k-Almost Newton-Conformal Ricci Solitons on Hypersurfaces Within Golden Riemannian Manifolds with Constant Golden Sectional Curvature

Abstract

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions for total geodesicity with adequate restrictions on some smooth functions using mathematical operators. Furthermore, we go over some natural conclusions in which the gradient k-almost Newton-conformal Ricci soliton on the hypersurface of the Golden Riemannian manifold becomes compact. Finally, we establish a Schur’s type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature.

1. Introduction

In 1988, Hamilton [1] originally proposed the concept of Ricci flow. A Ricci soliton can be described as the limit reached by solutions of this flow. Additionally, significant attention has recently been directed towards classifying solutions that exhibit self-similarity within geometric flows.

In 2011, the immersed almost Ricci solitons on Riemannian manifolds were examined in [2]. The authors discovered that within a Riemannian manifold $M^{n+p}$ possessing a non-positive sectional curvature, $M^n$ cannot be minimal, as an almost Ricci soliton is identical to a Ricci soliton, and an integrable norm on $M^n$ is maintained by the vector field V. Furthermore, Wylie [3] explained that compactness is a necessary property of a Riemannian manifold with a shrinking soliton. Additionally, these immersions lack minimality if $M^{n+p}$ is characterized by a non-positive sectional curvature.

Cunha et al. [4] employed the Newton transformation $P_k$ alongside the second-order differential operator $L_k$, for the range $0\le k\le n$, to formulate the concept of the r-almost Newton–Ricci soliton (ANRS) in Riemannian manifolds. De et al. [5] and Siddiqi [6,7] also discussed the same concept on Lagrangian submanifolds and Legendrian submanifolds.

The conformal Ricci flow, an enhancement of the classical Ricci flow, was introduced by Fischer in [8]. This geometric flow modifies the equation’s unit volume constraint, replacing it with a scalar curvature constraint. The formula for conformal Ricci flow on a Riemannian manifold $(M^n,g)$ is given by the following:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}g=-2\mathit{S}-\left(P+{\displaystyle \frac{2}{n}}\right)g,r\left(g\right)=-1,\end{array}$$

(1)

where P denotes a non-dynamical scalar field (time-dependent scalar field), S denotes the Ricci tensor, $r\left(g\right)$ is the scalar curvature, and n is the dimension of the manifold $(M^n,g)$.

Einstein metrics with the Einstein constant are the steady states of the equations of conformal Ricci flow. The de-Turck method was used by Lu et al. [9] to reconstruct the conformal Ricci flow in terms of concrete parabolic elliptic PDEs. They explain the transient presence of the conformal Ricci flow on both asymptotically flat and compact manifolds.

In 2015, Basu and Bhattacharyya [10] introduced the notion of a conformal Ricci soliton. The corresponding equation is as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\mathfrak{L}}_{F}g+\mathit{S}+\left[\mathsf{\Lambda }-\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)\right]g=0.\end{array}$$

(2)

Here, ${\mathfrak{L}}_F$ denotes the Lie derivative operator along the vector field F, and $\mathsf{\Lambda }$ represents a real constant. P is a non-dynamical scalar field, S denotes the Ricci tensor, and n is the dimension of the manifold $(M^n,g)$.

If $(g,F,\mathsf{\Lambda }-(P+{\displaystyle \frac{2}{n}}))$ meets (2), it is classified as a conformal Ricci soliton on M [10]. Based on $\mathsf{\Lambda }<0$, $\mathsf{\Lambda }=0$, or $\mathsf{\Lambda }>0$, $(g,F,\mathsf{\Lambda }-(P+{\displaystyle \frac{2}{n}}))$ will exhibit the following:

(i)

Shrinking;

(ii)

Steady;

(iii)

Expanding, respectively.

If the vector field F in (2) is a gradient vector field associated with a smooth function $\psi$, then we obtain the conformal gradient Ricci soliton by using mathematical operators as in [11].

$$\begin{array}{c}\hfill \mathcal{H}ess\psi +2\mathit{S}+\left[\mathsf{\Lambda }-\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)\right]g=0.\end{array}$$

(3)

On the other hand, on a manifold, the polynomial structures were considered in [12], which served as the foundation of the golden structure [13]. Ref. [14] provided insight into integrability within the Riemannian manifold with a golden structure. The golden sectional curvature was recently studied in [15]. The same authors also examined the geometry of submanifolds within locally decomposable Golden Riemannian manifolds with constant golden sectional curvature.

In [16], the authors conceptualized the metallic structure in 2013, a generalization of the golden structure established on Riemannian manifolds. The researchers in [17,18,19,20,21,22] addressed various aspects of the metallic Riemannian manifolds regarding curvature.

The literature on almost Ricci solitons, shrinking solitons, and related subjects in Riemannian manifolds and Lagrangian submanifolds serves as inspiration for the current work. In this study, we analyze k-almost Newton-conformal Ricci solitons on a hypersurface within a Golden Riemannian manifold with constant golden sectional curvature. It is the first investigation of such solitons on a hypersurface of a Golden Riemannian manifold with constant golden sectional curvature.

2. Golden Riemannian Manifolds

Consider a $(1,1)$-tensor field $\phi$ on a manifold ${\overline{\mathcal{N}}}^m$. Then $\phi$ is a golden structure on ${\overline{\mathcal{N}}}^m$ if [12,23]

$$\begin{array}{c}\hfill {\phi }^2-\phi -I=0.\end{array}$$

In this case, $({\overline{\mathcal{N}}}^m,\phi )$ is called a golden manifold. Additionally, a Riemannian manifold $({\overline{\mathcal{N}}}^m,g)$ having a golden structure $\phi$ fulfilling

$$\begin{array}{c}\hfill g(\phi Y_1,Y_2)=g(Y_1,\phi Y_2)\end{array}$$

(4)

is said to be a Golden Riemannian manifold [13]. Substituting $Y_1$ with $\phi Y_1$ in (4), we have

$$\begin{array}{ccc}\hfill g(\phi Y_1,\phi Y_2)& =& g({\phi }^2{Y}_1,Y_2)\hfill \\ & =& g(\phi Y_1,Y_2)+g(Y_1,Y_2).\hfill \end{array}$$

When

$$\nabla \phi =0,$$

the Golden Riemannian manifold $({\overline{\mathcal{N}}}^m,g,\phi )$ becomes locally decomposable.

Let $({\overline{\mathcal{N}}}^m,g,\phi )$ be a locally decomposable Golden Riemannian manifold with constant golden sectional curvature c. Subsequently, we obtain [15]

$$\begin{array}{ccc}\hfill R\left(X_1,Y_1\right)Z_1& & ={\displaystyle \frac{c}{3}}\left\{g\left(Y_1,Z_1\right)X_1-g\left(X_1,Z_1\right)Y_1\right.\hfill \\ & & -g\left(Y_1,\phi Z_1\right)X_1-g\left(Y_1,Z_1\right)\phi X_1+2g\left(Y_1,\phi Z_1\right)\phi X_1\hfill \\ & & \left.+g\left(X_1,\phi Z_1\right)Y_1+g\left(X_1,Z_1\right)\phi Y_1-2g\left(X_1,\phi Z_1\right)\phi Y_1\right\},\hfill \end{array}$$

for every vector field $X_1,Y_1$ and $Z_1$ on ${\overline{\mathcal{N}}}^m$.

Consider a connected submanifold defined by

$${\psi }_1:{\mathcal{M}}^n⟶{\overline{\mathcal{N}}}^m.$$

Then, the Gauss equation is written as [24]

$$\begin{array}{ccc}\hfill \overline{R}(X_1,Y_1,Z_1,W_1)& =& R(X_1,Y_1,Z_1,W_1)-g(h(X_1,W_1),h(Y_1,Z_1))\hfill \\ & & +g(h(X_1,Z_1),h(Y_1,W_1)),\hfill \end{array}$$

$\forall X_1,Y_1,Z_1,W_1\in \Gamma \left(T\mathcal{M}\right)$, where the Riemannian curvature tensors of ${\overline{\mathcal{N}}}^m$ and ${\mathcal{M}}^n$ are $\overline{R}$ and R, respectively.

Consider an orthonormal frame $\left\{E_1,\dots ,E_n\right\}$ on $T\left(\mathcal{M}\right)$ and denote the Hilbert–Schmidt norm of $∥\mathcal{A}∥$.

Recall that on ${\overline{\mathcal{N}}}^m$, the scalar curvature r is given by [15]

$$r={\displaystyle \frac{c}{3}}\left\{n^2+{2\parallel \phi \parallel }^2-3n-2n\parallel \phi \parallel \right\}+n^2{\mathcal{H}}^2-{∥\mathcal{A}∥}^2,$$

(5)

where $\mathcal{H}$ denotes the mean curvature vector and

$$\parallel \phi \parallel =\sum _{i=1}^{n}g\left(\phi E_i,E_i\right).$$

Furthermore, the golden scalar curvature $r^G$ corresponding to the golden sectional curvature of the submanifold ${\mathcal{M}}^n$ in $({\overline{\mathcal{N}}}^m,g,\phi )$ is expressed by [15]

$$\begin{array}{c}\hfill r^G={\displaystyle \frac{c}{3}}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}+n^2{\mathcal{H}}^2-{∥\mathcal{A}∥}^2.\end{array}$$

(6)

3. $\mathit{k}$-Almost Newton-Conformal Ricci Soliton

We remark that $f:{\mathcal{M}}^n⟶{\overline{\mathcal{N}}}^{m=2n+1}$ is a linked and oriented hypersurface immersed in an $(m=2n+1)$-dimensional Golden Riemannian manifold possessing a uniform golden sectional curvature. Let $P_k$ be the k-th Newton transformation

$$P_k:\mathcal{X}\left({\mathcal{M}}^n\right)⟶\mathcal{X}\left({\mathcal{M}}^n\right),$$

described by the recursive relationship:

$$\begin{array}{c}\hfill P_k=\sum _{j=0}^k{(-1)}^{k-j}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{j}}\right){\mathcal{H}}_j{\mathcal{A}}^{k-j},\end{array}$$

(7)

for the identity operator $P_0=I$ and $0\le k\le m$. Here, ${\mathcal{A}}^j$ indicates the composition of $\mathcal{A}$ with itself j times, i.e., $({\mathcal{A}}^0=I)$.

We call ${\mathcal{M}}^n$ a k-almost Newton-conformal Ricci soliton (ANCRS) if, for each $0\le k\le m$, there exists

$$\psi :{\mathcal{M}}^n⟶\mathbb{R},$$

such that,

$$\begin{array}{c}\hfill S+P_k\circ \mathcal{H}ess\psi =\left[\mathsf{\Lambda }-\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)\right]g.\end{array}$$

(8)

In this context, $\psi$ and $\mathsf{\Lambda }$ are smooth functions defined on ${\mathcal{M}}^n$. The tensor field $P_k\circ \mathcal{H}ess\psi$ is given by

$$\begin{array}{c}\hfill P_k\circ \mathcal{H}ess\psi (X_1,Y_1)=g(P_k{\nabla }_{X_1}\nabla \psi ,Y_1),\end{array}$$

$\forall X_1,Y_1\in \mathcal{X}\left(\mathcal{M}\right)$.

When $k=0$, (8) simplifies to a gradient almost conformal Ricci soliton. The second fundamental form $\mathcal{A}$ of ${\mathcal{M}}^n$ possesses n algebraic invariants, identified as elementary symmetric functions, $r_k$, derived from principal curvatures ${\rho }_1,\dots ,{\rho }_n$. In immersion, k-th mean curvature ${\mathcal{H}}_k$ can be expressed as follows:

$${\mathcal{H}}_k={\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}^{-1}{r}_k({\rho }_{i_1}\dots {\rho }_{i_k}),$$

where ${\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}^{-1}$ is the binomial coefficient, and $r_k$ is the k-th elementary symmetric polynomial given by

$$\begin{array}{c}\hfill r_k=({\rho }_0\dots {\rho }_n)=\sum _{0\le i_1<\dots <i_k}{\rho }_{i_1}\dots {\rho }_{i_k}.r_0=1.\end{array}$$

Putting $k=0,$ we obtain ${\mathcal{H}}_1={\displaystyle \frac{1}{n}}Tr\left(\mathcal{A}\right)=\mathcal{H}$, the mean curvature of ${\mathcal{M}}^n$. $Tr$ denotes the trace operator.

Moreover, it follows that the characteristic polynomial of $\mathcal{A}$ is given by

$$\mathcal{P}\left(t\right)=det(\mathcal{A}-tI)=\sum _{j=0}^k{(-1)}^{k-j}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{j}}\right){\mathcal{H}}_j{t}^{k-j}.$$

Next, we write

$${\mathcal{L}}_k:C^{\infty }\left({\mathcal{M}}^n\right)⟶C^{\infty }\left({\mathcal{M}}^n\right)$$

as

$$\begin{array}{c}\hfill {\mathcal{L}}_{k}u=Tr(P_k\circ \mathcal{H}essu).\end{array}$$

The Laplacian operator ${\mathcal{L}}_0$ can be obtained for $k=0$. We also recall that

$$\begin{array}{ccc}\hfill di{v}_{\mathcal{M}}(P_k\nabla u)& =& \sum _{i=1}^{n}g(\left({\nabla }_{E_i}{P}_k\right)\left({\nabla }_u\right),E_i)+\sum _{i=1}^{n}g(P_k\left({\nabla }_{E_i}{\nabla }_u\right),E_i)\hfill \\ & =& g(di{v}_{\mathcal{M}}{P}_k,{\nabla }_u)+{\mathcal{L}}_{k}u,\hfill \end{array}$$

(9)

where

$$\begin{array}{ccc}\hfill di{v}_{\mathcal{M}}{P}_k& =& Tr(\nabla P_k)\hfill \\ & =& \sum _{i=1}^m\left({\nabla }_{E_i}{P}_k\right)\left(E_i\right).\hfill \end{array}$$

(10)

If the ambient space has constant sectional curvatures, Equation (9) takes on this form

$$\begin{array}{c}\hfill {\mathcal{L}}_{k}u=di{v}_{\mathcal{M}}(P_k\nabla u),\end{array}$$

as ${\mathrm{div}}_{\mathcal{M}}{P}_k=0$ (see [25]).

The traceless second fundamental form of the hypersurface is

$$\begin{array}{c}\hfill \Phi =\mathcal{A}-\mathcal{H}I,Tr(\Phi )=0,\end{array}$$

and

$$\begin{array}{ccc}\hfill {|\Phi |}^2& =& Tr\left({\Phi }^2\right)\hfill \\ & =& {∥\mathcal{A}∥}^2-n{\mathcal{H}}^2\hfill \\ & \ge & 0.\hfill \end{array}$$

${\mathcal{M}}^n$ is totally umbilical if and only if

$${\left|\Phi \right|}^2=0.$$

Example 1. 

We consider the usual immersion of ${\mathcal{M}}^n$ in ${\mathbb{S}}^{2n+1}\left(1\right)$, known to be totally geodesic. Here, $P_r=0$, $1\le r\le n$. By selecting, $\mathsf{\Lambda }=\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)$, an immersion that satisfies (8) can be obtained.

Example 2. 

Let ${\mathbb{S}}^m\left(1\right)$ be the unit sphere in the Euclidean space ${\mathbb{R}}^{n+1}$ and

$$\psi :{\mathbb{S}}^m\left(1\right)\hookrightarrow {\mathbb{R}}^{n+1}$$

the embedding with induced metric g on ${\mathbb{S}}^m\left(1\right)$; then $({\mathbb{S}}^m{\mathbb{J}}_i,g)$ is a Riemannian manifold with sectional curvature $c=1$.

Let $i:{\mathcal{M}}^n⟶{\mathbb{S}}^m\left(1\right)\subset {\mathbb{R}}^{n+1}$ be an immersion of an n-dimensional manifold ${\mathcal{M}}^n$ into a smooth unit sphere.

According to [26], with a constant $t\in {\mathbb{R}}^{n+1}$, select the functions ${\overline{f}}_t$ on ${\mathbb{R}}^{n+1}$ such that

$$\begin{array}{c}\hfill {\overline{f}}_l\left(t\right)=-g(t,l)+n+1and{\psi }_l\left(t\right)=-{\overline{f}}_l+c,{\overline{f}}_l:=i\ast {\tilde{f}}_l\in C^{\infty }\left(S^m\right),\end{array}$$

where $t=(t_1,.....t_m)\in {\mathbb{S}}^m$ is the position vector and $l\in {\mathbb{S}}^m\left(1\right)$, $t\ne 0$, $c\in {\mathbb{R}}^{n+1}$. We get that $\left({\mathbb{S}}^m,{\mathbb{J}}_i,g,\nabla {\psi }_l,{\mathsf{\Lambda }}_l-\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)\right)$ holds for

$$\begin{array}{c}\hfill Hess{\psi }_l+S=\left[{\mathsf{\Lambda }}_l-\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)\right]g.\end{array}$$

However, ${\mathbb{S}}^m$ is known to be totally umbilical, having a second fundamental form $A=I$ and a k-th mean curvature ${\mathcal{H}}_k=1$. More specifically, the Newton tensor for each $0\le k\le n$ is provided by

$$\begin{array}{c}\hfill P_k=\alpha I,\end{array}$$

according to which

$$\alpha =\sum _{j=0}^k{(-1)}^{k-j}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{j}}\right).$$

Therefore, by using the smooth function $\psi ={\alpha }^{-1}{\psi }_l$, we can obtain the Equation (8), which describes the hypersurface.

Furthermore, if $\mathcal{M}$ has constant scalar curvature, (8) can be simplified by using mathematical operators as follows:

$$\begin{array}{c}\hfill S+P_r\circ \mathcal{H}essf=\mu g,\end{array}$$

$\mu =\left[\mathsf{\Lambda }-\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)\right]$. For a further illustration of a gradient r-almost Newton–Einstein soliton, see Example 2 in [4].

4. Main Results

We will use the maximum principle [27] to prove our results. First, we fix

$$\begin{array}{c}\hfill {\mathcal{L}}^s\left(\mathcal{M}\right)=\left\{u:{\mathcal{M}}^n⟶\mathcal{R};{\int }_{\mathcal{M}}{\left|u\right|}^{s}dL<+\infty \right\},\end{array}$$

$\forall s\ge 1$.

We also use the subsequent result [25]:

Lemma 1. 

Consider ${\mathcal{M}}^n$ to be an oriented, non-compact, complete Riemannian manifold, and a smooth vector field U on it. The divergence $di{v}_{\mathcal{M}}U$ maintains a uniform sign throughout ${\mathcal{M}}^n$. If the magnitude $\left|U\right|$ belongs to ${\mathcal{L}}^1\left(\mathcal{M}\right)$, then it follows that $di{v}_{\mathcal{M}}U$ must be zero.

Theorem 1.2 of [26] can be further generalized as follows:

Theorem 1. 

Suppose $(g,\psi ,\mathsf{\Lambda },k)$ is a complete ANCRS on ${\mathcal{M}}^n$ in $({\overline{\mathcal{N}}}^m,g,\phi )$ with constant golden sectional curvature r and bounded $\mathcal{A}$. If

(i) 

$c\le 0$, $\mathsf{\Lambda }>0$, and $P>-{\displaystyle \frac{2}{n}}$, implies that ${\mathcal{M}}^n$ is not minimal; or

(ii) 

$c<0$, $\mathsf{\Lambda }>0$, and $P>-{\displaystyle \frac{2}{n}}$, implies that ${\mathcal{M}}^n$ is not minimal; or

(iii) 

when $c=0$, with $\mathsf{\Lambda }>0$, and $P>-{\displaystyle \frac{2}{n}}$, ${\mathcal{M}}^n$ is minimal and isometric to ${\mathbb{R}}^n$.

Here, $\psi :{\mathcal{M}}^n⟶\mathcal{R}$ denotes the potential function with $\left|\nabla \psi \right|\in {\mathcal{L}}^1\left(\mathcal{M}\right).$

Proof. 

From (9), it is clear that the type of ${\mathcal{L}}_k$ is divergent. Since the second fundamental form is bounded on ${\mathcal{M}}^n$, Equation (7) implies that $P_k$ has a bounded norm

$$\begin{array}{c}\hfill \left|P_k\nabla \psi \right|\le \left|P_k\right|\left|\nabla \psi \right|\in {\mathcal{L}}^1\left(\mathcal{M}\right).\end{array}$$

On the other hand, assume that ${\mathcal{M}}^n$ is minimal to demonstrate (1) and (2). Using (5) with $c\le 0(c<0)$, we find that $r\le 0(r<0)$ is fulfilled by ${\mathcal{M}}^n$. Therefore, the contraction of (8) implies

$${\mathcal{L}}_r\psi =n\mathsf{\Lambda }-(P+{\displaystyle \frac{2}{n}})-r>0$$

in both cases. This contradicts Lemma 1. This finishes the proof of (1) and (2).

Next, because ${\overline{\mathcal{N}}}^m$ has constant sectional curvature $c=0$ and using the minimality of ${\mathcal{M}}^n$, (5) reduces to

$$\begin{array}{c}\hfill r=-{∥\mathcal{A}∥}^2\le 0.\end{array}$$

Assume $\mathsf{\Lambda }\ge 0$. One gets

$${\mathcal{L}}_r\left(\psi \right)=n\mathsf{\Lambda }-(P+{\displaystyle \frac{2}{n}})-r\ge 0.$$

But, ${\mathcal{L}}_{r}u=di{v}_{\mathcal{M}}(P_k\nabla u)$ and $\left|P_k\nabla \psi \right|$$\in {\mathcal{L}}^1\left(\mathcal{M}\right)$. Then, in view of Lemma 1, we deduce that

$${\mathcal{L}}_r\psi =0.$$

Hence, one obtains

$$0\ge r=n\mathsf{\Lambda }-(P+{\displaystyle \frac{2}{n}})\ge 0,$$

or,

$$r=\mathsf{\Lambda }-({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}})=0.$$

One concludes that

$${∥\mathcal{A}∥}^2=0,$$

proving that ${\mathcal{M}}^n$ is totally geodesic and flat. □

Corollary 1. 

Suppose all considerations of Theorem 1 are true and the golden scalar curvature $r^G$ of $({\overline{\mathcal{N}}}^m,g,\phi )$ is expressed in terms of the golden sectional curvature and $\mathsf{\Lambda }>0$. Then

(i) 

$c\le 0$, $P>-{\displaystyle \frac{2}{n}}$, ${\mathcal{M}}^n$ is not minimal.

(ii) 

$c<0$, $P>-{\displaystyle \frac{2}{n}}$, ${\mathcal{M}}^n$ fails to be minimal.

(iii) 

$c=0$, $P>-{\displaystyle \frac{2}{n}}$, ${\mathcal{M}}^n$ is minimal and isometrically equivalent to ${\mathbb{R}}^n$.

The conclusion coming next and related to Theorem 3 of [28] is necessary for establishing our results.

Lemma 2. 

Let $({\overline{\mathcal{N}}}^m,g,\phi )$ be complete, endowed with a subharmonic function u that is non-negative and smooth. For $s>1$, u is constant if $u\in {\mathcal{L}}^s({\overline{\mathcal{N}}}^m)$.

Additionally, we have

Theorem 2. 

Let $(g,\psi ,\mathsf{\Lambda },k)$ be a complete ANCRS on ${\mathcal{M}}^n$ in $({\overline{\mathcal{N}}}^m,g,\phi )$, which has constant golden sectional curvature r and $\mathsf{\Lambda }>0$. If

(i) 

$r\le 0$, $P>-{\displaystyle \frac{2}{n}}$, then ${\mathcal{M}}^n$ is not minimal;

(ii) 

$r<0$, $P>-{\displaystyle \frac{2}{n}}$, then ${\mathcal{M}}^n$ is not minimal;

(iii) 

$r\le 0$, $P>-{\displaystyle \frac{2}{n}}$, and ${\mathcal{M}}^n$ is minimal, then ${\mathcal{M}}^n$ is flat and totally geodesic.

In this context, $\psi :{\mathcal{M}}^n⟶\mathcal{R}$ is regarded as non-negative and belongs to ${\mathcal{L}}^s\left(\mathcal{M}\right)$ for $s>1$, while $P_k$ is assumed to be upper-bounded when considering quadratic forms.

Proof. 

The starting point to demonstrate (1) is to consider the contradiction that ${\mathcal{M}}^n$ is minimal in relation to (5) and the hypothesis $\rho \le 0$. The contraction of (8) yields

$${\mathcal{L}}_k\psi =n\left[\mathsf{\Lambda }-\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)\right]-r>0.$$

Assuming that $P_k$ is bounded from above, there exists a positive constant that fulfills

$$\omega \Delta \psi \ge {\mathcal{L}}_k\psi >0.$$

As a result, we argue that $\psi$ is constant through Lemma 2, which is not expected. One is likely to acquire (2) and (3) by employing an identical process to Theorem 1. □

Corollary 2. 

Assume the hypotheses of Theorem 2 are true, and the golden scalar curvature $r^G$ of $({\overline{\mathcal{N}}}^m,g,\phi )$ is expressed in terms of the golden sectional curvature and $\mathsf{\Lambda }>0$.

(i) 

$K_M\le 0$, $P>-{\displaystyle \frac{2}{n}}$ implies ${\mathcal{M}}^n$ is not minimal;

(ii) 

$K_M<0$, $P>-{\displaystyle \frac{2}{n}}$ implies ${\mathcal{M}}^n$ is not minimal;

(iii) 

$K_M\le 0$, $P>-{\displaystyle \frac{2}{n}}$, and ${\mathcal{M}}^n$ being a minimal manifold imply that ${\mathcal{M}}^n$ is necessarily flat and totally geodesic.

We extend Theorem 1.5 of [2] for $U=\nabla \psi$ in our subsequent result. We also give the conditions for an ANCRS on a hypersurface ${\mathcal{M}}^n$ of $({\overline{\mathcal{N}}}^m,g,\phi )$ with golden scalar curvature $r^G$ corresponding to the golden sectional curvature to be totally umbilical, assuming ${\mathcal{M}}^n$ possesses a bounded second fundamental form. Consequently, we derive the following:

Theorem 3. 

For $({\overline{\mathcal{N}}}^m,g,\phi )$, consider $(g,\psi ,\mathsf{\Lambda },k)$ a complete ANCRS situated on the hypersurface ${\mathcal{M}}^n$, possessing a uniform golden sectional curvature r, characterized by its golden sectional curvature, with a bounded second fundamental form. Consequently,

(i) 

$\mathsf{\Lambda }\ge \left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)+{\displaystyle \frac{c}{3}}\left\{n-3+{2\parallel \phi \parallel }^2-2\parallel \phi \parallel \right\}{\mathcal{H}}^2$, then ${\mathcal{M}}^n$ is totally geodesic with

$$\mathsf{\Lambda }=\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)-{\displaystyle \frac{c}{3}}\left\{n-3+{2\parallel \phi \parallel }^2-2\parallel \phi \parallel \right\},$$

and the scalar curvature is given by

$$r={\displaystyle \frac{c}{3}}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\}$$

(ii) 

${\mathcal{M}}^n$ is compact and

$$\mathsf{\Lambda }\ge \left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)+{\displaystyle \frac{c}{3}}\left\{n-3+{2\parallel \phi \parallel }^2-2\parallel \phi \parallel \right\}{\mathcal{H}}^2,$$

then ${\mathcal{M}}^n$ is isometric to a Euclidean sphere.

(iii) 

${\mathcal{M}}^n$ is totally umbilical with

$$\mathsf{\Lambda }\ge n[{\displaystyle \frac{c}{3}}\left\{2\parallel \phi \parallel (\parallel \phi \parallel -1)-2\right\}+{\mathcal{H}}^2].$$

Here, $\psi :{\mathcal{M}}^n⟶\mathbb{R}$ represents the potential function with $\left|\nabla \psi \right|\in {\mathcal{L}}^1\left(\mathcal{M}\right).$

Proof. 

By using (8) and (5), we derive

$$\begin{array}{ccc}\hfill {\mathcal{L}}_r\psi & =& n\left[\mathsf{\Lambda }-\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)-{\displaystyle \frac{c}{3}}\left\{n-3+2{\parallel \phi \parallel }^2-2\parallel \phi \parallel \right\}-n{\mathcal{H}}^2\right]-{∥\mathcal{A}∥}^2.\hfill \end{array}$$

(11)

It is evident that on ${\mathcal{M}}^n$, ${\mathcal{L}}_r\psi$ is a non-negative function for our study on $\mathsf{\Lambda }$. According to Lemma 1, ${\mathcal{L}}_k\psi$ vanishes identically. Consequently, ${\mathcal{M}}^n$ is totally geodesic from (11), and we get

$$\mathsf{\Lambda }={\displaystyle \frac{c}{3}}\left\{n-3+{2\parallel \phi \parallel }^2-2\parallel \phi \parallel \right\}+\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right).$$

Moreover, it is clear from (5) that

$$r={\displaystyle \frac{c}{3}}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\},$$

which establishes (1).

Given its totally geodesic nature, the ambient space will take the form of a sphere ${\mathcal{S}}^m$ (if ${\mathcal{M}}^n$ is compact). Furthermore, ${\mathcal{M}}^n$ is isometric to ${\mathcal{S}}^m$, i.e., (2). As a consequence of (11), we get

$$\begin{array}{c}\hfill {\mathcal{L}}_k\psi =n\left[\mathsf{\Lambda }\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)-n\left({\displaystyle \frac{c}{3}}\left\{2\parallel \phi \parallel (\parallel \phi \parallel -1)-2\right\}+{\mathcal{H}}^2\right)\right]+{\left|\Phi \right|}^2.\end{array}$$

Consequently, it follows from our assumption on $\mathsf{\Lambda }$ that

$${\mathcal{L}}_k\psi \ge 0.$$

So, from Lemma (1), it follows that one has

$${\mathcal{L}}_k\psi =0.$$

This demonstrates the totally umbilical nature of ${\mathcal{M}}^n$. □

Corollary 3. 

Suppose that the hypotheses of Theorem 3 are true and $r^G$ of $({\overline{\mathcal{N}}}^m,g,\phi )$ is expressed in terms of the golden sectional curvature. Then

(i) 

$\mathsf{\Lambda }\ge \left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)+{\displaystyle \frac{c}{3}}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}+n^2{\mathcal{H}}^2$ is totally geodesic with

$$\mathsf{\Lambda }=\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)-{\displaystyle \frac{c}{3}}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\},$$

and

$$r^G={\displaystyle \frac{c}{3}}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\};$$

(ii) 

${\mathcal{M}}^n$ is compact and

$$\mathsf{\Lambda }\ge \left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)+{\displaystyle \frac{c}{3}}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}+n^2{\mathcal{H}}^2$$

is isometric to a Euclidean sphere;

(iii) 

${\mathcal{M}}^n$ is totally umbilical provided

$$\mathsf{\Lambda }\ge n\left[{\displaystyle \frac{c}{3}}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -{\displaystyle \frac{3}{n}}-1\right\}+{\mathcal{H}}^2\right].$$

5. Compact Gradient ANCRS

A few natural outcomes, assuming $\mathsf{\Lambda }$ is constant, and that a compact ANCRS on the hypersurface ${\mathcal{M}}^n$ of $({\overline{\mathcal{N}}}^m,g,\phi )$ serves as the foundation for the main conclusions drawn in this section. Furthermore, the subsequent lemmas provide final results.

Lemma 3 

([25]). Let $\overline{\mathcal{M}}$ be non-compact and the support of ψ be compact, or $\overline{\mathcal{M}}$ be compact without a boundary. Then

(i) 

${\displaystyle {\int }_{\overline{\mathcal{M}}}{\mathcal{L}}_k\left(\psi \right)d\overline{\mathcal{M}}=0}$,

(ii) 

${\displaystyle {\int }_{\overline{\mathcal{M}}}\psi {\mathcal{L}}_k\left(\psi \right)d\overline{\mathcal{M}}=-{\int }_{\overline{\mathcal{M}}}g(P_r\nabla \psi ,\nabla \psi )}$

hold.

Dealing with the known traceless second fundamental form of ${\mathcal{M}}^n$ in $({\overline{\mathcal{N}}}^m,g,\phi )$, referred to as $\Phi =\mathcal{A}-\mathcal{H}I$, will serve our objectives. Note that

$$\mathrm{tr}\Phi =0,$$

and

$${|\Phi |}^2=\mathrm{tr}\left({\Phi }^2\right)={\left|\mathcal{A}\right|}^2-n{\mathcal{H}}^2\ge 0,$$

with equality holding if and only if ${\overline{\mathcal{M}}}^{2n+1}$ is totally umbilical.

We review Yau’s Lemma, which relates to Theorem 3 of [28], to finish this section.

Lemma 4. 

Let u be a non-negative, smooth subharmonic function on the complete Riemannian manifold $M^n$. If u belongs to ${\mathcal{L}}^p\left(\mathcal{M}\right)$ for some $p>1$, then u is necessarily a constant.

For every $p\ge 1$, we denote

$${\mathcal{L}}^p\left(\mathcal{M}\right)=\{u:{\mathcal{M}}^n\to \mathbb{R}{\int }_{\mathcal{M}}{\left|u\right|}^{p}d\mathcal{M}<+\infty \}.$$

Theorem 4. 

Let ${\mathcal{M}}^n$ be a compact gradient ANCRS immersed in a Golden Riemannian manifold $({\overline{\mathcal{N}}}^m,g,\phi )$ with constant golden sectional curvature r, such that $P_k$ is bounded from above or from below (in the sense of quadratic forms). If any of the below conditions are true,

(i) 

either ${\displaystyle \frac{nP}{2}}>-1$ and $r\ge 0$ and $\mathsf{\Lambda }\ge 0$ or $r\le 0$ and $\mathsf{\Lambda }\le 0$; or

(ii) 

${\displaystyle \frac{n}{2P}}<-1$ and $r\ge 0$ and $\mathsf{\Lambda }\le 0$ or $r\le 0$ and $\mathsf{\Lambda }\ge 0$; or

(iii) 

either $r\ge 2n\mathsf{\Lambda }-(nP+2)$ or $r\le 2n\mathsf{\Lambda }-(nP+2)$,

then $\mathcal{M}$ has to be trivial and of constant scalar curvature. Here, $P_k$ (in the sense of quadratic forms) is bounded from above or below.

Proof. 

Lemma 3 and (8) yield

$$0={\int }_{\mathcal{M}}{\mathcal{L}}_k\psi ={\int }_{\mathcal{M}}\left[\mathsf{\Lambda }n-\left({\displaystyle \frac{nP}{2}}+1\right)-r\right].$$

Consequently, when (i) or (ii) holds, one discovers

$$r=\mathsf{\Lambda }=0$$

and (8) yields

$${\mathcal{L}}_k\psi =0.$$

Given that $P_k$ has upper and lower bounds, there exists a constant $C>0$ to ensure

$$0={\mathcal{L}}_k\psi \le C\Delta \psi ,\mathrm{or}0={\mathcal{L}}_k\psi \ge -C\Delta \psi ,$$

respectively. Therefore, the function $\psi$ is subharmonic. By Hopf’s theorem, one can infer that $\psi$ is a constant function because $\mathcal{M}$ is compact. As a result, $\mathcal{M}$ is trivial.

Subsequently, (iii) follows exactly in the same way as (i) and (ii). □

Theorem 5. 

Consider a compact gradient ANCRS ${\mathcal{M}}^n$ immersed in $({\overline{\mathcal{N}}}^m,g,\phi )$, which has constant golden sectional curvature r, such that $P_r$ is bounded from above or from below and ${\displaystyle \frac{nP}{2}}\ne -1$. Then, ${\mathcal{M}}^n$ is trivial in case $\mathcal{M}$ has constant golden sectional curvature.

Proof. 

Lemma 3 and (8) allow us to write

$$\begin{array}{ccc}\hfill {\int }_{\mathcal{M}}{|n\mathsf{\Lambda }-({\displaystyle \frac{nP}{2}}+1)-r|}^2& =& {\int }_{\mathcal{M}}(n\mathsf{\Lambda }-({\displaystyle \frac{nP}{2}}+1)-r){\mathcal{L}}_k\psi \hfill \\ & =& (n\mathsf{\Lambda }-({\displaystyle \frac{nP}{2}}+1)-r){\int }_{\mathcal{M}}{\mathcal{L}}_k\psi =0.\hfill \end{array}$$

In turn, we conclude

$${\mathcal{L}}_k\psi =0$$

and

$$r=2n\mathsf{\Lambda }-\left(nP+2\right).$$

By assuming that $P_r$ has upper or lower bounds, we continue as in the proof of Theorem 4 to get that ${\mathcal{M}}^n$ is trivial. □

Moving on, we derive the subsequent corollaries based on Theorems 4 and 5.

Corollary 4. 

Let all the hypotheses of Theorem 4 hold, and the golden scalar curvature $r^G$ of ${\overline{\mathcal{N}}}^m$ is expressed corresponding to the golden sectional curvature. If any of the below statements holds

(i) 

either ${\displaystyle \frac{nP}{2}}>-1$ with $r^G\ge 0$ and $\mathsf{\Lambda }\ge 0$, or $r^G\le 0$ and $\mathsf{\Lambda }\le 0$; or

(ii) 

${\displaystyle \frac{n}{2P}}<-1$, coupled with $r^G\ge 0$ and $\mathsf{\Lambda }\le 0$, or $r^G\le 0$ and $\mathsf{\Lambda }\ge 0$; or,

(iii) 

$r^G$ is either greater than or equal to $2n\mathsf{\Lambda }-(nP+2)$, or less than or equal to $2n\mathsf{\Lambda }-(nP+2)$,

then ${\mathcal{M}}^n$ has constant golden scalar curvature $r^G$ and is trivial.

Proof. 

Since the ambient space has constant golden sectional curvature, by Equation (10), the operator ${\mathcal{L}}_k$ is a divergent type. Regarding (i) and (ii), Equation (6) jointly with assumption $c\le 0$ $(c<0)$ implies that golden scalar curvature $r^G$ of ${\mathcal{M}}^n$ satisfies $r^G\le 0$ ($r^G<0).$ Thus, from (8), we have

$$\begin{array}{ccc}\hfill {\mathcal{L}}_k& =& n\mathsf{\Lambda }-(P+{\displaystyle \frac{2}{n}})-r^G>0\hfill \end{array}$$

in both cases, which contradicts Lemma 1. Hence, $0\ge r^G=n\mathsf{\Lambda }-(P+{\displaystyle \frac{2}{n}})\ge 0,$ that is,

$$r^G=n\mathsf{\Lambda }-(P+{\displaystyle \frac{2}{n}}).$$

This completes the proof of the assertions. □

Corollary 5. 

Assuming the hypotheses of Theorem 5 are valid and the golden scalar curvature $r^G$ of ${\overline{\mathcal{N}}}^m$ aligns with the golden sectional curvature, then if $\mathcal{M}$ has a constant golden sectional curvature, ${\mathcal{M}}^n$ is trivial.

Proof. 

In view of (6), Lemma 3, and (8), we can easily prove the corollary. □

6. Some Applications

The Schur inequality [29], typically associated with positive real numbers, has generalizations and extensions to various mathematical contexts, including Riemannian manifolds and submanifolds. In this context, it relates quantities like sectional curvature, mean curvature, and scalar curvature of a submanifold to those of the ambient manifold. These inequalities provide insights into the geometry of submanifolds and their relationships with the surrounding space. Some Schur-type inequalities relate the mean curvature of a submanifold to its average value.

For example, Schur-type inequalities on sub-static manifolds and sub-static submanifolds naturally arise in general relativity [7].

The classical Schur’s lemma states that the scalar curvature of an Einstein manifold of dimension $n\ge 3$ must be constant. The applications of k-th mean curvatures of closed hypersurfaces in space forms and k-scalar curvatures for closed locally conformally flat manifolds can be expressed in terms of Schur-type inequalities involving the Ricci scalar. Moreover, Schur-type inequalities appear more generally in general relativity in relation to the so-called null convergence condition with Ricci and scalar curvature.

It was recently shown by De Lellis and Topping [29] that a closed Riemannian manifold satisfies

$${\int }_{\mathcal{M}}{(r-\overline{r})}^{2}d{v}_g\le {\displaystyle \frac{4n(n-1)}{{(n-2)}^2}}{\int }_{\mathcal{M}}{\left|S-{\displaystyle \frac{r}{n}}g\right|}^{2}d{v}_g.$$

(12)

The average of r over Riemannian manifolds $({\mathcal{M}}^n,g)$ is indicated by the symbol $\overline{r}$. Furthermore, if and only if $({\mathcal{M}}^n,g)$ is Einstein, the equality holds in the inequality (12).

We prove a Schur-type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature.

Theorem 6. 

Consider a compact gradient ANCRS ${\mathcal{M}}^n$ immersed in a Golden Riemannian manifold $({\overline{\mathcal{N}}}^m,g,\phi )$ with constant golden sectional curvature r, such that $P_k$ is bounded from above or from below and ${\displaystyle \frac{nP}{2}}\ne -1$. Then

$${\int }_M|r-\overline{r}{|}^2\le {\displaystyle \frac{2nC}{(n-2)(n-1)}}{\stackrel{\circ }{\parallel \mathrm{S}\parallel }}_{{\mathcal{L}}^2}{\parallel {\nabla }^{2}f-{\displaystyle \frac{\Delta f}{n}}g\parallel }_{{\mathcal{L}}^2},$$

where $\stackrel{\circ }{\mathrm{S}}$ is the traceless Ricci tensor, and $\overline{r}$ stands for the average of r on $\mathcal{M}$.

Proof. 

We recall from the contracted Bianchi identity that

$$\mathrm{div}\mathrm{S}+{\displaystyle \frac{1}{2}}\nabla r=0,$$

and, hence, that

$$\mathrm{div}\stackrel{\circ }{\mathrm{S}}=-{\displaystyle \frac{n-2}{2n}}\nabla r.$$

Since $\mathcal{M}$ is compact, using our assumption on $P_k$, we get the following:

$$\begin{array}{ccc}\hfill {\int }_{\mathcal{M}}{|n\mathsf{\Lambda }-({\displaystyle \frac{nP}{2}}+1)+(n-1)r|}^2& =& {\int }_{\mathcal{M}}[n\mathsf{\Lambda }-({\displaystyle \frac{nP}{2}}+1)S]L_{r}f\hfill \\ \hfill & =& {\int }_{\mathcal{M}}[n\mathsf{\Lambda }-({\displaystyle \frac{nP}{2}}+1)+(n-1)r]\mathrm{div}(P_k\nabla f)\hfill \\ \hfill & =& -(n-1){\int }_{M}g(\nabla r,P_k\nabla f)\le C(n-1){\int }_{\mathcal{M}}g(\nabla r,\nabla f)\hfill \\ \hfill & =& -{\displaystyle \frac{2nC(n-1)}{n-2}}{\int }_{\mathcal{M}}g(\mathrm{div}\stackrel{\circ }{\mathrm{S}},\nabla f)\hfill \\ \hfill & =& {\displaystyle \frac{2nC(n-1)}{n-2}}{\int }_{\mathcal{M}}g(\stackrel{\circ }{\mathrm{S}},{\nabla }^{2}f)\hfill \\ \hfill & =& {\displaystyle \frac{2nC(n-1)}{n-2}}{\int }_{\mathcal{M}}g(\stackrel{\circ }{\mathrm{S}},{\nabla }^{2}f-{\displaystyle \frac{\Delta f}{n}}g)\hfill \\ \hfill & \le & {\displaystyle \frac{2nC(n-1)}{n-2}}{\stackrel{\circ }{\parallel \mathrm{S}\parallel }}_{{\mathcal{L}}^2}{\parallel {\nabla }^{2}f-{\displaystyle \frac{\Delta f}{n}}g\parallel }_{{\mathcal{L}}^2},\hfill \end{array}$$

where we used $g(\stackrel{\circ }{\mathrm{S}},g)=0$. Since $\mathcal{M}$ is compact, we have

$$n\mathsf{\Lambda }-({\displaystyle \frac{nP}{2}}+1)=-(n-1)\overline{r}.$$

Therefore,

$${(n-1)}^2{\int }_{\mathcal{M}}|r-\overline{r}{|}^2={\int }_{\mathcal{M}}{|n\mathsf{\Lambda }-({\displaystyle \frac{nP}{2}}+1)+(n-1)S|}^2,$$

or equivalently,

$${(n-1)}^2{\int }_{\mathcal{M}}|r-\overline{r}{|}^2\le {\displaystyle \frac{2nC(n-1)}{n-2}}{\stackrel{\circ }{\parallel \mathrm{S}\parallel }}_{{\mathcal{L}}^2}{\parallel {\nabla }^{2}f-{\displaystyle \frac{\Delta f}{n}}g\parallel }_{{\mathcal{L}}^2},$$

i.e.,

$${\int }_{\mathcal{M}}|r-\overline{r}{|}^2\le {\displaystyle \frac{2nC}{(n-2)(n-1)}}{\stackrel{\circ }{\parallel \mathrm{S}\parallel }}_{{\mathcal{L}}^2}{\parallel {\nabla }^{2}f-{\displaystyle \frac{\Delta f}{n}}g\parallel }_{{\mathcal{L}}^2}.$$

(13)

This completes the proof. □

Remark 1. 

Observe that in the above theorem, if ${\mathcal{M}}^n$ is Einstein, then both sides of (13) vanish, so the equality holds. It would be an interesting problem to prove the rigidity.

Similarly, we turn up the Schur-type inequality for k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with golden scalar curvature.

Corollary 6. 

Consider a compact gradient ANCRS ${\mathcal{M}}^n$ immersed in a Golden Riemannian manifold $({\overline{\mathcal{N}}}^m,g,\phi )$ with golden scalar curvature $r^G$, such that $P_k$ is bounded from above or from below and ${\displaystyle \frac{nP}{2}}\ne -1$. Then

$${\int }_M|r^G-\overline{r^G}{|}^2\le {\displaystyle \frac{2nC}{(n-2)(n-1)}}{\stackrel{\circ }{\parallel \mathrm{S}\parallel }}_{{\mathcal{L}}^2}{\parallel {\nabla }^{2}f-{\displaystyle \frac{\Delta f}{n}}g\parallel }_{{\mathcal{L}}^2}.$$

7. Conclusions

k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature is the framework of this research. We obtained the triviality criteria for a compact gradient k-almost Newton-conformal Ricci soliton. Our analysis focused on the hypersurfaces of Golden Riemannian manifolds with constant golden sectional curvature, a bounded second fundamental form, and determined the conditions under which a k-almost Newton-conformal Ricci soliton on the hypersurface is totally umbilical. It has also been established that the steady k-almost Newton-conformal Ricci soliton admits a complete k-almost Newton-conformal Ricci soliton on the hypersurface of Golden Riemannian manifolds with constant golden sectional curvature. Furthermore, we deduced from Hopf’s strong maximum principle that the immersed k-almost Newton-conformal Ricci soliton in Golden Riemannian manifolds with constant golden sectional curvature is compact and totally geodesic. Additionally, our findings contribute to the understanding that the Euclidean sphere ${\mathcal{S}}^m$ is isometric to the immersed k-almost Newton-conformal Ricci soliton in Golden Riemannian manifolds with constant golden sectional curvature. Finally, we derived a Schur-type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature.