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
possessing a non-positive sectional curvature,
cannot be minimal, as an almost Ricci soliton is identical to a Ricci soliton, and an integrable norm on
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
is characterized by a non-positive sectional curvature.
Cunha et al. [
4] employed the Newton transformation
alongside the second-order differential operator
, for the range
, 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
is given by the following:
where
P denotes a non-dynamical scalar field (time-dependent scalar field),
S denotes the Ricci tensor,
is the scalar curvature, and
n is the dimension of the manifold
.
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:
Here, denotes the Lie derivative operator along the vector field F, and represents a real constant. P is a non-dynamical scalar field, S denotes the Ricci tensor, and n is the dimension of the manifold .
If
meets (
2), it is classified as a conformal Ricci soliton on
M [
10]. Based on
,
, or
,
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
, then we obtain the conformal gradient Ricci soliton by using mathematical operators as in [
11].
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
-tensor field
on a manifold
. Then
is a golden structure on
if [
12,
23]
In this case,
is called a golden manifold. Additionally, a Riemannian manifold
having a golden structure
fulfilling
is said to be a Golden Riemannian manifold [
13]. Substituting
with
in (
4), we have
When
the Golden Riemannian manifold
becomes locally decomposable.
Let
be a locally decomposable Golden Riemannian manifold with constant golden sectional curvature
c. Subsequently, we obtain [
15]
for every vector field
and
on
.
Consider a connected submanifold defined by
Then, the Gauss equation is written as [
24]
, where the Riemannian curvature tensors of
and
are
and
R, respectively.
Consider an orthonormal frame on and denote the Hilbert–Schmidt norm of .
Recall that on
, the scalar curvature
r is given by [
15]
where
denotes the mean curvature vector and
Furthermore, the golden scalar curvature
corresponding to the golden sectional curvature of the submanifold
in
is expressed by [
15]
3. -Almost Newton-Conformal Ricci Soliton
We remark that
is a linked and oriented hypersurface immersed in an
-dimensional Golden Riemannian manifold possessing a uniform golden sectional curvature. Let
be the
k-th Newton transformation
described by the recursive relationship:
for the identity operator
and
. Here,
indicates the composition of
with itself
j times, i.e.,
.
We call
a
k-almost Newton-conformal Ricci soliton (ANCRS) if, for each
, there exists
such that,
In this context,
and
are smooth functions defined on
. The tensor field
is given by
.
When
, (
8) simplifies to a gradient almost conformal Ricci soliton. The second fundamental form
of
possesses
n algebraic invariants, identified as elementary symmetric functions,
, derived from principal curvatures
. In immersion,
k-th mean curvature
can be expressed as follows:
where
is the binomial coefficient, and
is the
k-th elementary symmetric polynomial given by
Putting we obtain , the mean curvature of . denotes the trace operator.
Moreover, it follows that the characteristic polynomial of
is given by
The Laplacian operator
can be obtained for
. We also recall that
where
If the ambient space has constant sectional curvatures, Equation (
9) takes on this form
as
(see [
25]).
The traceless second fundamental form of the hypersurface is
and
is totally umbilical if and only if
Example 1.
We consider the usual immersion of in , known to be totally geodesic. Here, , . By selecting, , an immersion that satisfies (8) can be obtained. Example 2.
Let be the unit sphere in the Euclidean space andthe embedding with induced metric g on ; then is a Riemannian manifold with sectional curvature . Let be an immersion of an n-dimensional manifold into a smooth unit sphere.
According to [26], with a constant , select the functions on such thatwhere is the position vector and , , . We get that holds for However, is known to be totally umbilical, having a second fundamental form and a k-th mean curvature . More specifically, the Newton tensor for each is provided byaccording to which Therefore, by using the smooth function , we can obtain the Equation (8), which describes the hypersurface. Furthermore, if has constant scalar curvature, (8) can be simplified by using mathematical operators as follows:. 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
.
We also use the subsequent result [
25]:
Lemma 1.
Consider to be an oriented, non-compact, complete Riemannian manifold, and a smooth vector field U on it. The divergence maintains a uniform sign throughout . If the magnitude belongs to , then it follows that must be zero.
Theorem 1.2 of [
26] can be further generalized as follows:
Theorem 1.
Suppose is a complete ANCRS on in with constant golden sectional curvature r and bounded . If
- (i)
, , and , implies that is not minimal; or
- (ii)
, , and , implies that is not minimal; or
- (iii)
when , with , and , is minimal and isometric to .
Here, denotes the potential function with
Proof. From (
9), it is clear that the type of
is divergent. Since the second fundamental form is bounded on
, Equation (
7) implies that
has a bounded norm
On the other hand, assume that
is minimal to demonstrate (1) and (2). Using (
5) with
, we find that
is fulfilled by
. Therefore, the contraction of (
8) implies
in both cases. This contradicts Lemma 1. This finishes the proof of (1) and (2).
Next, because
has constant sectional curvature
and using the minimality of
, (
5) reduces to
But,
and
. Then, in view of Lemma 1, we deduce that
One concludes that
proving that
is totally geodesic and flat. □
Corollary 1.
Suppose all considerations of Theorem 1 are true and the golden scalar curvature of is expressed in terms of the golden sectional curvature and . Then
- (i)
, , is not minimal.
- (ii)
, , fails to be minimal.
- (iii)
, , is minimal and isometrically equivalent to .
The conclusion coming next and related to Theorem 3 of [
28] is necessary for establishing our results.
Lemma 2.
Let be complete, endowed with a subharmonic function u that is non-negative and smooth. For , u is constant if .
Additionally, we have
Theorem 2.
Let be a complete ANCRS on in , which has constant golden sectional curvature r and . If
- (i)
, , then is not minimal;
- (ii)
, , then is not minimal;
- (iii)
, , and is minimal, then is flat and totally geodesic.
In this context, is regarded as non-negative and belongs to for , while is assumed to be upper-bounded when considering quadratic forms.
Proof. The starting point to demonstrate (1) is to consider the contradiction that
is minimal in relation to (
5) and the hypothesis
. The contraction of (
8) yields
Assuming that
is bounded from above, there exists a positive constant that fulfills
As a result, we argue that
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 of is expressed in terms of the golden sectional curvature and .
- (i)
, implies is not minimal;
- (ii)
, implies is not minimal;
- (iii)
, , and being a minimal manifold imply that is necessarily flat and totally geodesic.
We extend Theorem 1.5 of [
2] for
in our subsequent result. We also give the conditions for an ANCRS on a hypersurface
of
with golden scalar curvature
corresponding to the golden sectional curvature to be totally umbilical, assuming
possesses a bounded second fundamental form. Consequently, we derive the following:
Theorem 3.
For , consider a complete ANCRS situated on the hypersurface , possessing a uniform golden sectional curvature r, characterized by its golden sectional curvature, with a bounded second fundamental form. Consequently,
- (i)
, then is totally geodesic with and the scalar curvature is given by - (ii)
then is isometric to a Euclidean sphere.
- (iii)
is totally umbilical with
Here, represents the potential function with
Proof. By using (
8) and (
5), we derive
It is evident that on
,
is a non-negative function for our study on
. According to Lemma 1,
vanishes identically. Consequently,
is totally geodesic from (
11), and we get
Moreover, it is clear from (
5) that
which establishes (1).
Given its totally geodesic nature, the ambient space will take the form of a sphere
(if
is compact). Furthermore,
is isometric to
, i.e., (2). As a consequence of (
11), we get
Consequently, it follows from our assumption on
that
So, from Lemma (1), it follows that one has
This demonstrates the totally umbilical nature of . □
Corollary 3.
Suppose that the hypotheses of Theorem 3 are true and of is expressed in terms of the golden sectional curvature. Then
- (i)
is totally geodesic with - (ii)
is isometric to a Euclidean sphere;
- (iii)
is totally umbilical provided
5. Compact Gradient ANCRS
A few natural outcomes, assuming is constant, and that a compact ANCRS on the hypersurface of serves as the foundation for the main conclusions drawn in this section. Furthermore, the subsequent lemmas provide final results.
Lemma 3
([
25])
. Let be non-compact and the support of ψ be compact, or be compact without a boundary. Then- (i)
,
- (ii)
hold.
Dealing with the known traceless second fundamental form of
in
, referred to as
, will serve our objectives. Note that
and
with equality holding if and only if
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 . If u belongs to for some , then u is necessarily a constant.
For every
, we denote
Theorem 4.
Let be a compact gradient ANCRS immersed in a Golden Riemannian manifold with constant golden sectional curvature r, such that is bounded from above or from below (in the sense of quadratic forms). If any of the below conditions are true,
- (i)
either and and or and ; or
- (ii)
and and or and ; or
- (iii)
either or ,
then has to be trivial and of constant scalar curvature. Here, (in the sense of quadratic forms) is bounded from above or below.
Proof. Consequently, when (i) or (ii) holds, one discovers
and (
8) yields
Given that
has upper and lower bounds, there exists a constant
to ensure
respectively. Therefore, the function
is subharmonic. By Hopf’s theorem, one can infer that
is a constant function because
is compact. As a result,
is trivial.
Subsequently, (iii) follows exactly in the same way as (i) and (ii). □
Theorem 5.
Consider a compact gradient ANCRS immersed in , which has constant golden sectional curvature r, such that is bounded from above or from below and . Then, is trivial in case has constant golden sectional curvature.
Proof. Lemma 3 and (
8) allow us to write
By assuming that has upper or lower bounds, we continue as in the proof of Theorem 4 to get that 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 of is expressed corresponding to the golden sectional curvature. If any of the below statements holds
- (i)
either with and , or and ; or
- (ii)
, coupled with and , or and ; or,
- (iii)
is either greater than or equal to , or less than or equal to ,
then has constant golden scalar curvature and is trivial.
Proof. Since the ambient space has constant golden sectional curvature, by Equation (
10), the operator
is a divergent type. Regarding (i) and (ii), Equation (
6) jointly with assumption
implies that golden scalar curvature
of
satisfies
(
Thus, from (
8), we have
in both cases, which contradicts Lemma 1. Hence,
that is,
This completes the proof of the assertions. □
Corollary 5.
Assuming the hypotheses of Theorem 5 are valid and the golden scalar curvature of aligns with the golden sectional curvature, then if has a constant golden sectional curvature, 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 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
The average of
r over Riemannian manifolds
is indicated by the symbol
. Furthermore, if and only if
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 immersed in a Golden Riemannian manifold with constant golden sectional curvature r, such that is bounded from above or from below and . Thenwhere is the traceless Ricci tensor, and stands for the average of r on . Proof. We recall from the contracted Bianchi identity that
and, hence, that
Since
is compact, using our assumption on
, we get the following:
where we used
. Since
is compact, we have
Therefore,
or equivalently,
i.e.,
This completes the proof. □
Remark 1.
Observe that in the above theorem, if
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 immersed in a Golden Riemannian manifold with golden scalar curvature , such that is bounded from above or from below and . Then