Classification Results of f-Biharmonic Immersion in T-Space Forms

Abstract

We investigate f-biharmonic submanifolds in T-space form, where we analyze different scenarios and provide necessary and sufficient conditions for f-biharmonicity. We also derive a non-existence result for f-biharmonic submanifolds where $\xi$ and $\varphi \Omega$ are tangents. Finally, we derive the Chen–Ricci inequality for submanifolds of T-space forms and provide the conditions under which this inequality becomes equality.

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 $(N,g)$ be a Riemannian manifold with $dim\left(N\right)=2n+s$. We say that N is an S-manifold if there exists an f-structure $\varphi$ on N of rank $2n$, along with s global vector fields ${\xi }_1,\dots ,{\xi }_s$ (structure vector fields), such that the following conditions are satisfied [17,18].

There are dual 1-forms $u^1,\dots ,u^s$ corresponding to ${\xi }_1,\dots ,{\xi }_s$, and they satisfy

$$\varphi {\xi }_i=0, u^i\circ \varphi =0,{\varphi }^2=-I+\sum _{i=1}^s{u}^i\otimes {\xi }_i.$$

The following metric relations hold:

$$\overline{g}(\varphi U,\varphi V)=\overline{g}(U,V)-\sum _{i=1}^s{u}^i\left(U\right)u^i\left(V\right)$$

$$\overline{g}(U,{\xi }_i)=u^i\left(U\right)$$

for any $U,V\in TN$, where $i=1,\dots ,s$.

In a T-manifold N, the following additional conditions are satisfied:

$$\left({\nabla }_U\varphi \right)V=0$$

and

$${\nabla }_U{\xi }_i=0$$

for any vector fields $U,V\in TN$.

Let $\overline{D}$ denote the distribution determined by $-{\varphi }^2$, and let ${\overline{D}}^{\perp }$ be its complementary distribution. The space ${\overline{D}}^{\perp }$ is determined by ${\varphi }^2+I$ and is spanned by ${\xi }_1,\dots ,{\xi }_s$. If $U\in \overline{D}$, then $u^i\left(U\right)=0$ for all i, and if $U\in {\overline{D}}^{\perp }$, then $\varphi U=0$.

A plane section $\Pi$ in $T_{p}N$ (the tangent space of N at point p) is called a $\varphi$-section if $\Pi \perp {\overline{D}}^{\perp }$ and $\varphi (\Pi )=\Pi$. If the sectional curvature $\overline{K}(\Pi )$ is constant for all non-null vectors in $\Pi$, we say that N has constant $\varphi$-sectional curvature at point p. The function $\mu$, defined by $\mu \left(p\right)=\overline{K}(\Pi )$, is called the $\varphi$-sectional curvature of N.

We define a T-manifold N with constant $\varphi$-sectional curvature $\mu$ as a T-space form, denoted by $N\left(\mu \right)$. The corresponding curvature tensor $\overline{R}$ is expressed as follows [19,20]:

$$\begin{array}{cc}\hfill \overline{g}(\overline{S}(L_1,L_2)L_3,L_4)& ={\displaystyle \frac{\mu }{4}}[\overline{g}(L_1,L_3)\overline{g}(L_2,L_4)-\overline{g}(L_2,L_3)\overline{g}(L_1,L_4)-\overline{g}(L_1,L_3)\sum u^i\left(L_2\right)u^i\left(L_4\right)\hfill \\ \hfill & - \overline{g}(L_2,L_4)\sum u^i\left(L_3\right)u^i\left(L_1\right)+\overline{g}(L_1,L_4)\sum u^i\left(L_2\right)u^i\left(L_3\right)\hfill \\ \hfill & + \overline{g}(L_2,L_3)\sum u^i\left(L_1\right)u^i\left(L_4\right)\hfill \\ \hfill & +\left(\sum u^i\left(L_3\right)u^i\left(L_1\right)\right)\left(\sum u^i\left(L_2\right)u^i\left(L_4\right)\right)\hfill \\ \hfill & -\left(\sum u^i\left(L_4\right)u^i\left(L_1\right)\right)\left(\sum u^i\left(L_2\right)u^i\left(L_3\right)\right)\hfill \\ \hfill & + \overline{g}(L_4,\varphi L_1)\overline{g}(L_2,\varphi L_3)+\overline{g}(L_2,\varphi L_4)\overline{g}(L_1,\varphi L_3)\hfill \\ \hfill & - 2\overline{g}(L_1,\varphi L_2)\overline{g}(L_4,\varphi L_3)].\hfill \end{array}$$

(1)

When the parameter $s=0$ in the definition of a T-manifold N, the structure reduces to that of a Kähler manifold, whereas setting $s=1$ results in a cosymplectic manifold.

The expression of the Gauss equation for a submanifold M embedded in $N\left(\mu \right)$ is as follows:

$$\begin{array}{c}\hfill \overline{S}(L_1,L_2,L_3,L_4)=S(L_1,L_2,L_3,L_4)+g(\zeta (L_1,L_3),\zeta (L_2,L_4))-g(\zeta (L_1,L_4),\zeta (L_2,L_3))\end{array}$$

(2)

for any vectors $L_1,L_2,L_3,L_4$ tangential to M, where $S(L_1,L_2,L_3,L_4)=g(S(L_1,L_2)L_3,L_4)$. From Equations (1) and (2), we obtain

$$\begin{array}{cc}\hfill S(L_1,L_2,L_3,L_4)& ={\displaystyle \frac{\mu }{4}}\left[\overline{g}(L_1,L_3)\overline{g}(L_2,L_4)-\overline{g}(L_2,L_3)\overline{g}(L_1,L_4)-\overline{g}(L_1,L_3)\sum u^i\left(L_2\right)u^i\left(L_4\right)\right.\hfill \\ \hfill & - \overline{g}(L_2,L_4)\sum u^i\left(L_3\right)u^i\left(L_1\right)+\overline{g}(L_1,L_4)\sum u^i\left(L_2\right)u^i\left(L_3\right)\hfill \\ \hfill & + \overline{g}(L_2,L_3)\sum u^i\left(L_1\right)u^i\left(L_4\right)\hfill \\ \hfill & +\left(\sum u^i\left(L_3\right)u^i\left(L_1\right)\right)\left(\sum u^i\left(L_2\right)u^i\left(L_4\right)\right)\hfill \\ \hfill & -\left(\sum u^i\left(L_4\right)u^i\left(L_1\right)\right)\left(\sum u^i\left(L_2\right)u^i\left(L_3\right)\right)\hfill \\ \hfill & + \overline{g}(L_4,\varphi L_1)\overline{g}(L_2,\varphi L_3)+\overline{g}(L_2,\varphi L_4)\overline{g}(L_1,\varphi L_3)\hfill \\ \hfill & \left.- 2\overline{g}(L_1,\varphi L_2)\overline{g}(L_4,\varphi L_3)\right]\hfill \\ \hfill & - g(\zeta (L_1,L_3),\zeta (L_2,L_4))+g(\zeta (L_1,L_4),\zeta (L_2,L_3)).\hfill \end{array}$$

(3)

Henceforth, we consider the structure vector fields to be tangential to N and denote its dimension by $n+s$. We also assume $n\ge 2$. Let $L=D_1\oplus D_2$ represent the orthogonal distribution to ${\overline{D}}^{\perp }$ in $TN$, and we can express the tangent bundle as a direct orthogonal decomposition: $TN=L\oplus {\overline{D}}^{\perp }$.

Finally, we recall that for a smooth, positive, and well-defined function $f:M\to \mathbb{R}$, f-biharmonic maps are defined as the critical points of the f-bienergy functional for a mapping $F:(M,g)\to (N,h)$ between two Riemannian manifolds. The functional is given by

$$\begin{array}{c}\hfill E_{2,f}\left(F\right)={\int }_{M}f{\left|\tau \left(F\right)\right|}^2{v}_g,\end{array}$$

where $\tau \left(F\right)$ represents the tension field of the map F, and $v_g$ is the volume form of the manifold M.

The corresponding Euler–Lagrange equation, which must be satisfied by f-biharmonic maps, is as follows:

$$\begin{array}{c}\hfill {\tau }_{2,f}\left(F\right)=f{\tau }_2\left(F\right)+\Delta f\tau \left(F\right)+2{\nabla }_{gradf}^F\tau \left(F\right)=0.\end{array}$$

(4)

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 $\xi$ and $\varphi \Omega$ are tangential to the submanifold.

Theorem 1. 

Let $M^n$ be a submanifold of T-space form $N^{2n+s}$. Then, $M^n$ is a biharmonic submanifold of the T-space form $N^{2n+s}$ if and only if

$$\begin{array}{cc}\hfill & -{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)+{\displaystyle \frac{\Delta f}{f}}\Omega +2{\nabla }_{grad\left(lnf\right)}\Omega ={\displaystyle \frac{\mu }{4}}\{\left(n\right)\Omega -\left(n\right)\sum u^i(\Omega ){\xi }_i^{\perp }-|{\xi }_i^T{|}^2\Omega +3Nt\Omega \},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\left(n\right)}{2}}{grad\left(\right|\Omega |}^2)+2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-2{\Lambda }_{\Omega }grad\left(lnf\right)={\displaystyle \frac{\mu }{4}}\{2(n-1)\sum u^i(\Omega ){\xi }_i^T+3Pt\Omega \}.\hfill \end{array}$$

(6)

Proof. 

We use ${\nabla }^N$ and ∇ to refer to the Levi-Civita connections corresponding to N and M. Suppose ${\left\{U_i\right\}}_{i=1}^n$ constitutes a local geodesic frame at $p\in M$. Consequently, as $\tau \left(\mathbf{i}\right)=\left(n\right)\Omega$, it follows that at p,

$$\begin{array}{ccc}\hfill {\tau }_2\left(\mathbf{i}\right)& =& -\Delta \tau \left(\mathbf{i}\right)-tr{S}^N(d\mathbf{i},\tau \left(\mathbf{i}\right))d\mathbf{i}\hfill \\ \hfill & =& n\left\{\sum _{i=1}^n{\nabla }_{U_i}^N{\nabla }_{U_i}^N\Omega -\sum _{i=1}^n{S}^N\left(U_i,\Omega \right)U_i\right\}.\hfill \end{array}$$

(7)

Using the Weingarten Equation, one obtains

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{1}{n}}\Delta \tau \left(\mathbf{i}\right)& =& \sum _{i=1}^n{\nabla }_{U_i}^N{\nabla }_{U_i}^N\Omega =-{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)\hfill \\ \hfill & -& tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-tr\nabla {\Lambda }_{\Omega }(·,·).\hfill \end{array}$$

(8)

Moving forward, the computation of the tr $\nabla {\Lambda }_{\Omega }(·,·)$ at p results in

$$\begin{array}{c}\hfill tr\nabla {\Lambda }_{\Omega }(·,·)=\sum _{i,j=1}^n{U}_i\left(g\left({\nabla }_{U_j}^N{U}_i,\Omega \right)\right)U_j\end{array}$$

and then

$$\begin{array}{c}\hfill tr\nabla {\Lambda }_{\Omega }(·,·)=\sum _{i,j=1}^{n}g\left({\nabla }_{U_i}^N{\nabla }_{U_j}^N{U}_i,\Omega \right)U_j+tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·).\end{array}$$

Building on this, with the given representation of the curvature tensor field $R^N,$ we deduce

$$\begin{array}{ccc}\hfill tr\nabla {\Lambda }_{\Omega }(·,·)& =& \sum _{i,j=1}^{n}g\left({\nabla }_{U_j}^N{\nabla }_{U_i}^N{U}_i+S^N\left(U_i,U_j\right)U_i+{\nabla }_{\left[U_i,U_j\right]}^N{U}_i,\Omega \right)U_j\hfill \\ & +& tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)\hfill \\ & =& \sum _{i,j=1}^{n}g\left({\nabla }_{U_j}^N{\nabla }_{U_i}^N{U}_i,\Omega \right)U_j+\sum _{i,j=1}^{n}g\left(S^N\left(U_i,U_j\right)U_i,\Omega \right)U_j\hfill \\ & +& tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·).\hfill \end{array}$$

(9)

But,

$$\begin{array}{c}\hfill \sum _{i,j=1}^{n}g\left({\nabla }_{U_j}^N{\nabla }_{U_i}^N{U}_i,\Omega \right)U_j={\displaystyle \frac{n}{2}}grad\left({|\Omega |}^2\right)\end{array}$$

(10)

and

$$\begin{array}{ccc}\hfill \sum _{i,j=1}^{n}g\left(S^N\left(U_i,U_j\right)U_i,\Omega \right)U_j& =& \sum _{i,j=1}^{n}g\left(S^N\left(U_i,\Omega \right)U_i,U_j\right)U_j\hfill \\ \hfill & =& (\mathrm{tr}{S}^N(d\mathbf{i},\Omega )d\mathbf{i}{)}^{\top }.\hfill \end{array}$$

(11)

Plugging (9) and (10) into (8), we arrive at

$$\begin{array}{c}\hfill \mathrm{tr}\nabla {\Lambda }_{\Omega }(·,·)={\displaystyle \frac{n}{2}}grad\left({|\Omega |}^2\right)+{\left(tr{S}^N(d\mathbf{i},\Omega )d\mathbf{i}\right)}^{\top }+tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)\end{array}$$

and, therefore,

$$\begin{array}{ccc}\hfill \mathrm{tr}{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)+tr\nabla {\Lambda }_{\Omega }(·,·)& =& 2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)+{\displaystyle \frac{n}{2}}grad\left({|\Omega |}^2\right)\hfill \\ \hfill & +& {\left(tr{S}^N(d\mathbf{i},\Omega )d\mathbf{i}\right)}^{\top }.\hfill \end{array}$$

(12)

Define ${\left\{U_i\right\}}_{i=1}^n$ as a local orthonormal frame over M. In this case, ${\left\{U_i,\phi U_j,\xi \right\}}_{i,j=1}^n$ serves as a local orthonormal frame on N. Utilizing the expression for the curvature tensor field and the fact that $\Omega \in$$span\left\{\phi U_i;i=1,\dots ,n\right\}$, a direct computation yields the following result.

$$\begin{array}{ccc}\hfill tr{S}^N(d\mathbf{i},\Omega )d\mathbf{i}& =& \sum _{i=1}^n{S}^N(U_i,\Omega )U_i\hfill \\ & =& \sum _{i=1}^n{\displaystyle \frac{\mu }{4}}\{\tilde{g}(U_i,U_i)\Omega -\tilde{g}(\Omega ,U_i)x_i-\tilde{g}(U_i,U_i)\sum u^i(\Omega ){\xi }_i\hfill \\ & -& \sum u^i\left(U_i\right)u^i\left(U_i\right)\Omega +\tilde{\sum }{u}^i(\Omega )u^i\left(U_i\right)U_i\hfill \\ & +& \tilde{g}(\Omega ,U_i)\sum u^i\left(U_i\right){\xi }_i+\left(\sum u^i\left(U_i\right)u^i\left(U_i\right)\right)\left(\sum u^i(\Omega ){\xi }_i\right)\hfill \\ & -& \left(\sum {\xi }_i{u}^i\left(U_i\right)\right)\left(\sum u^i(\Omega )u^i\left(U_i\right)\right)\hfill \\ & +& \varphi U\tilde{g}(\Omega ,\varphi U_i)-\tilde{g}(U_i,\varphi U_i)\varphi \Omega -2\tilde{g}(U,\varphi \Omega )\varphi U_i\left)\right\}.\hfill \\ & =& {\displaystyle \frac{\mu }{4}}\{n\Omega -n\sum u^i(\Omega ){\xi }_i-{\left|{\xi }_i^{\top }\right|}^2\Omega +\sum u^i(\Omega ){\xi }_i^{\top }+3(Pt\Omega +Nt\Omega )\},\hfill \end{array}$$

which implies

$${\left(tr{S}^N(d\mathbf{i},\Omega )d\mathbf{i}\right)}^{\top }={\displaystyle \frac{\mu }{4}}\left\{-n\sum u^i(\Omega ){\xi }_i^{\top }+\sum u^i(\Omega ){\xi }_i^T+3Pt\Omega \right\}.$$

From (8), we have

$$\begin{array}{ccc}\hfill \Delta \tau \left(\mathbf{i}\right)& =& -n{\Delta }^{\perp }\Omega -ntrB\left(·,{\Lambda }_{\Omega }·\right)-2ntr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-{\displaystyle \frac{n}{2}}{grad\left(\right|\Omega |}^2)\hfill \\ \hfill & +& {\displaystyle \frac{\mu }{4}}\left\{-n\sum u^i(\Omega ){\xi }_i^{\top }+\sum u^i(\Omega ){\xi }_i^{\top }+3Pt\Omega \right\}\hfill \\ \hfill & =& -n{\Delta }^{\perp }\Omega -ntrB\left(·,{\Lambda }_{\Omega }·\right)-2ntr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-{\displaystyle \frac{n}{2}}{grad\left(\right|\Omega |}^2)\hfill \\ \hfill & +& {\displaystyle \frac{\mu }{4}}\left\{n(1-n)\sum u^i(\Omega ){\xi }_i^{\top }+3nPt\Omega \right\}.\hfill \end{array}$$

(13)

Now, from (7) and (13),

$$\begin{array}{ccc}\hfill {\tau }_2\left(\mathbf{i}\right)& =& -n{\Delta }^{\perp }\Omega -ntrB\left(·,{\Lambda }_{\Omega }·\right)-2ntr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-{\displaystyle \frac{n}{2}}{grad\left(\right|\Omega |}^2)\hfill \\ \hfill & +& {\displaystyle \frac{\mu }{4}}\{n^2\Omega -n(2-n)\sum u^i(\Omega ){\xi }_i^{\top }-n^2\sum u^i(\Omega ){\xi }_i-{\left|{\xi }_i^{\top }\right|}^2\Omega \hfill \\ \hfill & +& 6nPt\Omega +3nPt\Omega \}.\hfill \end{array}$$

(14)

For an isometric immersion, (4) becomes

$$\begin{array}{c}\hfill {\tau }_2\left(\mathbf{i}\right)+n{\displaystyle \frac{\Delta f}{f}}\Omega -2n{\Lambda }_{\Omega }grad\left(lnf\right)+2n{\nabla }_{grad\left(lnf\right)}^{\perp }\Omega =0.\end{array}$$

(15)

Using (14) in (15), we arrive at

$$\begin{array}{cc}\hfill & -n{\Delta }^{\perp }\Omega -ntrB\left(·,{\Lambda }_{\Omega }·\right)-2ntr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-{\displaystyle \frac{n}{2}}{grad\left(\right|\Omega |}^2)\\ \hfill & +{\displaystyle \frac{\mu }{4}}\left\{n\Omega -n\sum u^i(\Omega ){\xi }_i-{\left|{\xi }_i^{\top }\right|}^2\Omega -\sum u^i(\Omega ){\xi }_i^{\top }+3Nt\Omega +3Pt\Omega \right\}\\ \hfill & +n(\Delta f)\Omega +2n{\nabla }_{gradf}^{\perp }\Omega -2n{\Lambda }_{\Omega }gradf=0.\end{array}$$

(16)

Comparing tangential and normal components in (16), we arrive at the conclusion. □

Corollary 1. 

Let $M^n$ be a submanifold of T-space form $N^{2n+s}$.

(1) 

The submanifold M is f-biharmonic under the invariance condition if and only if

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)+{\displaystyle \frac{\Delta f}{f}}\Omega +2{\nabla }_{grad\left(lnf\right)}\Omega ={\displaystyle \frac{\mu }{4}}\{\left(n\right)\Omega \hfill \\ -\left(n\right)\sum u^i(\Omega ){\xi }_i^{\perp }-|{\xi }_i^T{|}^2\Omega \}.\hfill \\ {\displaystyle \frac{\left(n\right)}{2}}{grad\left(\right|\Omega |}^2)+2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-2{\Lambda }_{\Omega }grad\left(lnf\right)\hfill \\ ={\displaystyle \frac{\mu }{4}}\{2(n-1)\sum u^i(\Omega ){\xi }_i^{\top }+3Pt\Omega \}.\hfill \end{array}\right.\end{array}$$

(2) 

The submanifold M is f-biharmonic under the anti-invariance condition if and only if

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)+{\displaystyle \frac{\Delta f}{f}}\Omega +2{\nabla }_{grad\left(lnf\right)}\Omega ={\displaystyle \frac{\mu }{4}}\{\left(n\right)\Omega \hfill \\ -\left(n\right)\sum u^i(\Omega ){\xi }_i^{\perp }-|{\xi }_i^T{|}^2\Omega +3Nt\Omega \}.\hfill \\ {\displaystyle \frac{\left(n\right)}{2}}{grad\left(\right|\Omega |}^2)+2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-2{\Lambda }_{\Omega }grad\left(lnf\right)\hfill \\ ={\displaystyle \frac{\mu }{4}}\{2(n-1)\sum u^i(\Omega ){\xi }_i^T\}.\hfill \end{array}\right.\end{array}$$

Proof. 

1. For invariant f-biharmonic submanifolds, we take $N=0$ in Theorem 1 and obtain the result.

2. For anti-invariant f-biharmonic submanifolds, we take $T=0$ in Theorem 1 and obtain the result. □

Corollary 2. 

Let $M^n$ be a submanifold of T-space form $N^{2n+s}$.

 (1) 

If ${{\xi }_i}_{i=1}^s$ is normal to M, then M is f-biharmonic if and only if

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)+{\displaystyle \frac{\Delta f}{f}}\Omega +2{\nabla }_{grad\left(lnf\right)}\Omega ={\displaystyle \frac{\mu }{4}}\{\left(n\right)\Omega \hfill \\ -\left(n\right)\sum u^i(\Omega ){\xi }_i+3Nt\Omega \}.\hfill \\ {\displaystyle \frac{\left(n\right)}{2}}{grad\left(\right|\Omega |}^2)+2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-2{\Lambda }_{\Omega }grad\left(lnf\right)\hfill \\ =0.\hfill \end{array}\right.\end{array}$$

 (2) 

If ${{\xi }_i}_{i=1}^s$ is tangential to M, then M is f-biharmonic if and only if

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)+{\displaystyle \frac{\Delta f}{f}}\Omega +2{\nabla }_{grad\left(lnf\right)}\Omega ={\displaystyle \frac{\mu }{4}}\{(n-1)\Omega \hfill \\ +3Nt\Omega \}.\hfill \\ {\displaystyle \frac{\left(n\right)}{2}}{grad\left(\right|\Omega |}^2)+2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-2{\Lambda }_{\Omega }grad\left(lnf\right)\hfill \\ ={\displaystyle \frac{\mu }{4}}\left\{2(n-1)\sum u^i(\Omega ){\xi }_i+6Pt\Omega \right\}.\hfill \end{array}\right.\end{array}$$

 (3) 

If M is a hypersurface, then M is f-biharmonic if and only if

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)+{\displaystyle \frac{\Delta f}{f}}\Omega +2{\nabla }_{grad\left(lnf\right)}\Omega ={\displaystyle \frac{\mu }{4}}\{(n-3)\Omega \hfill \\ -(n-1)\sum u^i(\Omega ){\xi }_i^{\perp }-{\left|{\xi }_i^T\right|}^2\Omega \}.\hfill \\ {\displaystyle \frac{\left(n\right)}{2}}{grad\left(\right|\Omega |}^2)+2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-2{\Lambda }_{\Omega }grad\left(lnf\right)\hfill \\ ={\displaystyle \frac{\mu }{4}}\left\{2(n-1)\sum u^i(\Omega ){\xi }_i^T+3Pt\Omega \right\}.\hfill \end{array}\right.\end{array}$$

Proof. 

1. Since ${\xi }_{\alpha }:\forall \alpha =1,\dots ,s$ are normal to $M^n$, then the tangential component of ${\xi }_{\alpha }$ vanishes and we take $M^n$ as anti-invariant, $T=0$. Taking ${\xi }^t=O$ and $T=0$ in Theorem 1, we obtain the result.

2. Since ${\xi }_{\alpha }:\forall \alpha =1,\dots ,s$ are tangential to $M^n$, ${\xi }^{\perp }$ vanishes, and taking $u_{\alpha }(\Omega )=0$ in Theorem 1, we obtain the result. □

Theorem 2. 

Let $M^{2n-1+s}$ be a hypersurface within a T-space form $N^{2n+s}$, 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:

$$\begin{array}{c}\hfill {\left|B\right|}^2={\displaystyle \frac{\mu }{4}}\{n-4\}-{\displaystyle \frac{\Delta f}{f}}\end{array}$$

(17)

or, equivalently, if and only if

$$\begin{array}{c}\hfill {Scal}_M={\displaystyle \frac{\mu }{4}}\{(1-n)(n-2s)-(n-4)\}-n{\Omega }^2-{\displaystyle \frac{\Delta f}{f}}.\end{array}$$

(18)

Proof. 

Let $M^{2n-1+s}$ be a f-biharmonic hypersurface of $N^{2n+s}$ and $\forall {\xi }_{\alpha }:\alpha =1,\dots ,s$ tangential to M; then, $m=2n-1+s$ and $u_{\alpha }(\Omega )=0$, and since

$$\begin{array}{cc} & f^2\Omega =-\Omega +u_{\alpha }(\Omega ){\xi }_{\alpha }=-\Omega ,\hfill \\ & f(f\Omega )=-\Omega ,\hfill \end{array}$$

this implies

$$\begin{array}{cc} & Tt\Omega =-TS\Omega ,\hfill \\ & Nt\Omega =-\Omega -NS\Omega .\hfill \end{array}$$

Therefore, taking $u_{\alpha }(\Omega )=0$ and $S\Omega =0$ in (13) and (14), we have

$$-{\displaystyle \frac{2n-1+s}{2}}grad{|\Omega |}^2-2tr\left({\Lambda }_{{\Delta }^{\perp }\Omega }(.)\right)-2{\Lambda }_{\Omega }grad(lnf)=0$$

and

$$\begin{array}{c}-{\Delta }^{\perp }\Omega -tr\left(B.,{\Lambda }_{\Omega }·\right)+{\displaystyle \frac{\Delta f}{f}}\Omega +2\left({\Delta }_{\mathrm{grad}(lnf)}^{\perp }\Omega \right)={\displaystyle \frac{\mu }{4}}(n-4)\Omega .\hfill \end{array}$$

For constant mean curvature $\Omega$, we have

$${\Lambda }_{\Omega }grad(lnf)=0$$

and

$$\begin{array}{c}-tr\left(B.,{\Lambda }_{\Omega ·}\right)+{\displaystyle \frac{\Delta f}{f}}\Omega ={\displaystyle \frac{\mu }{4}}(n-4)\Omega ,\hfill \end{array}$$

which implies

$$tr\left(B.,{\Lambda }_{\Omega ·}\right)={\displaystyle \frac{\mu }{4}}(n-4)\Omega -{\displaystyle \frac{\Delta f}{f}}\Omega$$

or, equivalently,

$${\left|B\right|}^2={\displaystyle \frac{\mu }{4}}(n-4)\Omega -{\displaystyle \frac{\Delta f}{f}}\Omega .$$

Using Gauss’s equation, we have

$$\begin{array}{cc}\hfill Sca{l}_M& =\sum _{i,j}\left(\tilde{R}\left(e_i,e_j\right)e_j,e_i\right)-{\left|B\right|}^2-n{\Omega }^2\hfill \\ \hfill & ={\displaystyle \frac{\mu }{4}}{\{(1-n)(n-2s)-|B|}^2-n{\Omega }^2.\hfill \end{array}$$

Therefore, using Equation (17), we have

$$\begin{array}{c}\hfill Sca{l}_M={\displaystyle \frac{\mu }{4}}\{(1-n)(n-2s)-(n-4)\}-n{\Omega }^2-{\displaystyle \frac{\Delta f}{f}}\Omega ,\end{array}$$

which completes the proof. □

Theorem 3. 

Let $M^n$ be a submanifold of T-space form $N^{2n+s}$ with constant mean curvature Ω so that ξ and $\varphi \Omega$ are tangential. Further, we consider $F\left(f,q,\mu \right)$, the function defined on M by

$$F\left(f,n,\mu \right)={\displaystyle \frac{\mu }{4}}(n-1)-{\displaystyle \frac{\Delta f}{f}}.$$

This leads to the following observations.

(1) 

${inf}_M{\displaystyle \frac{\mu }{4}}(n-1)-{\displaystyle \frac{\Delta f}{f}}\le 0$ acts as an obstruction to f-biharmonicity.

(2) 

${inf}_M{\displaystyle \frac{\mu }{4}}(n-1)-{\displaystyle \frac{\Delta f}{f}}>0$ and M is proper f-biharmonic; then,

$${0<|\Omega |}^2\le {\displaystyle \frac{1}{n}}\underset{M}{inf}{\displaystyle \frac{\mu }{4}}(n-1)-{\displaystyle \frac{\Delta f}{f}}.$$

(19)

Proof. 

Because M is a proper f-biharmonic submanifold with constant mean curvature $\Omega$, and $\xi$ is tangential to it, Corollary 2 leads to the conclusion that

$$\begin{array}{cc} & -{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)+{\displaystyle \frac{\Delta f}{f}}\Omega +2{\nabla }_{grad\left(lnf\right)}\Omega \\ & ={\displaystyle \frac{\mu }{4}}\{(n-1)\Omega +3Nt\Omega \},\\ & {\displaystyle \frac{\left(n\right)}{2}}{grad\left(\right|\Omega |}^2)+2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-2{\Lambda }_{\Omega }grad\left(lnf\right)\\ & ={\displaystyle \frac{\mu }{4}}\{2(n-1)\sum u^i(\Omega ){\xi }_i+6Pt\Omega \}.\end{array}$$

Given that $\varphi \Omega$ is a tangent, this directly implies that $s\Omega =0$. Reapplying $\varphi$, it follows that ${\varphi }^2\Omega =Pt\Omega +Nt\Omega$. But, from ${\varphi }^2\Omega =-\Omega +u(\Omega )\xi$, and $\xi$ being tangential, we have ${\varphi }^2\Omega =-\Omega$. Therefore, equating the tangential and normal parts, we find that $Pt\Omega =0$ and $Nt\Omega =-\Omega$. Incorporating these results into the preceding equation, we obtain

$$\begin{array}{c}\hfill -{\Delta }^{\perp }\Omega -trB\left(·,{\Lambda }_{\Omega }·\right)={\displaystyle \frac{\mu }{4}}\{(n-1)\Omega \}-{\displaystyle \frac{\Delta f}{f}}\Omega ,\\ \hfill {\displaystyle \frac{\left(n\right)}{2}}{grad\left(\right|\Omega |}^2)+2tr{\Lambda }_{{\nabla }_{(·)}^{\perp }\Omega }(·)-2{\Lambda }_{\Omega }grad\left(lnf\right)\\ \hfill =0.\end{array}$$

Projecting the first equation onto $\Omega$, we obtain

$$-〈{\Delta }^{\perp }\Omega ,\Omega 〉+〈trB\left(·,{\Lambda }_{\Omega }·\right),\Omega 〉=\left({\displaystyle \frac{\mu }{4}}(n-1)-{\displaystyle \frac{\Delta f}{f}}\right){|\Omega |}^2.$$

Using the facts $〈trB\left(·,{\Lambda }_{\Omega }·\right),\Omega 〉={\left|{\Lambda }_{\Omega }\right|}^2,|\Omega |$ is a constant and the Böchner formula, i.e., ${\displaystyle \frac{1}{2}}\Delta {|\Omega |}^2=〈{\Delta }^{\perp }\Omega ,\Omega 〉-{\left|{\nabla }^{\perp }\Omega \right|}^2$ in the above equation, we have

$${\left|{\Lambda }_{\Omega }\right|}^2+{\left|{\nabla }^{\perp }\Omega \right|}^2=\left({\displaystyle \frac{\mu }{4}}(n-1)-{\displaystyle \frac{\Delta f}{f}}\right){|\Omega |}^2.$$

Now, this equation reduces to

$$\left({\displaystyle \frac{\mu }{4}}(n-1)-{\displaystyle \frac{\Delta f}{f}}\right){|\Omega |}^2={\left|{\Lambda }_{\Omega }\right|}^2+{\left|{\nabla }^{\perp }\Omega \right|}^4\ge {n|\Omega |}^2+{\left|{\nabla }^{\perp }\Omega \right|}^2\ge n{|\Omega |}^4.$$

By considering the Cauchy–Schwarz inequality, ${\left|{\Lambda }_{\Omega }\right|}^2\ge {\displaystyle \frac{1}{n}}tr\left({\Lambda }_{\Omega }\right)=n{|\Omega |}^4.$ This implies that

$$\left({\displaystyle \frac{\mu }{4}}(n-1)-{\displaystyle \frac{\Delta f}{f}}\right)\ge n{|\Omega |}^2.$$

Given that $|\Omega |$ 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 $M^m$ be a submanifold of a T-space form $N^{2n+s}\left(c\right)$ with constant f-sectional curvature c, where the structural vector fields ${\xi }_1,{\xi }_2,\dots ,{\xi }_s$ are tangential to $M^m$. Then, for each unit vector $X\in T_{p}M$, orthogonal to ${\xi }_1,{\xi }_2,\dots ,{\xi }_s$, we have

$$\begin{array}{c}\hfill Ric\left(X\right)\le {\displaystyle \frac{1}{4}}\left[m^2{\parallel \Omega \parallel }^2+\mu (m-s-1)+{\displaystyle \frac{3\mu }{2}}{\parallel PX\parallel }^2\right].\end{array}$$

(20)

Proof. 

Assume $M^m$ is a submanifold of the T-space form $N^{2n+s}\left(K\right)$, with constant f-sectional curvature K. At a point $p\in M$, take a unit vector $X\in T_{p}M$. Let $U_1,U_2,\dots ,U_{2n+s}$ be an orthonormal basis for $T_{p}N$, such that $U_1,U_2,\dots ,U_m$ are tangential to M at p. Assume $X=U_1$.

Putting $L_1,L_4=U_i$ and $L_2,L_3=U_j$ in (3), we have

$$\begin{array}{cc}\hfill S(U_i,U_j,U_j,U_i)& ={\displaystyle \frac{\mu }{4}}\left[\overline{g}(U_i,U_j)\overline{g}(U_j,U_i)-\overline{g}(U_j,U_j)\overline{g}(U_i,U_i)-\overline{g}(U_i,U_j)\sum u^i\left(U_j\right)u^i\left(U_i\right)\right.\hfill \\ \hfill & - \overline{g}(U_j,U_i)\sum u^i\left(U_j\right)u^i\left(U_i\right)+\overline{g}(U_i,U_i)\sum u^i\left(U_j\right)u^i\left(U_j\right)\hfill \\ \hfill & + \overline{g}(U_j,U_j)\sum u^i\left(U_i\right)u^i\left(U_i\right)\hfill \\ \hfill & +\left(\sum u^i\left(U_j\right)u^i\left(U_i\right)\right)\left(\sum u^i\left(U_j\right)u^i\left(U_i\right)\right)\hfill \\ \hfill & -\left(\sum u^i\left(U_i\right)u^i\left(U_i\right)\right)\left(\sum u^i\left(U_j\right)u^i\left(U_j\right)\right)\hfill \\ \hfill & + \overline{g}(U_i,\varphi U_i)\overline{g}(U_j,\varphi U_j)+\overline{g}(U_j,\varphi U_i)\overline{g}(U_i,\varphi U_j)\hfill \\ \hfill & \left.- 2\overline{g}(U_i,\varphi U_j)\overline{g}(U_i,\varphi U_j)\right]\hfill \\ \hfill & - g(\zeta (U_i,U_j),\zeta (U_j,U_i))+g(\zeta (U_i,U_i),\zeta (U_j,U_j)).\hfill \end{array}$$

(21)

By taking summation $1\le i<j\le m$, we derive

$${\displaystyle \frac{\mu }{4}}[m(m-1)+2s(1-m)+s(1-s)]-{\displaystyle \frac{3\mu }{4}}{\parallel P\parallel }^2=2\tau -{m\parallel \Omega \parallel }^2+{\parallel \zeta \parallel }^2,$$

where $\Omega$ denotes the mean curvature, $\zeta$ is the second fundamental form, and $\tau$ is the scalar curvature.

Mathematically, the second fundamental form $\zeta$ is expressed as

$${\zeta }_{ij}^r=g(\zeta (U_i,U_j),U_r),{\parallel \zeta \parallel }^2=\sum _{r=m+1}^{2n+s}\sum _{i,j=1}^{m}g(\zeta (U_i,U_j),\zeta (U_i,U_j)).$$

Additionally, the projection operator P satisfies

$${\parallel P\parallel }^2=\sum _{i,j=1}^m{g}^2\left(P(U_i,U_j)\right).$$

We express the mean curvature $\Omega \left(p\right)$ as follows:

$$\Omega \left(p\right)={\displaystyle \frac{1}{m}}\sum _{i=1}^m\zeta (U_i,U_i).$$

As a consequence of the Gauss equation, we obtain

$$m^2{\parallel \Omega \parallel }^2=2\tau +{\displaystyle \frac{1}{2}}\sum _{r=m+1}^{2n+s}\left[{({\zeta }_{11}^r+\dots +{\zeta }_{mm}^r)}^2-{({\zeta }_{11}^r-\dots -{\zeta }_{mm}^r)}^2\right]+2\sum _{r=m+1}^{2n+s}\sum _{i<j}{\left({\zeta }_{ij}^r\right)}^2$$

$$-{\displaystyle \frac{\mu }{4}}\left[m(m-1)+2s(1-m)+s(1-s)\right]+{\displaystyle \frac{3\mu }{4}}{\parallel P\parallel }^2.$$

The sectional curvature for M is expressed as

$$\sum _{2\le i<j\le m}{K}_{ij}={\displaystyle \frac{\mu }{8}}\left[(m-1)(m-2)+2s(2-m)+s(1-s)\right]+{\displaystyle \frac{3\mu }{8}}\left({\parallel P\parallel }^2-{\parallel P_{U_1}\parallel }^2\right).$$

Using the Gauss equation, we rewrite this as

$$\sum _{2\le i<j\le m}{K}_{ij}={\displaystyle \frac{\mu }{8}}\left[(m-1)(m-2)+2s(2-m)+s(1-s)\right]+{\displaystyle \frac{3\mu }{8}}\left({\parallel P\parallel }^2-{\parallel P_{U_1}\parallel }^2\right)$$

$$+\sum _{r=m+1}^{2n+s}\sum _{2\le i<j\le m}\left({\zeta }_{ij}^r{\zeta }_{ij}^r-{\left({\zeta }_{ij}^r\right)}^2\right).$$

Substituting this expression into the equation for $m^2{\parallel \Omega \parallel }^2$, we derive

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{n}^2{\parallel \Omega \parallel }^2& =2Ric\left(X\right)+{\displaystyle \frac{1}{2}}\sum _{r=m+1}^{2n+s}{\left({\zeta }_{11}^r-{\zeta }_{22}^r-\dots -{\zeta }_{mm}^r\right)}^2\hfill \\ \hfill & +\sum _{r=m+1}^{2n+s}\sum _{i,j=1}^m{\left({\zeta }_{ij}^r\right)}^2-{\displaystyle \frac{\mu }{2}}(m-s-1)-{\displaystyle \frac{3\mu }{4}}{\parallel PX\parallel }^2.\hfill \end{array}$$

(22)

This simplifies to

$${\displaystyle \frac{1}{2}}{n}^2{\parallel \Omega \parallel }^2\ge 2Ric\left(X\right)-{\displaystyle \frac{\mu }{2}}(m-s-1)-{\displaystyle \frac{3\mu }{4}}{\parallel PX\parallel }^2,$$

which gives the required inequality. □

Corollary 3. 

In Theorem 4, equality holds at a point $p\in M$ under the following conditions:

(1) 

$\Omega \left(p\right)=0$ and X is normal to $T_{p}M$.

(2) 

p is a totally geodesic point.

Proof. 

(1)

When $\Omega \left(p\right)=0$, we have ${\zeta }_{11}^r={\zeta }_{22}^r=\dots ={\zeta }_{mm}^r=0$. If X is normal to $T_{p}M$, then ${\zeta }_{12}^r={\zeta }_{13}^r=\dots ={\zeta }_{1m}^r=0$ for $r\in \{m+1,\dots ,2n+s\}$, and equality follows from (22).

(2)

If p is a totally geodesic point, then ${\zeta }_{ij}^r=0$ for all $i,j,r$, and equality holds as a result of (22).

□

Remark 1. 

(1) 

Setting $s=0$ yields all the results corresponding to Kähler manifolds.

(2) 

Setting $s=1$ 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.