1. Introduction
For nearly two thousand years, the concept of the golden ratio has fascinated scholars from diverse disciplines. The allure of the golden ratio extends beyond mathematics; it captivates biologists, artists, musicians, historians, architects, and psychologists alike. The golden ratio, known for its aesthetic harmony and proportionality, is extensively employed in iconic architectural structures and artworks, musical composition frameworks, harmonious frequency ratios, and human body measurements. It is likely fair to say that the golden ratio has inspired more scholars in various fields than any other number throughout the history of mathematics [
1].
Polynomial structures on a manifold, as discussed in [
2], were the foundation for the concept of golden structure. C. E. Hretcanu and M. Crasmareanu investigated some characteristics of the induced structure on an invariant submanifold within a golden Riemannian manifold in [
3]. In [
4], Crasmareanu and Hretcanu used a corresponding almost product structure to study the geometry of the golden structure on a manifold. In [
5], Hretcanu and Crasmareanu demonstrated that a golden structure also induces a golden structure on each invariant submanifold. The issue of integrability for golden Riemannian structures was examined by Gezer et al. in [
6]. Ozkan studied a golden semi-Riemannian manifold in [
7], where he defined the golden structure’s horizontal lift in the tangent bundle. Other structures on golden Riemannian manifolds have been studied by many authors (see, e.g., [
8,
9,
10]).
However, in the study of differential geometry, the theory of submanifolds is an intriguing subject. Its roots are in Fermat’s work on the geometry of surfaces and plane curves. Since then, it has been developed in different directions of differential geometry and mechanics. It is still a vibrant area of study that has contributed significantly to the advancement of differential geometry in the present era. Among all the submanifolds of an ambient manifold, there are two well known types: invariant submanifolds and anti-invariant submanifolds. The differential geometry of submanifolds in golden Riemannian manifolds was initially studied by Crasmareanu and Hretcanu. Certain characteristics of invariant submanifolds in a Riemannian manifold with a golden structure were investigated by Hretcanu and Crasmareanu in [
3], which have been advanced considerably since then. Various classifications of submanifolds within golden Riemannian manifolds have been established based on how their tangent bundles react to the golden structure of the ambient manifold and been explored by numerous geometers. Erdoğan and Yıldırım presented the idea of semi-invariant submanifolds within golden Riemannian manifolds as a generalization of both invariant and anti-invariant types, followed by an analysis of the geometry of their defining distributions [
11]. The properties and distributions associated with semi-invariant submanifolds in golden Riemannian manifolds were further explored by Gök, Keleş, and Kılıç [
12]. The notion of pointwise slant submanifolds and pointwise bi-slant submanifolds in golden Riemannian manifolds was introduced by Hretcanu and Blaga in [
13].
R. L. Bishop and B. O’Neill [
14] proposed the idea of a warped product. Warped-product CR submanifolds in a Kähler manifold, which consist of warped products of holomorphic and totally real submanifolds, were first studied by B.-Y. Chen in [
15,
16,
17,
18,
19]. Later, Riemannian manifolds with golden warped products were studied by Blaga and Hretcanu [
20], who also investigated submanifolds with pointwise semi-slant and hemi-slant warped products within locally golden Riemannian manifolds [
13].
The similarities between the geometries of semi-Riemannian submanifolds and their Riemannian counterparts are well established, yet the study of lightlike submanifolds presents unique challenges due to the intersection of their normal vector bundles with the tangent bundle. This complexity adds to the intrigue of researching lightlike geometry, which finds practical applications in mathematical physics, notably in general relativity and electromagnetism [
21]. Duggal and Bejancu were pioneers in the study of lightlike submanifolds within semi-Riemannian manifolds [
21]. In 2017, Poyraz and Yasar explored lightlike hypersurfaces in golden semi-Riemannian manifolds [
22] and further extended their research to define lightlike submanifolds of golden semi-Riemannian manifolds in 2019 [
23]. Subsequently, numerous researchers have investigated various types of lightlike submanifolds in golden semi-Riemannian manifolds, as evidenced by references [
24,
25,
26,
27,
28,
29], among others. Additionally, the concept of a lightlike hypersurface in meta-golden Riemannian manifolds was introduced by Erdoğan et al. [
30].
Finding the optimal inequality between the intrinsic and extrinsic invariants of a Riemannian submanifold is a key problem in submanifold geometry. Chen [
31,
32] developed the
invariants in this context, which are known today as Chen invariants. Utilizing these invariants, along with the mean curvature, which is the key extrinsic invariant of Riemannian submanifolds, he formulated sharp inequality relationships, which are well known as Chen inequalities. The study of Chen invariants and Chen inequalities across various submanifolds in diverse ambient spaces has been thoroughly pursued (see, e.g., [
32,
33,
34]). Research on Chen inequalities within golden Riemannian manifolds and golden-like statistical manifolds was conducted by Choudhary and Uddin [
35] and Bahadir et al. [
36], respectively.
Thanks to F. Casorati [
37], the widely accepted Gaussian curvature can be substituted with the Casorati curvature. Within the framework of Casorati curvatures for submanifolds across diverse ambient spaces, geometric inequalities have been formulated. The rationale behind the introduction of this curvature by F. Casorati is that it disappears precisely when both principal curvatures of a surface in
are zero, aligning more closely with the typical understanding of curvature. Numerous scholars have explored Casorati curvatures to derive sharp inequalities for specific submanifolds in varied golden ambient spaces (refer to [
38,
39,
40,
41], etc.). Moreover, in 1979, P. Wintgen [
42] introduced a significant geometric inequality involving Gauss curvature, normal curvature, and square mean curvature known as Wintgen’s inequality [
43]. For slant, invariant, C-totally real, and Lagrangian submanifolds in golden Riemannian spaces, generalized Wintgen-type inequalities were introduced by Choudhary et al. [
44].
Motivated by the previously mentioned advancements in the subject, the purpose of this paper is to provide a comprehensive survey of the latest progress on golden Riemannian manifolds achieved over the past decade.
3. Submanifolds Immersed in Riemannian Manifolds with Golden Structure
3.1. Invariant Submanifolds with Golden Structure
Among all the submanifolds of an ambient manifold, invariant submanifolds are a common class. It is commonly known that practically every property of an ambient manifold is inherited by an invariant submanifold. As a result, invariant submanifolds represent a dynamic and productive area of study that has greatly influenced the advancement of modern differential geometry. Numerous articles concerning invariant submanifolds in golden Riemannian manifolds have been published.
In [
3], Hretcanu and Crasmareanu studied invariant submanifolds in golden Riemannian manifolds and proved the following three propositions:
Proposition 2. Let N be an n-dimensional submanifold of a golden Riemannian manifold of codimension r and let , be the structure on N induced by structure , where is the 1 form, represents tangent vector fields, and is an matrix of a real function on N. Then, an essential and sufficient requirement for N to be invariant is that the induced structure on N be a golden Riemannian structure whenever is non-trivial.
Proposition 3. If is an invariant submanifold of codimension r in a golden Riemannian manifold with and is the structure on N induced by (where is the Levi–Civita connection defined on N with respect to g), then the Nijenhuis torsion tensor field of vanishes identically on N.
Proposition 4. Let N be an invariant submanifold of codimension r in a golden Riemannian manifold with and let be the induced structure on N. If normal connection on normal bundle vanishes identically , then components , and of the Nijenhuis torsion tensor field of for the structure induced on N have the following forms:
for any . Remark 3 ([
3])
. Under the conditions of Proposition 4, if , where A is the shape operator for , then components , and vanish identically on N. Inspired by [
3], in [
12], Gök et al. demonstrated the local decomposability of any invariant submanifold of a golden Riemannian manifold and came up with a definition of invariance for submanifolds in a golden Riemannian manifold. They also determined the prerequisites that must be met for any invariant submanifold to be totally geodesic.
Theorem 3 ([
12])
. Let N be an invariant submanifold of a locally decomposable golden Riemannian manifold . Then, N is a locally decomposable golden Riemannian manifold whenever the induced structure () on N is non-trivial. Theorem 4 ([
12])
. Let N be a submanifold of a golden Riemannian manifold . Then, N is an invariant submanifold if and only if there exists a local orthonormal frame of the normal bundle () such that it consists of eigenvectors of the golden structure (ϕ). Remark 4. The result of N as a totally geodesic invariant submanifold was also obtained in [12] by Gök et al. 3.2. Anti-Invariant Submanifolds in Golden Riemannian Manifolds
Some properties of an anti-invariant submanifold of a golden Riemannian manifold were studied in [
54], and some necessary prerequisites for any submanifold in a locally decomposable golden Riemannian manifold to be anti-invariant were obtained. Any anti-invariant submanifold (
N) of a golden Riemannian manifold
is a submanifold such that the golden structure (
) of the ambient manifold (
) carries each tangent vector of the submanifold (
N) into its corresponding normal space in the ambient manifold (
), that is,
for each point
(see [
54]).
Theorem 5 ([
54])
. Let N be an n-dimensional submanifold of a -dimensional locally decomposable golden Riemannian manifold . Then, for any , N is an anti-invariant submanifold whenever . In addition, the submanifold (N) is totally geodesic. Theorem 6 ([
11])
. Let N be an n-dimensional submanifold of a -dimensional locally decomposable golden Riemannian manifold . If for any , then N is an anti-invariant submanifold. Furthermore, the submanifold (N) is totally geodesic. Remark 5. In [54], M. Gök et al. also obtained results with respect to the existence of an orthonormal frame of an anti-invariant submanifold of a locally decomposable golden Riemannian manifold. Gök and Kılıç [
55] studied a non-invariant submanifold of a locally decomposable golden Riemannian manifold in a case in which the rank of the set of tangent vector fields of the structure on the submanifold induced by the golden structure of the ambient manifold is less than or equal to the co-dimension of the submanifold.
Theorem 7 ([
55])
. Let N be a submanifold of codimension r in a locally decomposable golden Riemannian manifold . If the tangent vector fields () are linearly independent and , then N is a totally geodesic submanifold. Theorem 8 ([
55])
. Let N be a submanifold of codimension r in a locally decomposable golden Riemannian manifold , where is an eigenvalue of the matrix . If for any and then N is totally geodesic. Theorem 9 ([
55])
. Let N be a submanifold of codimension r in a locally decomposable golden Riemannian manifold . If for any is constant, and N is totally umbilical, then N is totally geodesic. Remark 6. (i)
Gök and Kılıç [55] also obtained some results on the non-invariant submanifold if the tangent vector fields of the induced structure are linearly dependent.(ii)
The stability problem of certain anti-invariant submanifolds in golden Riemannian manifolds was discussed by the same authors in [56].(iii)
Effective relations for certain induced structures on a submanifold of codimension 2 in golden Riemannian manifolds were obtained in [57]. 3.3. Slant Submanifolds of Golden Riemannian Manifolds
Given a golden Riemannian manifold
, let
be one of its submanifolds and
g be the induced metric on
N. Then, we can write
for any
, where
and
are the tangent and transversal components of
, respectively.
A submanifold of a golden Riemannian manifold is referred to as a slant submanifold if, at any point (x), each nonzero vector tangent to N and the angle between and , as represented by , are independent of the selection of and . It can also be seen that N is a -invariant (resp. -anti-invariant) submanifold if the slant angle is (resp. ). The term proper slant (or -slant proper) submanifold refers to a slant submanifold that is neither anti-invariant nor invariant.
Using a golden Riemannian manifold, Uddin and Bahadir [
45] establish the following characterization of slant submanifolds.
Theorem 10 ([
45])
. Assume that a golden Riemannian manifold has a submanifold . Consequently, N is a slant submanifold if and only if is a constant such thatAdditionally, if θ represents the slant angle of N, then .
Corollary 1 ([
45])
. Take a golden Riemannian manifold and let be a submanifold of it. After that, N is a slant submanifold if and only if exists and ensureswhere and θ are slant angles of N. Remark 7. In [45] Uddin and Bahadir also derived some results of ϕ-invariant and ϕ-anti-invariant submanifolds of a golden Riemannian manifold and provided some examples of such submanifolds. 3.4. Semi-Invariant Submanifolds of Golden Riemannian Manifolds
Definition 6 ([
58])
. Given a golden Riemannian manifold , consider N to be a real submanifold of . If N is equipped with a pair of orthogonal distributions that meet the given conditions, it can be deemed a semi-invariant submanifold of .(i)
(ii) The distribution (D) is invariant, i.e., for each ;
(iii) The distribution () is anti-invariant, i.e., for each
For any , a semi-invariant submanifold (N) is considered invariant and anti-invariant if and , respectively.
The following results were obtained by Erdogan et al. for semi-invariant submanifolds of the golden Riemannian manifold investigated in [
58].
Theorem 11 ([
58])
. Assume that N is a semi-invariant submanifold of , the golden Riemannian manifold. Consequently, the distribution (D) is integrable if and only ifwhere h is the second fundamental form and and .
Theorem 12 ([
58])
. Let N be a semi-invariant submanifold of the golden Riemannian manifold . Then, distribution D is integrable if and only ifhas no components in D, where A is a shape operator and for every and . Remark 8. The conditions for distributions D and of semi-invariant submanifolds of golden Riemannian manifolds to be a totally geodesic foliation were examined in [58]. The condition for a semi-invariant submanifold (N) to be totally geodesic was also covered. Readers can also refer to [45] for conclusions of a similar kind for semi-invariant submanifolds of a golden Riemannian manifold. Remark 9. M. Gök et al. [59] proposed specific characterizations for every submanifold of a golden Riemannian manifold to be semi-invariant in terms of canonical structures on the submanifold as a consequence of the ambient manifold’s golden structure. Totally umbilical, semi-invariant submanifolds of golden Riemannian manifolds were studied in [
11].
Theorem 13 ([
11])
. Let N be a totally umbilical submanifold of a golden Riemannian manifold (). Then, distribution D is always integrable. Theorem 14 ([
11])
. Let N be a totally umbilical submanifold of a golden Riemannian manifold . Then, is integrable. Remark 10. Moreover, the properties of semi-invariant submanifolds and totally umbilical, semi-invariant submanifolds of golden Riemannian manifolds with constant sectional curvatures were studied by Sahin et al. in [60]. 3.5. Skew Semi-Invariant Submanifolds
In [
61], Ahmad and Qayyoom studied skew semi-invariant submanifolds in a golden Riemannian manifold and in the a locally golden Riemannian manifold.
Definition 7 ([
61])
. A submanifold (N) of a golden Riemannian manifold () is defined as a skew semi-invariant submanifold if there exists an integer (k) and constant functions () defined on N with values in the range of such that(i)
Each is a distinct eigenvalue of withfor , and (ii)
The dimensions of and are independent of . Remark 11. The tangent bundle of N has the following decomposition: If then N is a semi-invariant submanifold. Also, if and () are trivial, then N is an invariant (or anti-invariant) submanifold of
Definition 8 ([
61])
. A submanifold (N) of a locally golden Riemannian manifold () is defined as a skew semi-invariant submanifold of order 1 if N is a skew semi-invariant submanifold with . In this case, we havewhere and are constant. A skew semi-invariant submanifold of order 1 is proper if and . Remark 12. Some lemmas for proper skew semi-invariant submanifolds of a locally golden Riemannian manifold were also discussed in [61]. 3.6. Pointwise Slant Submanifolds in Golden Riemannian Manifolds
The notion of slant submanifolds in almost Hermitian manifolds was first introduced by the first author in [
62,
63,
64]. Later, the first author and Garay [
65] extended the notion of slant submanifolds to pointwise slant submanifolds in almost Hermitian manifolds. Hretcanu and Blaga [
13] defined the notion of pointwise slant submanifolds of golden Riemannian manifolds as follows.
A submanifold
N of a golden Riemannian manifold
is referred to as a
pointwise slant [
13] if, at every point (
), the angle (
) between
and
(called the Wirtinger angle) is consistent, regardless of the nonzero tangent vector (
), but it depends on
. The Wirtinger angle is a real-valued function (
; called a Wirtinger function) verifying
for any
and
. If the Wirtinger function (
) of a pointwise slant submanifold of a golden Riemannian manifold is globally constant, it is referred to as a
slant submanifold.
Proposition 5 ([
13])
. In a golden Riemannian manifold , if N is an isometrically immersed submanifold and T is the map, then N is a pointwise slant submanifold if and only iffor some real-valued function () for . Proposition 6 ([
13])
. Let N be a submanifold of a golden Riemannian manifold that is isometrically immersed. Given N as a pointwise slant submanifold and as its Wirtinger angle, thenfor any and any . 3.7. Pointwise Bi-Slant Submanifolds in Golden Riemannian Manifolds
Consider N as an immersed submanifold within a golden Riemannian manifold . We define N as a pointwise bi-slant submanifold of if there exist two orthogonal distributions (D and ) on N such that
(i) ;
(ii) and ;
(iii) Distributions D and are pointwise slant with slant functions of and , respectively, for . The pair of slant functions is referred to as the bi-slant function.
A pointwise bi-slant submanifold (
N) is called
proper if its bi-slant functions (
) and neither
or
are constant on
N. Specifically, if
and
, then
N is called a pointwise semi-slant submanifold; if
and
, then
N is called a pointwise hemi-slant submanifold.
and
are verified by distributions
D and
on
N if
N is a pointwise bi-slant submanifold of
[
13].
Remark 13. Some examples of pointwise bi-slant submanifolds in golden Riemannian manifolds were given in [13], where Blaga and Hretcanu provided fundamental lemmas for pointwise bi-slant, pointwise semi-slant, and pointwise hemi-slant submanifolds in locally golden Riemannian manifolds. 3.8. CR Submanifolds of a Golden Riemannian Manifold
A submanifold (N) within a golden Riemannian manifold () is called a CR submanifold if there exists a differentiable distribution () on N that meets the following criteria:
(i) D is holomorphic, meaning for every ; and
(ii) The orthogonal complementary distribution (
) is completely real, i.e.,
for each
. If
(or
), then the CR submanifold (
N) is a holomorphic submanifold (or a totally real submanifold). If
, then the CR submanifold is an anti-holomorphic submanifold (or a generic submanifold). A submanifold is considered a proper CR submanifold if it is neither holomorphic nor totally real [
66].
The authors of [
66] defined and studied CR submanifolds of a golden Riemannian manifold.
Proposition 7 ([
66])
. Let N be a CR submanifold of a locally golden Riemannian manifold (). Then,for and , where is the complementary orthogonal sub-bundle of in ; is the unit normal vector field; and , and W are vector fields. Lemma 1 ([
66])
. Consider N a CR submanifold of , a locally golden Riemannian manifold. Given any , then Remark 14. The integral condition of the D of CR submanifolds of a golden Riemannian manifold was also discussed in [66]. The following outcomes were attained by Ahmad and Qayyoom in [
66] from their study of totally umbilical CR submanifolds of golden Riemannian manifolds:
Lemma 2 ([
66])
. Assume that N is a totally umbilical CR submanifold of , a locally golden Riemannian manifold. Then, either H, the mean curvature vector, is perpendicular to or the totally real distribution ( is one-dimensional. Theorem 15 ([
66])
. For a locally golden Riemannian manifold (), let N be a totally umbilical CR submanifold. Therefore, for every CR-section π, i.e., the CR-sectional curvature of vanishes. 9. Chen Invariants and Inequalities
Let be a Riemannian n manifold. Let us choose a local field of an orthonormal frame () on . denotes the sectional curvatures of of the plane section spanned by and .
The
scalar curvature (
) of
at
p is defined by
Similarly, if
is an
ℓ-dimensional linear subspace of
with
, then the scalar curvature (
) of
is defined as follows:
where
is an orthonormal basis of
.
9.1. Chen Invariants
Let n be a positive integer (). For a positive integer (), let denote the set consisting of k tuples of integers () such that Furthermore, .
For a given point (
p) in a Riemannian
n-manifold
and each
, in [
31,
86,
87], the first author introduced the following invariants:
where
run over all
k mutually orthogonal subspaces of
such that
and
. In particular, we have
(a) ;
(b) , where K is the sectional curvature;
(c) .
Remark 37. is known today as the first Chen invariant among all of the invariants ().
9.2. Chen Inequalities
For each
, we set
The first author proved the following optimal universal inequalities (see [
32,
86,
87,
88]).
Theorem 74. Let N be an n-dimensional submanifold of a Riemannian manifold (). Then, for each point () and each k-tuple (), we havewhere is the squared mean curvature of N and is the maximum of the sectional curvature function of restricted to 2-plane sections of the tangent space () at p. The equality case of inequality (15) holds at if and only if the following conditions hold: (a)
There is an orthonormal basis () at p such that the shape operators of N in at p take the following form:where I is an identity matrix and is a symmetric submatrix such that(b)
For mutual orthogonal subspaces () satisfying at p, we have for and , where An important case of Theorem 74 is presented as follows.
Theorem 75 ([
86,
87])
. For an n-dimensional submanifold (N) of a real-space form () of constant curvature (c), we haveThe equality case of inequality (18) holds at a point if and only if there is an orthonormal basis () such that the shape operators at p take the forms of (16) and (17).
10. Inequalities in Golden Riemannian Manifolds
Following Chen’s inequalities, many researchers have studied Chen-type inequalities within golden Riemannian manifolds.
10.1. Chen-Type Inequality in Golden Riemannian Manifolds
The following findings about Chen-type inequalities for slant submanifolds in golden Riemannian manifolds were discovered by Uddin and Choudhary in [
35].
Theorem 76 ([
35]).
The following inequality holds for any proper θ-slant submanifold () that is isometrically immersed in a locally golden product manifold (). For the equality case, consider the following.
Theorem 77 ([
35]).
When all conditions of the above Theorem 76 are met, equality in Equation (19) is achieved at if and only if , and the shape operator (A) has the following form:for . Then, an inequality involving is calculated as follows.
Theorem 78 ([
35]).
In each proper θ-slant submanifold () immersed in , the following inequality is true:whereAdditionally, the equality sign in (21) holds at a point if and only if there exists an orthonormal basis () and A such thatfor , where satisfyand is a symmetric submatrix satisfying Remark 38. Additionally, in [35], Uddin and Choudhary deduced a special case of Theorems 76 and 78 for ϕ-invariant submanifolds () immersed in a locally golden product manifold () and inequalities for a Ricci curvature tensor. 10.2. Casorati Curvature in Golden Riemannian Manifolds
In 1890, Casorati [
37] introduced what is now termed Casorati curvature for surfaces in a Euclidean 3-space
. Casorati favored this curvature over Gaussian curvature because the latter may vanish for surfaces that intuitively seem curved, whereas the former only vanishes at planar points. The Casorati curvature (
C) of a submanifold in a Riemannian manifold is generally defined as the normalized squared norm of the second fundamental form. Decu et al. introduced normalized Casorati curvatures
and
in 2007 (refer to [
89]), aligning with the essence of
invariants. In 2008, they extended normalized Casorati curvatures to generalized normalized
Casorati curvatures (
and
) in [
90]. Concurrently, they were able to ascertain the optimal inequality concerning the (intrinsic) scalar curvature and the (extrinsic)
Casorati curvature.
Let us recall the Weingarten and Gauss formulas in this context. For a Riemannian manifold
and a Riemannian submanifold (
N) isometrically immersed in
, where
and ∇ are the Levi–Civita connections on
and
N respectively, and
h represents the second fundamental form of
N, the Weingarten and Gauss formulas are expressed as follows:
∀
and
, where
denotes the shape operator of
N associated with
and
represents the connection in the normal bundle. The relationship between
and
h can then be recalled.
The Gaussian formula is written as
for any vector fields tangent to
N, such as
, and
W. Assume that the local orthonormal tangent frame is
and the local orthonormal normal frame is
. The definition of the scalar curvature is
and the normalized scalar curvature (
) is defined as
For
N, the mean curvature vector (
H) is
The Casorati curvature (
C) of
N is defined by
Let
be an
l-dimensional subspace of
, and assume that
. The scalar curvature of
for an orthonormal basis
can be expressed as
Assume that a hyperplane of
is
. Then, the normalized
Casorati curvatures
and
are expressed by
The generalized, normalized Casorati curvatures of N contain the following expression for any real number ().
and if
,
with
[
40].
In modern differential geometry, the study of Casorati curvatures is a highly active research subject. Many researchers have obtained intriguing findings on Casorati curvatures in golden Riemannian manifolds.
Choudhary and Park obtained the following results in [
41] regarding
Casorati curvatures of slant submanifolds of locally golden space forms.
Theorem 79 ([
41]).
Given an -dimensional locally golden product space of the form , let N be a n-dimensional θ-slant proper submanifold. Then, we have the following:(i)
The curvature expressed by , which is the generalized, normalized δ Casorati curvature, satisfiesfor any real number (r) such that .(ii)
The generalized, normalized δ Casorati curvature () satisfiesfor any real number ().Furthermore, if and only if N is an invariantly quasi-umbilical submanifold with a trivial normal connection in (), then the equalities in relations (22) and (23) hold such that the shape operators (), with respect to some orthonormal tangent frame () and orthonormal normal frame (), have the following forms: In golden Riemannian space forms, Choudhary and Park [
48] also obtained sharp inequalities for
-invariant and
-anti-invariant submanifolds as a consequence of Theorem 79.
Theorem 80 ([
41]).
Consider N as an n-dimensional invariant submanifold of a locally golden product space of the form . Then, we have the following:(i)
The generalized, normalized δ Casorati curvature () satisfiesfor any real number (r) such that .(ii)
The generalized normalized δ Casorati curvature satisfiesfor any real number ().Furthermore, the equalities in relations (25) and (26) hold if and only if N is an invariantly quasi-umbilical submanifold with a trivial normal connection in such that the shape operators (), take the following forms for some orthonormal tangent frame () and orthonormal normal frame (): Theorem 81 ([
41]).
Let N be an n-dimensional anti-invariant submanifold within -dimensional locally golden product space of the form . Thus,(i)
The generalized normalized δ Casorati curvature () satisfiesfor any real number (r) such that . Then, the following conditions hold:(ii)
The generalized normalized δ Casorati curvature () satisfiesfor any real number ().Additionally, the equalities hold in relations (28) and (29) if and only if N is an invariantly quasi-umbilical submanifold with a trivial normal connection in such that the shape operators ) take the following forms for some orthonormal tangent frame () and orthonormal normal frame (): Theorem 82 ([
41]).
Assume that the locally golden product space form of dimension is . For any n-dimensional θ-slant proper submanifold (N) of ,(i)
The normalized δ-Casorati curvature () satisfies(ii)
The normalized δ Casorati curvature satisfiesRegarding any invariant submanifold (N) of , we have(i)
The normalized δ Casorati curvature () satisfies(ii)
The normalized δ Casorati curvature satisfiesFor any n-dimensional anti-invariant submanifold (N) of ,(i)
The normalized δ Casorati curvature () satisfies(ii)
The normalized δ Casorati curvature satisfiesAdditionally, (31), (33), and (35) hold as equalities if and only if the submanifold () is invariantly quasi-umbilical with a trivial normal connection in . In this case, the shape operators () with respect to some orthonormal tangent frame () and orthonormal normal frame () fulfill the following requirements: Furthermore, (32), (34), and (36) hold as equalities if and only if the submanifold () is invariantly quasi-umbilical with a trivial normal connection in such that the shape operators () with respect to some orthonormal tangent frame () and orthonormal normal frame () satisfy the following: Remark 39. Some sharp inequalities for slant submanifolds immersed in golden Riemannian space forms with a semi-symmetric metric connection were deduced by Lee et al. in [91]. Furthermore, they characterized submanifolds in the equality case. They concluded by talking about these inequalities for a few special submanifolds. Geometric inequalities for the Casorati curvatures on submanifolds on golden Riemannian manifolds with constant golden sectional curvature were established by Choudhary and Mihai in [
40]. Let
be a locally decomposable golden Riemannian manifold with a constant golden sectional curvature. Next, on a submanifold (
N), the following are the optimal inequalities for
and
:
Theorem 83 ([
40]).
Considering M as an n-dimensional Riemannian manifold isometrically immersed in under the condition of , we have:(i)
satisfiesif ;(ii)
satisfiesif .Furthermore, if and only if N is invariantly quasi-umbilical does the equality holds in (39) or (40). There exists an orthonormal tangent frame () and an orthonormal normal frame () such that have the following forms and the normal connection of N in is trivial. Theorem 84 ([
40]).
Let N be an isometrically immersed n-dimensional submanifold in . Then,(i)
satisfiesfor .(ii) satisfiesfor . Furthermore, achieves the following forms, and the equality holds in (42) or (43) if and only if N meets the equality’s criteria as stated in Theorem 83. Remark 40. In [40], Choudhary and Mihai established the consequences of Theorems 83 and 84 and obtained inequality cases for Casorati curvature on an anti-invariant submanifold in . Remark 41. Regarding sharp inequalities concerning δ Casorati curvatures for slant submanifolds of golden Riemannian space forms, we refer to [38]. 10.3. Wintgen-Type Inequality in Golden Riemannian Manifolds
In the context of four-dimensional Euclidean space, Wintgen inequality represents a critical geometric inequality that incorporates Gaussian curvature, normal curvature, and squared mean curvature, all of which are intrinsic invariants. In 1979, P. Wintgen [
42] formulated this inequality to show that for any surface (
) in
, the Gaussian curvature (
K), the normal curvature (
), and the squared mean curvature (
) meet the following condition:
The equality is valid if and only if the ellipse of curvature of
in
is a circle. This result was further generalized by I. V. Guadalupe et al. in [
92] for an arbitrary codimension of
m in real-space forms (
) as follows:
They also discussed the conditions under which equality is achieved.
De Smet, Dillen, Verstraelen, and Vrancken [
93] proposed an inequality for submanifolds in real-space forms, referred to as the generalized Wintgen inequality or DDVV conjecture, which extends the Wintgen inequality. This conjecture was independently proven by Ge and Tang in [
94]. Different researchers have obtained DDVV inequality for various classes of submanifolds in various ambient manifolds in recent years. For slant, invariant, C-totally real, and Lagrangian submanifolds in golden Riemannian space forms, researchers obtained generalized Wintgen-type inequalities in [
44] and discussed the equality cases.
For slant submanifolds, the generalized Wintgen inequality is outlined as follows:
Theorem 85 ([
44]).
Let N be an n-dimensional θ-slant proper submanifold of a locally golden product space of the form . Then, we have the following:For scalar normal curvature, is used.
Choudhary et al. [
44] established the generalized Wintgen inequality for an invariant submanifold of golden Riemannian space forms with the aid of Theorem 85.
Theorem 86 ([
44]).
Consider N an n-dimensional invariant submanifold within a locally golden product space of the form . Then, The generalized Wintgen inequality of a locally golden product space form for a Lagrangian submanifold is outlined as follows.
Theorem 87 ([
44]).
Suppose N is a Lagrangian submanifold in a locally golden product space of the form . Then,