1. Introduction
Biharmonic maps have been a subject of mathematical research since as early as 1862, attracting significant attention due to their relevance in various applications. In the early studies, Maxwell and Airy were among the pioneers who used biharmonic maps to develop mathematical models for elasticity, which laid the foundation for further exploration in the field. Over the years, biharmonic maps have continued to gain prominence, and in 1988, J. Eells and L. Lemaire [
1] conducted the first formal and systematic study on these maps, generalizing the well-known concept of harmonic maps. Their work marked a pivotal moment in the study of biharmonic maps, bringing new insights into this area of differential geometry.
Biharmonic maps can be understood as a generalization of harmonic maps, where the energy functional is replaced by a more complex biharmonic energy functional. While harmonic maps minimize this energy, biharmonic maps, in contrast, are critical points of the biharmonic energy, making them a more intricate and subtle extension of harmonic maps. Biharmonic maps can be linked to differential equations, functional inequalities, and quantumgeometry [
2,
3].
Building on this foundational work, several further advancements have been made. For instance, in a notable contribution, Lu introduced the concept of f-biharmonic maps [
4], offering a natural extension of biharmonic maps. In his work, Lu computed the equations governing f-biharmonic conformal mappings between manifolds of the same dimension and studied the f-biharmonic map equation, broadening the theoretical framework initially established by Eells and Lemaire.
Further exploration of f-biharmonic maps has been followed, notably by Ye-Lin Ou, who in [
5] examined f-biharmonic curves in real space forms. Ou’s work demonstrated that f-biharmonic submanifolds are not only a generalization of biharmonic submanifolds but also extend to minimal submanifolds. Two key observations arose from this research:
When the function f is constant, f-biharmonic submanifolds are exactly the same as biharmonic submanifolds.
Any minimal submanifold can be classified as f-biharmonic for any positive function defined on it, making f-biharmonic submanifolds a natural extension of both biharmonic and minimal submanifolds.
The study of f-biharmonic submanifolds has also been expanded to more complex geometries. For example, in their work [
6], C. Ozgur and S. Guvenca focused on the f-biharmonic Legendre curves in Sasakian space forms, contributing to a deeper understanding of how these maps behave in specialized geometric settings. Similarly, numerous researchers have investigated f-biharmonic submanifolds within generalized Sasakian space forms and complex space forms [
7,
8,
9], exploring their behavior in different contexts.
The concept of f-biharmonic submanifolds has also been extended in product spaces. For example, Karaca and Ozgur [
10] advanced the work of previous researchers by examining these submanifolds in the context of product spaces, particularly building on prior findings on biharmonic submanifolds by Karaca [
11]. This body of work highlights the richness and variety of biharmonic and f-biharmonic submanifolds in various geometric structures. Additionally, studies on the toeplitz quantization of geometric structures, geometric flow and curvatures, etc. can provide new insights into the behavior of biharmonic maps [
2,
12,
13].
More recently, N. A. Rehman [
14] conducted investigations into f-biharmonic mappings on S-space forms, further extending the theory into new domains and offering a broader understanding of how f-biharmonic maps operate in these specialized forms. These concepts naturally connect to other important geometric structures, such as conformal Ricci solitons and curvature tensors, which play fundamental roles in understanding the geometry of Lorentzian, Riemannian manifolds and warped product spaces [
3,
15,
16].
Motivated by these developments, we aim to explore the properties and behaviors of f-biharmonic and bi-f-harmonic submanifolds in the context of T-space forms. These T-space forms provide an intriguing geometric setting in which to examine how these submanifolds behave under different conditions, building on the prior work in Sasakian and generalized space forms.
In particular, in this paper, we investigate f-biharmonic submanifolds in T-space forms, a class of manifolds that generalizes Sasakian and cosymplectic manifolds while retaining significant structural properties. Our study aims to achieve the following objectives:
Classification: Establish necessary and sufficient conditions for the existence of f-biharmonic submanifolds in T-space forms.
Non-existence results: Derive conditions under which f-biharmonic submanifolds cannot exist due to constraints on tangential structural vector fields.
Chen–Ricci inequality: Develop a curvature-based inequality for submanifolds in T-space forms and analyze cases of equality.
The results obtained in this paper contribute to the broader understanding of biharmonic and f-biharmonic mappings and their interaction with the curvature properties of the ambient manifold. Furthermore, this study opens avenues for exploring applications in mathematical physics, particularly in elasticity theory, general relativity, and geometric analysis.
The structure of this paper is as follows:
In
Section 2, we introduce the necessary preliminaries on biharmonic and f-biharmonic submanifolds in T-space forms.
Section 3 establishes key results concerning the classification and conditions for f-biharmonicity.
Section 4 derives Chen–Ricci inequalities and explores their implications. Finally, in
Section 5, we present our conclusions and discuss possible future research directions.
2. Preliminaries
Let
be a Riemannian manifold with
. We say that
N is an
S-manifold if there exists an
f-structure
on
N of rank
, along with
s global vector fields
(structure vector fields), such that the following conditions are satisfied [
17,
18].
There are dual 1-forms
corresponding to
, and they satisfy
The following metric relations hold:
for any
, where
.
In a
T-manifold
N, the following additional conditions are satisfied:
and
for any vector fields
.
Let denote the distribution determined by , and let be its complementary distribution. The space is determined by and is spanned by . If , then for all i, and if , then .
A plane section in (the tangent space of N at point p) is called a -section if and . If the sectional curvature is constant for all non-null vectors in , we say that N has constant -sectional curvature at point p. The function , defined by , is called the -sectional curvature of N.
We define a
T-manifold
N with constant
-sectional curvature
as a
T-space form, denoted by
. The corresponding curvature tensor
is expressed as follows [
19,
20]:
When the parameter in the definition of a T-manifold N, the structure reduces to that of a Kähler manifold, whereas setting results in a cosymplectic manifold.
The expression of the Gauss equation for a submanifold
M embedded in
is as follows:
for any vectors
tangential to
M, where
. From Equations (
1) and (
2), we obtain
Henceforth, we consider the structure vector fields to be tangential to N and denote its dimension by . We also assume . Let represent the orthogonal distribution to in , and we can express the tangent bundle as a direct orthogonal decomposition: .
Finally, we recall that for a smooth, positive, and well-defined function
, f-biharmonic maps are defined as the critical points of the f-bienergy functional for a mapping
between two Riemannian manifolds. The functional is given by
where
represents the tension field of the map
F, and
is the volume form of the manifold
M.
The corresponding Euler–Lagrange equation, which must be satisfied by
f-biharmonic maps, is as follows:
This equation governs the behavior of
f-biharmonic maps and serves as the foundation for our exploration in this paper.
3. Geometric Aspects of f-Biharmonic Submanifolds
In this section, we study f-biharmonic submanifolds within T-space forms, exploring various cases and establishing the necessary and sufficient conditions for f-biharmonicity. Additionally, we prove a non-existence theorem for f-biharmonic submanifolds in which both and are tangential to the submanifold.
Theorem 1. Let be a submanifold of T-space form . Then, is a biharmonic submanifold of the T-space form if and only if
Proof. We use
and ∇ to refer to the Levi-Civita connections corresponding to
N and
M. Suppose
constitutes a local geodesic frame at
. Consequently, as
, it follows that at
p,
Using the Weingarten Equation, one obtains
Moving forward, the computation of the tr
at
p results in
and then
Building on this, with the given representation of the curvature tensor field
we deduce
But,
and
Plugging (
9) and (
10) into (
8), we arrive at
and, therefore,
Define
as a local orthonormal frame over
M. In this case,
serves as a local orthonormal frame on
N. Utilizing the expression for the curvature tensor field and the fact that
, a direct computation yields the following result.
which implies
From (
8), we have
Now, from (
7) and (
13),
For an isometric immersion, (
4) becomes
Using (
14) in (
15), we arrive at
Comparing tangential and normal components in (
16), we arrive at the conclusion. □
Corollary 1. Let be a submanifold of T-space form .
- (1)
The submanifold M is f-biharmonic under the invariance condition if and only if - (2)
The submanifold M is f-biharmonic under the anti-invariance condition if and only if
Proof. 1. For invariant f-biharmonic submanifolds, we take in Theorem 1 and obtain the result.
2. For anti-invariant f-biharmonic submanifolds, we take in Theorem 1 and obtain the result. □
Corollary 2. Let be a submanifold of T-space form .
- (1)
If is normal to M, then M is f-biharmonic if and only if - (2)
If is tangential to M, then M is f-biharmonic if and only if - (3)
If M is a hypersurface, then M is f-biharmonic if and only if
Proof. 1. Since are normal to , then the tangential component of vanishes and we take as anti-invariant, . Taking and in Theorem 1, we obtain the result.
2. Since are tangential to , vanishes, and taking in Theorem 1, we obtain the result. □
Theorem 2. Let be a hypersurface within a T-space form , possessing a non-vanishing constant mean curvature, and assume that the vector field ξ lies tangential to M. Then, M is classified as proper biharmonic if and only if the following condition holds:or, equivalently, if and only if Proof. Let
be a f-biharmonic hypersurface of
and
tangential to
M; then,
and
, and since
this implies
Therefore, taking
and
in (13) and (14), we have
and
For constant mean curvature
, we have
and
which implies
or, equivalently,
Using Gauss’s equation, we have
Therefore, using Equation (
17), we have
which completes the proof. □
Theorem 3. Let be a submanifold of T-space form with constant mean curvature Ω so that ξ and are tangential. Further, we consider , the function defined on M byThis leads to the following observations. - (1)
acts as an obstruction to f-biharmonicity.
- (2)
and M is proper f-biharmonic; then,
Proof. Because
M is a proper
f-biharmonic submanifold with constant mean curvature
, and
is tangential to it, Corollary 2 leads to the conclusion that
Given that
is a tangent, this directly implies that
. Reapplying
, it follows that
. But, from
, and
being tangential, we have
. Therefore, equating the tangential and normal parts, we find that
and
. Incorporating these results into the preceding equation, we obtain
Projecting the first equation onto
, we obtain
Using the facts
is a constant and the Böchner formula, i.e.,
in the above equation, we have
Now, this equation reduces to
By considering the Cauchy–Schwarz inequality,
This implies that
Given that remains a positive constant, this establishes the validity of the two assertions in the theorem. □
4. Ricci Curvature Inequality
The investigation of the Chen–Ricci inequalities has become a promising research area in differential geometry that provides new information on the relationship between the intrinsic curvature and extrinsic curvature of submanifolds. These inequalities, rooted in the pioneering work of B.-Y. Chen [
21], pose significant geometric constraints on the Ricci curvature, the mean curvature, and the other invariants of submanifolds with the embedding of Riemannian manifolds. This inspired a multitude of generalizations across diverse ambient spaces. These developments have enriched our understanding of curvature-related inequalities and their geometric implications [
22,
23,
24,
25,
26,
27,
28,
29,
30].
This section seeks to discuss the basic feature of the Chen–Ricci inequalities in T-space forms.
Theorem 4. Let be a submanifold of a T-space form with constant f-sectional curvature c, where the structural vector fields are tangential to . Then, for each unit vector , orthogonal to , we have Proof. Assume is a submanifold of the T-space form , with constant f-sectional curvature K. At a point , take a unit vector . Let be an orthonormal basis for , such that are tangential to M at p. Assume .
Putting
and
in (
3), we have
By taking summation
, we derive
where
denotes the mean curvature,
is the second fundamental form, and
is the scalar curvature.
Mathematically, the second fundamental form
is expressed as
Additionally, the projection operator
P satisfies
We express the mean curvature
as follows:
As a consequence of the Gauss equation, we obtain
The sectional curvature for
M is expressed as
Using the Gauss equation, we rewrite this as
Substituting this expression into the equation for
, we derive
This simplifies to
which gives the required inequality. □
Corollary 3. In Theorem 4, equality holds at a point under the following conditions:
- (1)
and X is normal to .
- (2)
p is a totally geodesic point.
Proof. - (1)
When
, we have
. If
X is normal to
, then
for
, and equality follows from (
22).
- (2)
If
p is a totally geodesic point, then
for all
, and equality holds as a result of (
22).
□
Remark 1. - (1)
Setting yields all the results corresponding to Kähler manifolds.
- (2)
Setting yields all the results corresponding to cosymplectic manifolds.
5. Conclusions
In this paper, we conducted a detailed study of f-biharmonic submanifolds in T-space forms, providing classification results and establishing key conditions for their existence. By analyzing different scenarios, we formulated necessary and sufficient conditions for f-biharmonicity and derived a non-existence theorem under specific geometric constraints. Furthermore, we introduced a Chen–Ricci inequality for submanifolds of T-space forms and identified the conditions under which this inequality attains equality.
The findings of this paper have several important implications. The classification results contribute to the broader field of biharmonic submanifolds and their generalizations, particularly in non-trivial geometric structures such as Sasakian and cosymplectic manifolds. The derivation of Chen–Ricci inequalities extends classical results in Riemannian geometry, offering new perspectives on the interplay between curvature and biharmonicity.
From an applied mathematics perspective, our results have potential implications in areas such as applications in elasticity theory and the study of biharmonic functions in material science, insights into the role of biharmonic maps in spacetime models, and contributions to stability and energy minimization in differential geometry.
Future research directions include the following:
Extending the analysis to more general pseudo-Riemannian and Lorentzian settings, with potential applications in relativistic physics.
Investigating higher-order biharmonic equations in T-space forms to explore additional geometric constraints.
Applying the developed techniques to submanifolds of nearly Kähler and nearly Sasakian manifolds, expanding the scope of f-biharmonic research.
This work advances our understanding of biharmonic submanifolds in specialized geometric settings, paving the way for further theoretical and applied explorations in this field.