d i v ω)}^{2} - \int_{N^{m}} R i c (ω, ω) .

Employing inequality (3.1) in above equation, we conclude

ρ^{2} {∥T - \frac{τ}{m} I∥}^{2} = 0 .

However,

ρ \neq 0

on connected

N^{m}

, gives

T = \frac{τ}{m} I .

(3.6)

Taking covariant derivative in above equation, we have

(\nabla_{X} T) (Y) = \frac{1}{m} X (τ) Y

and using a frame

\{F_{1}, . ., F_{m}\}

(N^{m}, g)

in above equation, we get

\sum_{j = 1}^{m} (\nabla_{E_{j}} T) (E_{j}) = \frac{1}{m} \nabla τ .

Using equation (2.5) in this equation, we arrive at

\frac{1}{2} \nabla τ = \frac{1}{m} \nabla τ

and as

m > 2

, we conclude

\nabla τ = 0

. Hence, the scalar curvature

τ

is a constant and it is a nonzero constant. Now, equations (2.7) and (3.6) imply

\frac{τ}{m} ω = \frac{τ}{m} \nabla ρ - τ \nabla ρ,

that is,

ω = - (m - 1) \nabla ρ

(3.7)

and it gives

d i v ω = - (m - 1) Δ ρ

, which in view of equation (3.2) implies

τ ρ = - (m - 1) Δ ρ

, that is,

- (m - 1) ρ Δ ρ = τ ρ^{2} .

Integrating above equation by parts, we arrive at

(m - 1) \int_{N^{m}} {∥\nabla ρ∥}^{2} = τ \int_{N^{m}} ρ^{2} .

Since,

ρ

is a nonconstant, from above equation, we conclude the constant

τ > 0

. We put

τ = m (m - 1) α

for a positive constant

α

. Now, differentiating equation (3.7) and using equations (2.2) and (3.6), we conclude

\nabla_{X} \nabla ρ = - α ρ X, X \in Θ (T N^{m}),

where

ρ

is nonconstant function and

α > 0

is a constant. Hence,

H e s s (ρ) = - α ρ g

, that is,

(N^{m}, g)

is isometric to the sphere

S^{m} (α)

(cf. [14,15]).

Conversely, suppose that

(N^{m}, g)

is isometric to the sphere

S^{m} (α)

. Then, we know that a nonzero constant vector field

b

on the ambient Euclidean space

R^{m + 1}

induces a vector field

ω

on the sphere

S^{m} (α)

, which by equation (1.4) is a

ρ

-Ricci vector field. Clearly, the scalar curvature of

S^{m} (α)

is given by

τ = m (m - 1) α \neq 0

. We claim that the function

ρ

is nonzero and nonconstant. If

ρ = 0

, then by equation (1.4), we have

f = 0

, which in view of equation (1.3) implies

ω = 0

, and this in turn will imply that the constant vector field

b = 0

. This is a contrary to the assumption that

b

is a nonzero constant vector field. Hence,

ρ \neq 0

. Now, suppose

ρ

is a constant, then by equation (1.4), f is a constant and by equation (1.3), we have

d i v ω = - m \sqrt{α} f

, which by Stokes’s Theorem on compact

S^{m} (α)

, would imply

f = 0

. This in turn by virtue of equation (1.4) implies

ρ = 0

, which is a contradiction as seen above. Hence, the function

ρ

is nonzero and nonconstant.

Next, using equations (1.3) and (1.4), we have

d i v ω = m (m - 1) α ρ

(3.8)

and it gives

\int_{S^{m} (α)} {(d i v ω)}^{2} = m^{2} {(m - 1)}^{2} α^{2} \int_{S^{m} (α)} ρ^{2} .

(3.9)

Now, using equation (1.4), we have

\nabla ρ = - \frac{1}{(m - 1) \sqrt{α}} \nabla f,

(3.10)

which on using equation (1.3), gives

\nabla ρ = - \frac{1}{m - 1} ω .

Taking divergence in above equation and using equation (3.8), we conclude

Δ ρ = - m α ρ

, that is,

ρ Δ ρ = - m α ρ^{2}

integrating this equation by parts, we conclude

\int_{S^{m} (α)} {∥\nabla ρ∥}^{2} = m α \int_{S^{m} (α)} ρ^{2} .

Treating this equation with equation (3.9), we conclude

\int_{S^{m} (α)} {(d i v ω)}^{2} = m {(m - 1)}^{2} α \int_{S^{m} (α)} {∥\nabla ρ∥}^{2} .

(3.11)

Also, using equations (1.3) and (3.10), we have

ω = - (m - 1) \nabla ρ

and it changes the equation (3.11) to

\int_{S^{m} (α)} {(d i v ω)}^{2} = m α \int_{S^{m} (α)} {∥ω∥}^{2} .

Finally, using

R i c (ω, ω) = (m - 1) {∥ω∥}^{2}

in above equation, we conclude

\int_{S^{m} (α)} R i c (ω, ω) = \frac{m - 1}{m} \int_{S^{m} (α)} {(d i v ω)}^{2}

and this finishes the proof. □

Next, we consider a closed

ρ

-Ricci vector field on a compact and connected Riemannian manifold

(N^{m}, g)

such that the smooth function

ρ

is a nontrivial solution of the FM-equation and find yet another characterization of the sphere

S^{m} (α)

. Indeed we prove the following:

Theorem 2.

An m-dimensional complete and simply connected Riemannian manifold

(N^{m}, g)

with scalar curvature

τ > 0

admits a closed ρ-Ricci vector field ω such that the function ρ is a nontrivial solution of the FM-equation and the length of covariant derivative of ω satisfies

{∥\nabla ω∥}^{2} \leq \frac{1}{m} τ^{2} ρ^{2},

if and only if, τ is a positive constant

τ = m (m - 1) α

and

(N^{m}, g)

is isometric to

S^{m} (α)

Proof.

Suppose

(N^{m}, g)

is an m-dimensional complete and simply connected Riemannian manifold with scalar curvature

τ > 0

, and it admits a closed

ρ

-Ricci vector field

ω

, where

ρ

is nontrivial solution of the FM-equation (1.6) and the length of covariant derivative of

ω

satisfies

{∥\nabla ω∥}^{2} \leq \frac{1}{m} τ^{2} ρ^{2} .

(3.12)

For

ρ

, we define the operator

B_{ρ}

B_{ρ} X = \nabla_{X} \nabla ρ, X \in Θ (T N^{m}),

then

B_{ρ}

is a symmetric operator related to

H e s s (ρ)

H e s s (ρ) (X, Y) = g (B_{ρ} X, Y), X, Y \in Θ (T N^{m}) .

(3.13)

As,

ρ

is a nontrivial solution of the FM-equation, using equations (3.13) and (1.6), we have

ρ T X = B_{ρ} X - (Δ ρ) X,

which in view of equation (1.7) becomes

B_{ρ} X = ρ T X - \frac{τ}{m - 1} ρ X .

(3.14)

Note that owing to the fact that

ρ

is a nontrivial solution of the FM-equation on

(N^{m}, g)

, the scalar curvature

τ

is a constant and we put

τ = m (m - 1) α

for a constant

α

. Using equation (3.14), we have

B_{ρ} X + α ρ X = ρ T X - (m - 1) α ρ X, X \in Θ (T N^{m}) .

Now, using equation (2.2) in above equation, we get

B_{ρ} X + α ρ X = \nabla_{X} ω - (m - 1) α ρ X, X \in Θ (T N^{m}) .

Taking a local frame

\{F_{1}, . ., F_{m}\}

(N^{m}, g)

, by above equation, we conclude

\begin{matrix} {∥B_{ρ} + α ρ I∥}^{2} & = & \sum_{j = 1}^{m} g (B_{ρ} F_{j} + α ρ F_{j}, B_{ρ} F_{j} + α ρ F_{j}) \\ = & \sum_{j = 1}^{m} g (\nabla_{F_{j}} ω - (m - 1) α ρ F_{j}, \nabla_{F_{j}} ω - (m - 1) α ρ F_{j}) \\ = & {∥\nabla ω∥}^{2} + m {(m - 1)}^{2} α^{2} ρ^{2} - 2 (m - 1) α ρ (d i v ω) . \end{matrix}

Now, using equating (2.2), we have

d i v ω = τ ρ = m (m - 1) α ρ

and inserting it in above equation, we arrive at

{∥B_{ρ} + α ρ I∥}^{2} = {∥\nabla ω∥}^{2} - m {(m - 1)}^{2} α^{2} ρ^{2},

that is,

{∥B_{ρ} + α ρ I∥}^{2} = {∥\nabla ω∥}^{2} - \frac{1}{m} τ^{2} ρ^{2} .

Using inequality (3.12) in above equation results in

B_{ρ} = - α ρ I,

that is,

H e s s (ρ) = - α ρ g .

(3.15)

Note that as

τ > 0

, the constant

α > 0

and

ρ

being a nontrivial solution,

ρ

is a nonconstant function. Hence, by equation (3.15), the complete and simply connected Riemannian manifold

(N^{m}, g)

is isometric to the sphere

S^{m} (α)

(cf. [14,15]).

Conversely, suppose that

(N^{m}, g)

is isometric to the sphere

S^{m} (α)

. Then, by equation (1.7), the function f is a solution of FM-equation on the sphere

S^{m} (α)

, which has a closed

ρ

-Ricci vector field

ω

. The solution f of FM-equation is related to

ρ

by equation (1.4), that is,

f = - (m - 1) \sqrt{α} ρ .

(3.16)

In the proof of Theorem-1, we have seen that

ρ

is a nonconstant function on

S^{m} (α)

. Moreover, using equation (3.16), we have

Δ f = - (m - 1) \sqrt{α} Δ ρ, H e s s (f) = - (m - 1) \sqrt{α} H e s s (ρ)

and the equation (1.7) takes the form

- (m - 1) \sqrt{α} (Δ ρ) g + f R i c = - (m - 1) \sqrt{α} H e s s (ρ),

which in view of equation (3.16) changes to

(Δ ρ) g + ρ R i c = H e s s (ρ) .

Hence,

ρ

is a nontrivial solution of the FM-equation on the sphere

S^{m} (α)

. Now, the Ricci operator T of the sphere

S^{m} (α)

is given by

T = (m - 1) α I

and therefore equation (2.2) on

S^{m} (α)

\nabla_{X} ω = (m - 1) α ρ X, X \in Θ (T S^{m} (α)) .

Using the expression for the scalar curvature

τ = m (m - 1) α

for the sphere

S^{m} (α)

, we have

\nabla_{X} ω = \frac{τ}{m} ρ X, X \in Θ (T S^{m} (α)) .

This proves

{∥\nabla ω∥}^{2} = \frac{1}{m} τ^{2} ρ^{2}

and completes the proof. □

4. Conclusions

In previous section, we have used a closed

ρ

-Ricci vector field

ω

on an m-dimensional Riemannian manifold

(N^{m}, g)

to find two different characterizations of a m-sphere

S^{m} (α)

. The scope of studying

ρ

-Ricci vector fields on a Riemannian manifold is quite modest. We observe that, in previous section, we restricted the

ρ

-Ricci vector field

ω

to be closed that simplified the expression for the covariant derivative of

ω

. It will be interesting to investigate whether, we could get similar results after removing the restriction that the

ρ

-Ricci vector field

ω

is closed. It will be interesting future topic to study the geometry of an m-dimensional Riemannian manifold

(N^{m}, g)

that admits a

ρ

-Ricci vector field

ω

, which need not be closed. In order to simplify the findings on an m-dimensional Riemannian manifold

(N^{m}, g)

admitting a

ρ

-Ricci vector field

ω

, which is not necessarily closed, we could impose the restriction on the Ricci operator T of

(N^{m}, g)

to be Codazzi type tensor, namely it satisfies

(\nabla_{X} T) (Y) = (\nabla_{Y} T) (X), X, Y \in Θ (T N^{m}) .

Note that above restriction on

(N^{m}, g)

is slightly stronger than demanding the scalar curvature is a constant.

Author Contributions

writing—original draft preparation, S.D.; writing—review and editing, H.A.; All authors have read and agreed to the published version of the manuscript.

Funding

Not Applied

Data Availability Statement

Not applied

Acknowledgments

Researchers Supporting Project number (RSPD2023R860), King Saud University, Riyadh, Saudi Arabia.

Conflicts of Interest

The authors declare no conflict of interest.

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Ricci Vector Fields

Abstract

1. Introduction

2. Preliminaries

3. Characterizing spheres via ρ -Ricci fields

4. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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3. Characterizing spheres via $ρ$ -Ricci fields