Beyond the compass: Exploring geometric constructions via a circle arc template and a straightedge

1.   Introduction

Ricci flow is a technique widely used in differential geometry, geometric topology, and geometric analysis. The Yamabe flow that deforms the metric of a Riemannian manifold $M$ is presented by the equations ^[1,2]

where $r(t)$ is the scalar curvature of $M$, and $t$ indicates the time. For a $2$-dimensional manifold, (1.1) is equivalent to the Ricci flow presented by

where $S$ is the Ricci tensor of $M$. However, in cases of dimension $> 2$, these flows do not coincide, since the Yamabe flow preserves the conformal class of $g(t)$ but the Ricci flow does not in general. The Ricci flow has been widely used for dealing with numerous significant problems, such as deformable surface registration in vision, parameterization in graphics, cancer detection in medical imaging, and manifold spline construction in geometric modeling. For more utilization in medical and engineering fields, see ^[3]. It has also been used in theoretical physics, particularly; in the study of the geometry of spacetime in the context of general relativity. In this area, it has been applied to understand the behavior of black holes and the large-scale structure of the universe.

The self-similar solutions of (1.1) and (1.2) are called the Ricci and Yamabe solitons, respectively ^[4,5]. They are respectively expressed by the following equations:

and

where $£_{F}$ is the Lie derivative operator along the smooth vector field $F$ on $M$, $\Lambda \in \mathbb R$.

In 2010, Blair ^[6] defined the concept of $*$-Ricci tensor $S^*$ in contact metric manifolds $M$ as:

for any vector fields $X$ and $Y$ on $M$, here $Q^*$ is the $*$-Ricci operator, $R$ is the curvature tensor, and $\varphi$ is a $(1, 1)$ tensor field. It is to be noted that the notion of the $*$-Ricci tensor on complex manifolds was introduced by Tachibana ^[7]. Later, Hamada ^[8] studied $*$-Ricci flat real hypersurfaces of complex space forms.

If we replace $S$ with $S^*$ in (1.3), then we recover the expression of $*$-Ricci soliton, proposed and defined by Kaimakamis and Panagiotidou ^[9] as follows:

Definition 1.1. On a Riemannian $($or a semi-Riemannian$)$ $M$, the metric $g$ is called a $*$-Ricci soliton; if

holds and $\Lambda \in \mathbb R$.

In 2019, a modern class of geometric flows, namely, the Ricci–Yamabe (RY) flow of type $(\rho, \sigma)$ was established by Güler and Crasmareanu ^[10]; and is defined by

for $\rho, \sigma \in \mathbb R$.

The RY flow can be Riemannian, semi-Riemannian, or singular Riemannian due to the involvement of the scalars $\rho$ and $\sigma$. This kind of different choice is useful in differential geometry and physics, especially in general relativity theory (i.e., a new bimetric approach to space–time geometry ^[11,12]).

A Ricci–Yamabe soliton (RYS) is a solution of RY flow; if it depends only on one parameter group of diffeomorphism and scaling. A Riemannian manifold $M$ is said to admit an RYS if

where $\rho, \sigma \in \mathbb R$.

If $F = grad f$, $f \in C^{\infty}(M)$, then the RYS is called the gradient Ricci–Yamabe soliton (gradient RYS), and then (1.5) takes the form

where $\nabla^2f$ indicates the Hessian of $f$. A RYS of types $(\rho, 0)$ and $(0, \sigma)$ is respectively known as $\rho$-Ricci soliton and $\sigma$-Yamabe soliton.

A manifold $M$ is said to admit a $*$-Ricci–Yamabe soliton ($*$-RYS) if

holds, where $\rho, \sigma$, $\Lambda \in \mathbb R$ and $r^*$ is the $*$-scalar curvature tensor of $M$.

In a similar way to (1.7), the gradient $*$-Ricci–Yamabe soliton (gradient $*$-RYS) is defined by

A $*$-RYS is said to be shrinking if $\Lambda < 0$, steady if $\Lambda = 0$, or expanding if $\Lambda > 0$. A $*$-RYS is called a

$(i)$ $*$-Yamabe soliton if $\rho = 0, \sigma = 1$,

$(ii)$ $*$-Ricci soliton if $\rho = 1, \sigma = 0$,

$(iii)$ $*$-Einstein soliton if $\rho = 1, \sigma = -1$,

$(iv)$ $*$-$\rho$-Einstein soliton if $\rho = 1, \sigma = -2\rho$.

Note that the $*$-Ricci–Yamabe soliton is the generalization of the aforementioned cases $(i)$–$(iv)$. Thus, the research on $*$-Ricci–Yamabe soliton is more significant and promising.

On the other hand, the Lorentzian manifold, which is one of the most important subclasses of pseudo-Riemannian manifolds, plays a key role in the development of the theory of relativity and cosmology ^[13]. In 1989, Matsumoto ^[14] proposed the notion of $LP$-Sasakian manifolds, while the same notion was independently studied by Mihai and Rosca ^[15] in 1992, and they contributed several important results on this manifold. Later, this manifold was studied by many researchers. Recently, in 2021, Haseeb and Prasad proposed and studied $LP$-Kenmotsu manifolds ^[16] as a subclass of Lorentzian paracontact manifolds.

Since the turn of the 21st century, the study of Ricci solitons and their generalizations has become highly significant due to their wide uses in various fields of science, engineering, computer science, medical etc. Here we are going to mention some works on Ricci solitons and their generalizations that were carried out by several authors, such as: the geometric properties of Einstein, Ricci and Yamabe solitons were studied by Blaga ^[17] in 2019; Deshmukh and Chen ^[18] find the sufficient conditions on the soliton vector field, where the metric of a Yamabe soliton is of constant scalar curvature; Chidananda et al. ^[19] have studied Yamabe and Riemann solitons in $LP$-Sasakian manifolds; the study of $LP$-Kenmotsu manifolds and $\epsilon$-Kenmotsu manifolds admitting $\eta$-Ricci solitons have been carried out by Haseeb and Almusawa ^[20], and Haseeb and De ^[21]; in ^[22], the authors studied conformal Ricci soliton and conformal gradient Ricci solitons on Lorentz-Sasakian space forms; RYSs have been studied by Haseeb et al. ^[23], Singh and Khatri ^[24], Suh and De ^[25], Yoldas ^[26], Zhang et al. ^[27]. The study of Ricci solitons and their generalizations has been extended to $*$-Ricci solitons and their generalizations on various manifolds and has been explored by the authors: Dey ^[28], Dey et al. ^[29], Ghosh and Patra ^[30], Haseeb and Chaubey ^[31], Haseeb et al. ^[32], and Venkatesha et al. ^[33]. Recently, Azami et al. ^[34] investigated perfect fluid spacetimes and perfect fluid generalized Roberston–Walker spacetimes.

2.   Preliminaries

A differentiable manifold $M$ (dim $M = n$) with the structure $(\varphi, \zeta, \eta, g)$ is named a Lorentzian almost paracontact metric manifold; in case $\varphi$: a $(1, 1)$-tensor field, $\zeta$: a contravariant vector field, $\eta$: a 1-form, and $g$: a Lorentzian metric $g$ satisfy ^[13]

We define the 2-form $\varPhi$ on $M$ as

for any $X, Y\in \chi(M)$, where $\chi(M)$ is the Lie algebra of vector fields on $M$. If

here $d$ is an exterior derivative, then $(M, \varphi, \xi, \eta, g)$ is called a paracontact metric manifold.

Definition 2.1. A Lorentzian almost paracontact manifold $M$ is called a Lorentzian para-Kenmotsu $($in brief, LP-Kenmotsu$)$ manifold if

for any $X$ and $Y$ on $M$ ^[16,23,35].

In the case of an $LP$-Kenmotsu manifold, we have

where $\nabla$ indicates the Levi–Civita connection with respect to $g$.

Furthermore, in an $LP$-Kenmotsu manifold of dimension $n$ (in brief, $(LP$-$K)_n$), the following relations hold ^[16]:

for any $X, Y, Z\in\chi(M)$.

Lemma 2.1. ^[36] In an $(LP$-$K)_n$, we have

for any $X$ on $(LP$-$K)_n$.

Lemma 2.2. ^[37] In an $(LP$-$K)_n$, we have

for any $Y, Z$ on $(LP$-$K)_n$, and $a$ is the trace of $\varphi$.

Lemma 2.3. In an $(LP$-$K)_n$, the eigenvalue of the $*$-Ricci operator $Q^{*}$corresponding to the eigenvector ${\xi}$ is zero, i.e., $Q^*{\xi} = 0.$

Proof. From (2.19), we have

which, by putting $Y = {\xi}$; and using (2.1), (2.3), and (2.16) gives $Q^*{\xi} = 0.$

Lemma 2.4. The $*$-Ricci operator $Q^{*}$ in an $(LP$-$K)_n$ satisfies the following identities:

for any $X$ on $(LP$-$K)_n$.

Proof. By the covariant differentiation of $Q^*{\xi} = 0$ with respect to $X$ and using (2.9), (2.21) and $Q^*{\xi} = 0$, we obtain (2.22). Next, differentiating (2.21) covariantly with respect to ${\xi}$ and using (2.8)–(2.10), (2.18), and (2.21), we obtain (2.23).

Lemma 2.5. In an $(LP$-$K)_n$, we have ^[23]

for any $X$ on $M$, and $Dr$ stands for the gradient of $r$.

Remark 2.1. From the equation $(2.24)$, it is observed that if $r$ of an $(LP$-$K)_n$ is constant, then $r = n(n-1)$.

3.   $*$-RYS on $(LP$-$K)_n$

In this section, we first prove the following result:

Theorem 3.1. In an $(LP$-$K)_n$ admitting a $*$-RYS, the scalar curvature $r$ of the manifold satisfies the Poisson's equation $\Delta r = \Psi,$ where $\Psi = \frac{4(n-1)\Lambda }{\sigma}+2 r(n-3)+(n-1)\{h-2 a^2-2(n^2-4n+1)\}, \ \sigma \neq 0.$

Proof. Let the metric of an $(LP$-$K)_n$ be a $*$-RYS, then in view of (2.19), $(1.8)$ takes the form

for any $X$, $Y$ on $M$.

Taking the covariant derivative of (3.1) with respect to $Z$, we find

As $g$ is parallel with respect to $\nabla$, then the following formula ^[38]

turns to

Due to the symmetric property of $£_V \nabla$, we have

which, by using (3.2), becomes

By putting $Y = {\xi}$ and using (2.3), (2.1), (2.5), (2.17), and (2.18), the preceding equation gives

By using the relation (2.20) in (3.3), we have

The covariant derivative of (3.4) with respect to $X$ leads to

where (2.1), (2.4), and (3.4) are used.

Again, in ^[38], we have

which, by setting $Z = {\xi}$ and using (3.5), becomes

Now, by putting $Y = {\xi}$ in (3.6) and using (2.5), (2.1), (2.17), and (2.18), we have

For the $h$ constant, we assume that $D a^2 = h {\xi}$, and hence we deduce the following values:

In light of (3.8), (3.7) reduces to

By contracting (3.9) over $X$, we lead to

where (2.26) is used and $\Delta$ symbolizes the Laplacian operator of $g$.

The Lie derivative of (2.15) along $F$ gives

By setting $X = Y = {\xi}$ in (3.1) and using (2.3), (2.1), (2.5), and (2.16), we have

The Lie derivative of $1+g({\xi}, {\xi}) = 0$ gives

Now, combining (3.10)–(3.13), we deduce

where $\Psi = \frac{4(n-1)\Lambda }{\sigma}+2 r(n-3)+(n-1)\{h-2 a^2-2(n^2-4n+1)\}, \ \sigma \neq 0.$

For the smooth functions $\theta$ and $\Psi$, an $(LP$-$K)_n$ satisfies Poisson's equation if $\theta = \Psi$ holds. In the case; where $\theta = 0$, Poisson's equation transforms into Laplace's equation. This completes the proof of our theorem.

A function $\upsilon \in C^{\infty}(M)$ is said to be subharmonic if $\Delta \upsilon\geq 0$, harmonic if $\Delta \upsilon = 0$, and superharmonic if $\Delta \upsilon\leq 0$. Thus, from (3.14), we state the following corollaries:

Corollary 3.1. In an $(LP$-$K)_n$ admitting a $*$-RYS, we have

Corollary 3.2. In an $(LP$-$K)_n$ admitting a $*$-RYS, the scalar curvature $r$ of the manifold satisfies the Laplace equation if and only if

Let an $(LP$-$K)_n$ admit a $*$-RYS, and if the scalar curvature $r$ of the manifold satisfies Laplace's equation, then (3.15) holds. If this value of $r$ is constant, then by virtue of Remark 2.1, we find $\Lambda = -\frac{\sigma}{2}(n+\frac{h}{2}-a^2-1).$ Thus, we have:

Corollary 3.3. In an $(LP$-$K)_n$ admitting a $*$-RYS, we have

4.   Gradient $*$-RYS on $(LP$-$K)_n$

This section explores the properties of gradient $*$-RYS on $(LP$-$K)_n$.

Let $M$ be an $(LP$-$K)_n$ with $g$ as a gradient $*$-RYS. Then (1.9) can be written as

for all $X$ on $(LP$-$K)_n$, where $D$ indicates the gradient operator of $g$.

The covariant differentiation of (4.1) along $Y$ leads to

By virtue of (4.2) and the curvature identity $R(X, Y) = [\nabla_{X}, \nabla_{Y}]-\nabla_{[X, Y]}$, we find

By contracting (4.3) along $X$, we have

From (2.15), we have

Thus, from (4.4) and (4.5), it follows that

Now, we divide our study into two cases, as follows:

Case I. Let $\rho = (n-1)\sigma$. For this case, we prove the following theorem:

Theorem 4.1. Let an $(LP$-$K)_n$ admit a gradient $*$-RYS and $\rho = (n-1)\sigma$. Then, $(LP$-$K)_n$ possesses a constant scalar curvature $r^*$.

Proof. Let $\rho = (n-1)\sigma$. Then, from (4.6), we obtain

The covariant derivative of (4.7) along $X$ gives

where (2.9), (2.19), (4.1), and (4.7) are used.

By setting $X = {\xi}$ in (4.8), and then using (4.7) and (2.1), we infer that

which informs us that $r^*$ is constant. This completes the proof.

Corollary 4.1. Let an $(LP$-$K)_n$ admit a gradient $*$-RYS and $\rho = (n-1)\sigma$. Then, we have:

Now, from (4.8) and (4.9), we conclude that

This indicates that the gradient function $f$ of the gradient $*$-RYS is constant on $(LP$-$K)_n$. Thus, we have

Corollary 4.2. Let an $(LP$-$K)_n$ admit a gradient $*$-RYS and $\rho = (n-1)\sigma$. Then the function $f$ is constant, and hence the gradient $*$-RYS is trivial. Moreover, $(LP$-$K)_n$ is an Einstein manifold.

Case II. Let $\rho\neq (n-1)\sigma$. For this case, we prove the following theorem:

Theorem 4.2. Let an $(LP$-$K)_n$ admit a gradient $*$-RYS and $\rho\neq (n-1)\sigma$. Then the gradient of the potential function $f$ is pointwise collinear with ${\xi}$.

Proof. Replacing $X = {\xi}$ in (4.3), then using (2.12) and Lemma 2.5, we have

Also from (2.12), we have

By equating the last two equations, we have

By contracting $Y$ in (4.11), we have

Now, from (2.20), (2.24), (2.25), and (3.8), we easily find

This implies

where $g({\xi}, Da^2) = -h$.

By using (4.14) in (4.12), we obtain

From (4.11) and (4.15), it follows that

The inner product of (4.16) with ${\xi}$ and using (2.2), (2.1), (2.5), and (2.15) gives

where we assumed $Da^2 = h {\xi}$.

From (4.17), we conclude that $Df = k{\xi}$, where $k$ is a smooth function given by

This completes the proof.

5.   Example

We consider the 3-dimensional manifold $M = \{(u_1, u_2, u_3)\in \mathbb R^3, u_3 > 0 \},$ where $(u_1, u_2, u_3)$ are the standard coordinates in $\mathbb R^3$. Let $e_1, e_2,$ and $e_3$ be the vector fields on $\mathcal{M}$ given by

which are linearly independent at each point of $M$.

Define a Lorentzian metric $g$ on $M$ such that

Let $\eta$ be the 1-form on $M$ defined by $\eta(X) = g(X, e_3) = g(X, {\xi})$ for all $X$ on $M$; and let $\varphi$ be the (1, 1)-tensor field on $M$ is defined as

By applying the linearity of $\varphi$ and $g$, we have

for all $X$, $Y$ on $M$.

Let $\nabla$ be the Levi–Civita connection with respect to the Lorentzian metric $g$. Thus, we have

With the help of Koszul's formula, we easily calculate

One can also easily verify that

Hence, $M$ is a Lorentzian para-Kenmotsu manifold of dimension 3. By using $(5.2)$, we obtain

Now, with the help of the above components of the curvature tensor, it follows that

Thus, the manifold is of constant curvature.

From $(5.3)$, we get $S(Y, Z) = 2g(Y, Z)$. This implies that

Now, by putting $Y = {\xi}$ in (2.21), then using (5.1) and (5.4), we obtain $Q^*\mathcal {\xi} = 0.$ This proves Lemma 2.3. Next, by contracting $S(Y, Z) = 2g(Y, Z)$, we find $r = 6$. Since $r$ is constant, therefore, (2.24) leads to ${\xi}(r) = 0 \implies r = 6$, where $n = 3$. This proves Remark 2.1.

6.   Conclusions

In the present study, we obtain certain important results on $(LP$-$K)_n$ admitting a $*$-RYS and a gradient $*$-RYS. First, we prove that the scalar curvature $r$ of $(LP$-$K)_n$ admitting a $*$-RYS satisfies Poission's equation, and we discuss the conditions for the $*$-RYS to be shrinking, steady, and expanding. Furthermore, we deal with the study of gradient $*$-RYS on $(LP$-$K)_n$ in two cases: $(i)$ $\rho = (n-1)\sigma$; in this case, we showed that the gradient function $f$ of the gradient $*$-RYS is constant, and hence the gradient $*$-RYS is trivial. Moreover, $(LP$-$K)_n$ is an Einstein manifold; $(ii$) $\rho\neq(n-1)\sigma$, in this case, we proved that if an $(LP$-$K)_n$ admits a gradient $*$-RYS, then the gradient of the potential function $f$ is pointwise collinear with ${\xi}$.

Author contributions

Abdul Haseeb: Conceptualization, investigation, methodology, writing–original draft; Fatemah Mofarreh: Investigation, methodology, writing–review & editing; Sudhakar Kumar Chaubey: Conceptualization, methodology, writing–review & editing; Rajendra Prasad: Conceptualization, investigation, writing–review & editing. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

The authors are really thankful to the learned reviewers for their careful reading of our manuscript and their insightful comments and suggestions that have improved the quality of our manuscript. The author Fatemah Mofarreh (F. M.), expresses her thankful to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Also, the author Sudhakar Kumar Chaubey was supported by the Internal Research Funding Program of the University of Technology and Applied Sciences, Shinas, Oman.

Funding

The author Fatemah Mofarreh (F. M.), expresses her thankful to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflicts of interest.